multiple positive solutions of conjugate boundary value problems with singularities the existence of one or several positive solutions for some nth order differential equations with conjugate boundary conditions is obtained the approach is to employ the well known results on the existence of positive solutions for hammerstein integral equations obtained recently by the author this avoids utilizing the theory of fixed point index for compact maps defined on cones directly new properties on the kernels corresponding to the boundary value problems are provided and then employed to prove new properties of nonzero positive solutions for these boundary value problems and new inequalities on functions satisfying the conjugate boundary conditions our results improve and generalize many recent results

the green s function for k n k conjugate boundary value problems and its applications abstract a new integral representation of the green s function for k n k conjugate boundary value problems is obtained its applications are also given

a fast algorithm for sparse multichannel blind deconvolution abstractwe have addressed blind deconvolution in a multichannel framework recently a robust solution to this problem based on a bayesian approach called sparse multichannel blind deconvolution smbd was proposed in the literature with interesting results however its computational complexity can be high we have proposed a fast algorithm based on the minimum entropy deconvolution which is considerably less expensive we designed the deconvolution filter to minimize a normalized version of the hybrid l1 l2 norm loss function this is in contrast to the smbd in which the hybrid l1 l2 norm function is used as a regularization term to directly determine the deconvolved signal results with synthetic data determined that the performance of the obtained deconvolution filter was similar to the one obtained in a supervised framework similar results were also obtained in a real marine data set for both techniques

velocity analysis based on minimum entropy seismic resolution is determined by the sparsity of reflection events together with the dispersion of the wavelets representing those events in this paper minimum entropy me norms are introduced as a measure of spatial resolving power it is shown that the lateral dispersion of inverted diffractor responses inverted spatial wavelets increases with increasing velocity error using this property minimum entropy velocity analysis meva is proposed to extract velocity information from diffraction energy meva can be successfully applied to zero offset including poststack data and common offset data with a sufficient amount of diffraction energy in addition meva can be used as an alternative to existing cmp velocity estimation techniques

nonlinear multichannel impedance inversion by total variation regularization abstractthe analysis of acoustic impedance ai allows for the mapping of seismic reflection data to lithology and hence it plays an important role in the interpretation of poststack seismic data the ai is obtainable from the inversion of the earth reflectivity series efficient deconvolution methods have been developed for recovering the reflectivity series from band limited poststack data which are multiple free zero offset and migrated however calculation of the ai from the reflectivity when considering the spatial correlation of the impedance parameters demands the solution of a constrained nonlinear inverse problem two efficient algorithms are proposed for solving the nonlinear impedance problem in multichannel form with the total variation tv constraint to recover impedance maps with blocky structures the first uses the continuous earth model for reflectivity which allows linearizing the problem in the logarithm domain the second uses the discrete layered earth model for reflectivit

feasibility of approximate broadband estimation of acoustic impedance profile from first principles and band limited reflection data abstractbecause seismic reflection data are band limited acoustic impedance profiles derived from them are nonunique the conventional inversion methods counter the nonuniqueness either by stabilizing the answer with respect to an initial model or by imposing mathematical constraints such as sparsity of the reflection coefficients by making a nominal assumption of an earth model locally consisting of a stack of homogeneous and horizontal layers we have formulated a set of linear equations in which the reflection coefficients are the unknowns and the recursively integrated seismic trace constitute the data drawing only on first principles the zoeppritz equation in this case the approach makes a frontal assault on the problem of reconstructing reflection coefficients from band limited data the local layer cake assumption and the strategy of seeking a singular value decomposition solution of the linear equations counter the nonuniqueness provided that the objective is to reconstruct a smooth version

seismic sparse spike deconvolution via toeplitz sparse matrix factorization abstractwe have developed a new sparse spike deconvolution ssd method based on toeplitz sparse matrix factorization tsmf a bilinear decomposition of a matrix into the product of a toeplitz matrix and a sparse matrix to address the problems of lateral continuity effects of noise and wavelet estimation error in ssd assuming the convolution model a constant source wavelet and the sparse reflectivity a seismic profile can be considered as a matrix that is the product of a toeplitz wavelet matrix and a sparse reflectivity matrix thus we have developed an algorithm of tsmf to simultaneously deconvolve the seismic matrix into a wavelet matrix and a reflectivity matrix by alternatively solving two inversion subproblems related to the toeplitz wavelet matrix and sparse reflectivity matrix respectively because the seismic wavelet is usually compact and smooth the fused lasso was used to constrain the elements in the toeplitz wavelet matrix moreover due to the limitations of computer memory large

root mean square velocities and recovery of the acoustic impedance the loss of low frequency information in reflection seismograms causes serious difficulties when attempting to generate a full band impedance profile information about the low frequency velocity structure is available from rms stacking velocities we show how rms velocities can be inverted with additional point velocity constraints if they are available to construct either smooth or blocky velocity structures backus gilbert averages of the constructed velocity are then autoregressive solutions for recovering a full band reflectivity from band limited seismograms our final result is therefore a full band acoustic impedance which is consistent with the seismic data section stacking velocities and available point constraints

from classical reflectivity to velocity inversion to full waveform inversion using phase modified and deconvolved reverse time migration images abstractto obtain a physical understanding of gradient based descent methods in full waveform inversion fwi we find a connection between the fwi gradient and the image provided by reverse time migration rtm the gradient uses the residual data as a virtual source and rtm uses the observed data directly as the boundary condition so the fwi gradient is similar to a time integration of the rtm image using the residual data which physically converts the phase of the reflectivity to the phase of the velocity therefore gradient based fwi can be connected to the classical reflectivity to velocity impedance inversion rvi we have developed a new fwi scheme that provides a self contained and physically intuitive derivation which naturally establishes a connection among the amplitude preserved rtm the zoeppritz equations amplitude variation with angle inversion and rvi and combines them into a single framework to produce a preconditioned inversion formula in this scheme the relative velocity upd

maximum likelihood deconvolution an optimization theory perspective a large number of deconvolution procedures have appeared in the literature during the last three decades including a number of maximum likelihood deconvolution mld procedures the major advantages of the mld procedures are 1 no assumption is required about the phase of the wavelet most of the classical deconvolution techniques assume a minimum phase wavelet an assumption that may not be appropriate for many data sets 2 mld procedures can resolve closely spaced events i e they are high resolution techniques and 3 they can efficiently handle modeling and measurement errors as well as backscatter effects i e reflections from small features a comparative study of six different mld procedures for estimating the input of a linear time invariant system from measurements which have been corrupted by additive noise was made by using a common framework developed from fundamental optimization theory arguments to date only the kormylo and the chi t algorithms can be recommended

direct multichannel predictive deconvolution the levinson principle generally can be used to compute recursively the solution of linear equations it can also be used to update the error terms directly this is used to do single channel deconvolution directly on seismic data without computing or applying a digital filter multichannel predictive deconvolution is used for seismic multiple attenuation in a standard procedure the prediction error filter matrices are computed with a levinson recursive algorithm using a covariance matrix of the input data the filtered output is the prediction errors or the nonpredictable part of the data starting with the classical levinson recursion wehave derived new algorithms for direct recursive calculationof the prediction errors without computing the data covariance matrix or computing the prediction error filters one algorithm generates recursively the one step forward and backward predic tion errors and the l step forward prediction error computing only the filter matrices with the highest index a numeri

seismic sparse layer reflectivity inversion using basis pursuit decomposition a basis pursuit inversion of seismic reflection data for reflection coefficients is introduced as an alternative method of incorporating a priori information in the seismic inversion process the inversion is accomplished by building a dictionary of functions representing reflectivity patterns and constituting the seismic trace as a superposition of these patterns basis pursuit decomposition finds a sparse number of reflection responses that sum to form the seismic trace when the dictionary of functions is chosen to be a wedge model of reflection coefficient pairs convolved with the seismic wavelet the resulting reflectivity inversion is a sparse layer inversion rather than a sparse spike inversion synthetic tests suggest that a sparse layer inversion using basis pursuit can better resolve thin beds than a comparable sparse spike inversion application to field data indicates that sparse layer inversion results in the potentially improved detectability and resolution of some thin layers and reveals a

resolution in seismic trace inversion by parameter estimation an alternative to the conventional time series approach to single trace modeling and inversion by convolution and inverse filtering is a parametric approach to obtain insight into the potential of the parametric approach the solution of the single trace forward problem is formulated in matrix terms for the nonlinear reflector lag time parameters this is achieved by linearization which is shown to be a valid approximation over a sufficiently large region the matrix forward operators are analyzed by means of the singular value decomposition svd the svd can be considered a generalization of the fourier transform of convolution operators on the basis of the svd analysis inverse operators are designed which combine stability with high resolving power a method to determine the resolving power of the parametric inverse operators is presented several examples show how wavelet bandwidth data noise level and model complexity influence the resolving power of the data for the reflection coefficient and

applications of high resolution time frequency transforms to attenuation estimation abstractattenuation estimates quantify the loss of energy of propagating seismic waves due to anelastic processes it is often carried out in the frequency domain the most well known methods for attenuation estimation such as the spectral ratio and frequency shift methods compare spectral shapes of waveforms along a given raypath they require broad spectra such as those obtained with the fourier transform and the continuous wavelet transform these methods are incompatible with high resolution time frequency transforms which drastically localize time frequency information on the other hand these transforms indicate stronger resistance to noise and can be used in combination with the peak frequency method to estimate attenuation we have applied high resolution transforms namely the synchrosqueezing transform basis pursuit and complete ensemble empirical mode decomposition to a synthetic wedge example and two seismic data set examples a seismic reflection profile and a vertical seismic profile

simultaneous multitrace impedance inversion with transform domain sparsity promotion abstractthe impedance inversion technique plays a crucial role in seismic reservoir properties prediction however most existing impedance inversion methods often suffer from spatial discontinuities and instability because each vertical profile is processed independently in the inversion we tested a transform domain sparsity promotion simultaneous multitrace impedance inversion method to address this issue the approach was implemented through minimizing a data misfit term and a transform domain sparsity constraint term that incorporates the 2d or 3d structural information into the inversion processing a 2d synthetic data example was applied to mainly explain the roles of the transform domain sparsity constraint we determined that the transform domain sparsity constraint can help in stabilizing the inversion reducing the influence of high wavenumber noise on the inverted result and exploring spatial continuities of structures furthermore a 3d field data example was used to examine the effectiven

a fast automatic multichannel blind seismic inversion for high resolution impedance recovery abstractthe inversion of seismic reflection data for acoustic impedance ai is a common and accepted method for the interpretation of poststack seismic data the original mathematical problem is nonlinear due to the nonlinearity of relation between the ai and the earth reflectivity series and also the insufficiency of information about the source wavelet furthermore the problem is ill posed due to the lack of low and high frequencies in the data we have developed a multichannel blind inversion and solved it for obtaining the ai model and the wavelet directly from seismic reflection data we found a solution to the overall problem by alternating between two subproblems corresponding to the ai and wavelet recovery having an estimation of the wavelet the algorithm directly inverts multichannel data for a high resolution ai model having blocky structures in the sense of total variation tv regularization while satisfying a priori low frequency information the advantages of the split bregman techniq

high resolution prestack seismic inversion using a hybrid fista least squares strategy abstracta new inversion method to estimate high resolution amplitude versus angle attributes ava attributes such as intercept and gradient from prestack data is presented the proposed technique promotes sparse spike reflectivities that when convolved with the source wavelet fit the observed data the inversion is carried out using a hybrid two step strategy that combines fast iterative shrinkage thresholding algorithm fista and a standard least squares ls inversion fista which can be viewed as an extension of the classical gradient algorithm provides sparse solutions by minimizing the misfit between the modeled and the observed data and the l1 norm of the solution fista is used to estimate the location in time of the main reflectors then ls is used to retrieve the appropriate reflectivity amplitudes that honor the data fista like other iterative solvers for l1 norm regularization does not require matrices in explicit form making it easy to apply economic in computational terms and ad

spectral decomposition of seismic data with continuous wavelet transform this paper presents a new methodology for computing a time frequency map for nonstationary signals using the continuous wavelet transform cwt the conventional method of producing a time frequency map using the short time fourier transform stft limits time frequency resolution by a predefined window length in contrast the cwt method does not require preselecting a window length and does not have a fixed time frequency resolution over the timefrequency space cwt uses dilation and translation of a wavelet to produce a time scale map a single scale encompasses a frequency band and is inversely proportional to the time support of the dilated wavelet previous workers have converted a time scale map into a time frequency map by taking the center frequencies of each scale we transform the time scale map by taking the fourier transform of the inverse cwt to produce a time frequency map thus a time scale map is converted into a time frequency map in which the amplitudes of individual frequencies rather than frequency bands are represented we refer to such a map as the time frequency cwt tfcwt we validate our approach with a nonstationary synthetic example and compare the results with the stft and a typical cwt spectrum two field examples illustrate that the tfcwt potentially can be used to detect frequency shadows caused by hydrocarbons and to identify subtle stratigraphic features for reservoir characterization

multichannel spatially correlated reflectivity inversion using block sparse bayesian learning abstractthe reflectivity inversion approach based on a variety of regularization terms was extensively developed and applied to image subsurface structure in recent years in addition multichannel reflectivity inversion or deconvolution considering the lateral continuity of reflection interfaces or reflectivity in adjacent channels has been developed however these processing operations seldom adaptively judge the stratal continuity or automatically alter the parameters of the corresponding algorithm to use the special correlation of the reflection information contained in the seismic data a multichannel spatially correlated reflectivity inversion using block sparse bayesian learning bsbl is introduced the method adopts a covariance matrix that describes the spatial relationship of reflectivity and simultaneously controls the temporal sparsity with an expectation maximization algorithm we can obtain the parameters of the multichannel reflectivity model including the mean i e the estimated mul

the global holder continuous solution of isentropic gas dynamics this paper considers the cauchy problem for the isentropic equations of gas dynamics in euler coordinates t u x 0 u t u 2 p x 0 and gives the holder continuous solution by applying the method of vanishing viscosity

global regularity and formation of singularities of solutions to first order quasilinear hyperbolic systems for the cauchy problem for strictly hyperbolic systems with general eigenvalues we obtain existence of global smooth solutions under certain conditions on the composition of the eigenvalues and the initial data on the other hand we give a sufficient condition which guarantees that singularities of the solution must occur in a finite time and describe certain applications the present paper includes the corresponding results in earlier papers by several authors as special cases

multichannel matching pursuit for seismic trace decomposition the technique of matching pursuit can adaptively decompose a seismic trace into a series of wavelets however the solution is not unique and is also strongly affected by data noise multichannel matching pursuit mcmp exploiting lateral coherence as a constraint might improve the uniqueness of the solution it extracts a constituent wavelet that has an optimal correlation coefficient to neighboring traces instead of to a single trace only according to linearity theory a wavelet shared by neighboring traces is the best match to the average of multiple traces and therefore it might effectively suppress the data noise and stabilize the performance it is found that the mcmp scheme greatly improves spatial continuity in decomposition and can generate a plausible time frequency spectrum with high resolution for reservoir detection

frequency time decomposition of seismic data using wavelet based methods spectral analysis is an important signal processing tool for seismic data the transformation of a seismogram into the frequency domain is the basis for a significant number of processing algorithms and interpretive methods however for seismograms whose frequency content vary with time a simple 1 d fourier frequency transformation is not sufficient improved spectral decomposition in frequency time ft space is provided by the sliding window short time fourier transform although this method suffers from the time frequency resolution limitation recently developed transforms based on the new mathematical field of wavelet analysis bypass this resolution limitation and offer superior spectral decomposition the continuous wavelet transform with its scale translation plane is conceptually best understood when contrasted to a short time fourier transform the discrete wavelet transform and matching pursuit algorithm are alternative wavelet transforms that map a seismogram into ft space decomposition into ft space of synthetic and calibrated explosive source seismic data suggest that the matching pursuit algorithm provides excellent spectral localization and reflections direct and surface waves and artifact energy are clearly identifiable wavelet based transformations offer new opportunities for improved processing algorithms and spectral interpretation methods

mixed phase deconvolution and wavelet estimation the seismic convolutional model describes seismic wave propagation as a linear system a seismic trace then consists of a wavelet or pulse convolved with the reflection response or reflectivity series plus additive noise an important problem in seismic data processing is the removal of the effect of the wavelet spiking deconvolution or inverse filtering in order to estimate the reflectivity series when the wavelet is known designing an optimal spiking filter by a least squares approach is straightforward for a mixed delay wavelet we must delay the position of the desired output spike this delay can later be removed so that the effective filter is no longer causal but operates on both past and future input data values with respect to the output position when the wavelet is unknown we must determine both the wavelet and the reflectivity series from the input data solving this ambiguous problem requires additional assumptions about the wavelet the reflectivity series and the noise common assumptions are that the reflectivity series is a stationary random process and that it is uncorrelated to the stationary random noise furthermore if it is assumed that the signal to noise ratio is large the autocorrelation function acf of the seismic trace can be used as an estimate of the acf of the wavelet if in addition it is assumed as it commonly is that the wavelet is minimum delay meaning that all roots of its associated polynomial have amplitude greater than 1 an optimal least squares inverse filter can be computed by solving the normal equations using a levinson algorithm the normal equations are also called the yule walker yw equations however problems can arise with minimum phase inverse filtering when the wavelet is not minimum delay in a tutorial on spiking deconvolution tle 1995 leinbach showed that applying a minimum phase filter to an air gun

blow up of solutions to the initial boundary value problem for quasilinear hyperbolic systems of conservation laws abstract this work is a continuation of our previous work z q shao d x kong y c li shock reflection for general quasilinear hyperbolic systems of conservation laws nonlinear anal 66 1 2007 93 124 in the present paper we study the global structure instability of the self similar solution u u x t containing shocks contact discontinuities and at least one rarefaction wave of the initial boundary riemann problem for general n n quasilinear hyperbolic systems of conservation laws on the quarter plane t x t 0 x 0 we prove the nonexistence of global piecewise c 1 solution to a class of the mixed initial boundary value problem for general n n quasilinear hyperbolic systems of conservation laws on the quarter plane t x t 0 x 0 our result indicates that the kind of riemann solution u u x t mentioned above is globally structurally unstable as an application of our result we consider the model proposed by aw and rascle a aw m rascle resurrection of second order models of traffic flow siam j appl math 60 2000 916 938 describing traffic flow on a road network following the work of garavello and piccoli m garavello b piccoli traffic flow on a road network using the aw rascle model comm partial differential equations 31 2 2006 243 275 we prove the nonexistence of global piecewise c 1 solution containing one rarefaction wave of the initial boundary value problem for this model

formation of singularities for certain hyperbolic systems in one space dimension we study the development of singularities of classical solutions of quasilinear strictly hyperbolic systems in one space dimension the systems are supposed to be in the diagonal form but we do not impose restrictions on the size of the initial data and in some cases we can replace the genuine nonlinearity by weaker conditions

mixed phase wavelet estimation using fourth order cumulants in recent years methods have been developed in the field of high order statistics that can reliably estimate a mixed phase wavelet from the noisy output of a convolutional process these methods use high order covariance functions of the data called cumulants which retain phase information and allow recovery of the wavelet the assumption is that the reflection coefficient series is a non gaussian stationary and statistically independent random process the method described in this paper uses the fourth order cumulant of the data and a moving average noncausal parametric model for the wavelet the fourth order moment function of this wavelet matches the fourth order cumulant of the data in a minimum mean squared error sense numerical simulations demonstrate that the method can accurately estimate mixed phase wavelets even when the reflectivity has a distribution close to gaussian three seismic data examples demonstrate possible uses of the method in the first two an average source wavelet is est

green s function and positive solutions for higher order ode abstract green s function for k n k conjugate boundary value problem 1 n k x n t t f x t a x i a 0 0 i k 1 x j b 0 0 j n k 1 is obtained and its integral representation is given as applications we obtain sufficient conditions for the existence of one two or three positive solutions

on the breakdown of solutions to n n quasilinear hyperbolic systems we shall be concerned with the development of singularities of the c1 solutions of the quasilinear n n hyperbolic system ut a u ux 0 in one space dimension generalizing a nonlinearity condition introduced by maccamy and mizel arch rat mech anal 25 1967 299 320 in this paper we prove some results of blow up for the first order derivatives without restrictions on the size of the initial data

stochastic relaxation gibbs distributions and the bayesian restoration of images we make an analogy between images and statistical mechanics systems pixel gray levels and the presence and orientation of edges are viewed as states of atoms or molecules in a lattice like physical system the assignment of an energy function in the physical system determines its gibbs distribution because of the gibbs distribution markov random field mrf equivalence this assignment also determines an mrf image model the energy function is a more convenient and natural mechanism for embodying picture attributes than are the local characteristics of the mrf for a range of degradation mechanisms including blurring nonlinear deformations and multiplicative or additive noise the posterior distribution is an mrf with a structure akin to the image model by the analogy the posterior distribution defines another imaginary physical system gradual temperature reduction in the physical system isolates low energy states annealing or what is the same thing the most probable states under the gibbs distribution the analogous operation under the posterior distribution yields the maximum a posteriori map estimate of the image given the degraded observations the result is a highly parallel relaxation algorithm for map estimation we establish convergence properties of the algorithm and we experiment with some simple pictures for which good restorations are obtained at low signal to noise ratios

recovery of the acoustic impedance from reflection seismograms this paper examines the problem of recovering the acoustic impedance from a band limited normal incidence reflection seismogram the convolutional model for the seismogram is adopted at the outset and it is therefore required that initial processing has removed multiples and recovered true amplitudes as well as possible in the first portion of the paper we investigate the effect of substituting the deconvolved seismic trace that is the band limited version of the reflectivity function into the standard recursion formula for the acoustic impedance the formalism of linear inverse theory is used to show that the logarithm of the normalized acoustic impedance estimated from the deconvolved seismogram is approximately an average of the true logarithm of the impedance moreover the averaging function is identical to that used in deconvolving the initial seismogram the advantage of these averages is that they are unique their disadvantage is that low frequency information which is crucial to making a g

deconvolution of sparse spike trains by iterated window maximization a new algorithm for deconvolution of sparse spike trains is presented to maximize a joint map criterion an initial configuration is iteratively improved through a number of small changes computational savings are achieved by precomputing and storing two correlation functions and by employing a window strategy the resulting formulas are simple intuitive and efficient in addition they allow much more complicated transitions than state space solutions such as kormylo and mendel s 1982 single most likely replacement algorithm this makes it possible to reduce significantly the probability that the algorithm terminates in a local maximum synthetic data examples are presented that support these claims

sparse reflectivity inversion for nonstationary seismic data abstractconventional reflectivity inversion methods are based on a stationary convolution model and theoretically require stationary seismic traces as input i e those free of attenuation and dispersion effects reflectivity inversion for nonstationary data which is typical for field surveys requires us to first compensate for the earth s q filtering effects by inverse q filtering however the attenuation compensation for inverse q filtering is inherently unstable and offers no perfect solution thus we presented a sparse reflectivity inversion method for nonstationary seismic data we referred to this method as nonstationary sparse reflectivity inversion nsri it makes the novel contribution of avoiding intrinsic instability associated with inverse q filtering by integrating the earth s q filtering operator into the stationary convolution model nsri also avoids time variant wavelets that are typically required in time variant deconvolution although nsri is initially designed for nonstationary

singularities of solutions for first order quasilinear hyperbolic systems in this paper the existence of global smooth solutions and the formation of singularities of solutions for strictly hyperbolic systems with general eigenvalues are discussed for the cauchy problem with essentially periodic small initial data or nonperiodic initial data a result of klainerman and majda is thus extended to the general case

smlr type blind deconvolution of sparse pulse sequences under a minimum temporal distance constraint we consider bayesian blind deconvolution bd of an unknown sparse sequence convolved with an unknown pulse our goal is to detect the positions where the sparse input sequence is nonzero and to estimate the corresponding amplitudes as well as the pulse shape for this task we propose a novel evolution of the single most likely replacement smlr algorithm our method uses a modified bernoulli gaussian prior that incorporates a minimum temporal distance constraint this prior simultaneously induces sparsity and enforces a prescribed minimum distance between the pulse centers the minimum distance constraint provides an effective way to avoid overfitting i e spurious detected pulses and improve resolution the proposed bd method overcomes certain weaknesses of the traditional smlr based bd method which is verified experimentally to result in improved detection estimation performance and reduced computational complexity our simulation results also demonstrate performance and complexity advantages relative to the iterated window maximization iwm algorithm and a recently proposed partially collapsed gibbs sampler method

wavelet radon domain dealiasing and interpolation of seismic data spatial aliasing is unavoidable in some seismic data and has serious effects on the performance of multichannel data processing and migration antialias filtering produces distortion of the signal through the removal of high frequency information in contrast dealiasing produces an unaliased estimate of the signal at all frequencies present in the original time series a new dealiasing algorithm is developed by exploiting the properties of seismic wavefields in the wavelet radon transform domain specifically the overlap of information between wavelet scales at the same frequency the effectiveness of the wavelet radon dealiasing algorithm is demonstrated through the processing of both synthetic and field seismic data

reconstruction of the velocity and density in a stratified acoustic half space using a short pulse point source the reconstruction of the velocity and density in a stratified half space using the reflection data generated by a short pulse point source is considered transverse spatial hankel transform is applied to the transient reflected and incident pressures on the surface the reflection kernel in the hankel transformed space is obtained through a deconvolution the time domain green function approach based on the wave splitting is then used to reconstruct the parameters this time domain approach to the inverse problem is exact and noniterative numerical results for some synthetical examples are presented to illustrate the hankel transform deconvolution and the reconstruction the finite difference time domain method is used to generate the transient reflection data in the physical space in a numerical simulation the reconstruction using obliquely incident plane waves is also considered in an appendix

existence of globally bounded continuous solutions for nonisentropic gas dynamics equations abstract in this paper we study the existence of globally bounded continuous solutions for the nonisentropic gas dynamics equations under some conditions on the initial data and the entropy function we prove some existence theorems for the cauchy problems and initial boundary value problems the sufficient conditions obtained here improve those obtained in the previous works because they are less restricted and the initial data are uniformly bounded

on the statistical analysis of dirty pictures may 7th 1986 professor a f m smith in the chair summary a continuous two dimensional region is partitioned into a fine rectangular array of sites or pixels each pixel having a particular colour belonging to a prescribed finite set the true colouring of the region is unknown but associated with each pixel there is a possibly multivariate record which conveys imperfect information about its colour according to a known statistical model the aim is to reconstruct the true scene with the additional knowledge that pixels close together tend to have the same or similar colours in this paper it is assumed that the local characteristics of the true scene can be represented by a nondegenerate markov random field such information can be combined with the records by bayes theorem and the true scene can be estimated according to standard criteria however the computational burden is enormous and the reconstruction may reflect undesirable largescale properties of the random field thus a simple iterative method of reconstruction is proposed which does not depend on these large scale characteristics the method is illustrated by computer simulations in which the original scene is not directly related to the assumed random field some complications including parameter estimation are discussed potential applications are mentioned briefly

on simultaneous signal estimation and parameter identification using a generalized likelihood approach a common approach to blind deconvolution of bernoulli gaussian processes consists of performing both signal restoration and hyperparameter identification through maximization of a single generalized likelihood criterion it is shown on a simple example that the resulting hyperparameter estimates may not converge toward any meaningful value therefore other more reliable approaches should be adopted whenever possible

positive solutions for k n k conjugate boundary value problems in banach spaces abstract the existence and multiplicity of positive solutions for a integral equation in banach spaces are obtained using these results we investigate the existence of multiple positive solutions for k n k conjugate boundary value problems the arguments are based upon a specially constructed cone and the fixed point theory in a cone for strict set contraction operators this work gives an unified approach applicable to many boundary value problems

multitrace impedance inversion with lateral constraints abstractwe have developed a lateral constraint to the inversion of 1d seismic impedance models to suppress the effect of data noise and improve the fidelity of formation boundaries in 2d models for situations with dips of less than 20 typical inversion frameworks rely on a 1d forward model with each 1d trace being inverted independently adjacent inversion models are combined together to form a 2d impedance model adding a lateral constraint improves the fidelity of the 2d impedance models while retaining much of the advantage of the low computational cost associated with typical 1d inversion schemes solving the 1d lateral constraint inversion 1d lci problem involves the simultaneous inversion of multiple 1d traces producing layered sections with lateral smoothed transition in addition to enforcing lateral continuity in the inversion model this algorithm allows for the inclusion of a priori knowledge from boreholes we determined the effectiveness of this algorithm on two synthetic models as well

multichannel sparse blind deconvolution on the sphere multichannel blind deconvolution is the problem of recovering an unknown signal f and multiple unknown channels x i from their circular convolution y i x i circledast f i 1 2 dots n we consider the case where the x i s are sparse and convolution with f is invertible our nonconvex optimization formulation solves for a filter h on the unit sphere that produces sparse output y i circledast h under some technical assumptions we show that all local minima of the objective function correspond to the inverse filter of f up to an inherent sign and shift ambiguity and all saddle points have strictly negative curvatures this geometric structure allows successful recovery of f and x i using a simple manifold gradient descent mgd algorithm the same approach is also applicable to blind gain and phase calibration with a fourier sensing matrix our algorithm and analysis require fewer assumptions than previous algorithms for the same problem our theoretical findings are complemented by numerical experiments which demonstrate superior performance of the proposed approach over the previous methods empirically our algorithm has low computation cost converging in a small number of iterations and low memory footprint solving only for the inverse filter of f

cauchy problem for weakly linearly degenerate hyperbolic systems in diagonal form abstract we study the cauchy problem for quasi linear strictly hyperbolic systems of diagonal form with weakly linearly degenerate characteristics the global existence and uniqueness of c1 solution with initial data in c b 1 r w 1 1 r are proved when the l 1 r norm of or the c 0 r norm of is small we also give some examples to show that if the initial data are not in w 1 1 r or not small then the solution to the cauchy problem may blow up in a finite time

transdimensional seismic inversion using the reversible jump hamiltonian monte carlo algorithm abstractprestack or angle stack gathers are inverted to estimate pseudologs at every surface location for building reservoir models recently several methods have been proposed to increase the resolution of the inverted models all of these methods however require that the total number of model parameters be fixed a priori we have investigated an alternate approach in which we allow the data themselves to choose model parameterization in other words in addition to the layer properties the number of layers is also treated as a variable in our formulation such transdimensional inverse problems are generally solved by using the reversible jump markov chain monte carlo rjmcmc approach which is a tool for model exploration and uncertainty quantification this method however has very low acceptance we have developed a two step method by combining rjmcmc with a fixed dimensional mcmc called hamiltonian monte carlo which makes use of gradient information to take large steps acceptance probability

multichannel blind deconvolution of seismic signals a new algorithm for simultaneous wavelet estimation and deconvolution of seismic reflection signals is given to remove the inherent ambiguity in this blind deconvolution problem we introduce relevant a priori information our major assumption is sparseness of the reflectivity which corresponds to a layered earth model this allows nonminimum phase wavelets to be recovered reliably and closely spaced reflectors to be resolved to combine a priori knowledge and data we use a bayesian framework and derive a maximum a posteriori estimate computing this estimate is a difficult optimization problem solved by a suboptimal iterative procedure the procedure alternates steps of wavelet estimation and reflectivity estimation the first step only requires a simple least squares fit while the second step is solved by the iterated window maximization algorithm proposed by kaaresen this enables better efficiency and optimality than established alternatives the resulting optimization method can easily handle multichannel models with only a moderate increase of the computational load lateral continuity of the reflectors is achieved by modeling local dependencies between neighboring traces major improvements in both wavelet and reflectivity estimates are obtained by taking the wavelet to be invariant across several traces the practicality of the algorithm is demonstrated on synthetic and real seismic data an application to multivariate well log segmentation is also given

sparse multichannel blind deconvolution abstractwe developed a sparse multichannel blind deconvolution smbd method the method is a modification of the multichannel blind deconvolution technique often called euclid deconvolution in which the multichannel impulse response of the earth is estimated by solving an homogeneous system of equations classical euclid deconvolution is unstable in the presence of noise and requires the correct estimation of the length of the seismic wavelet the proposed method on the other hand can tolerate moderate levels of noise and does not require a priori knowledge of the length of the wavelet smbd solves the homogeneous system of equations arising in euclid deconvolution by imposing sparsity on the unknown multichannel impulse response trivial solutions to the aforementioned homogeneous system of equations are avoided by seeking sparse solutions on the unit sphere we tested smbd with synthetic and real data examples synthetic examples were used to judge the viability of the method in terms of noise we f

tutorial spectral bandwidth extension invention versus harmonic extrapolation abstractvarious postprocessing methods can be applied to seismic data to extend the spectral bandwidth and potentially increase the seismic resolution frequency invention techniques including phase acceleration and loop reconvolution produce spectrally broadened seismic sections but arbitrarily create high frequencies without a physical basis tests in extending the bandwidth of low frequency synthetics using these methods indicate that the invented frequencies do not tie high frequency synthetics generated from the same reflectivity series furthermore synthetic wedge models indicate that the invented high frequency seismic traces do not improve thin layer resolution frequency invention outputs may serve as useful attributes but they should not be used for quantitative work and do not improve actual resolution on the other hand under appropriate circumstances layer frequency responses can be extrapolated to frequencies outside the band of the original data using spectral periodicities determined

stochastic sparse spike deconvolution sparse spike deconvolution can be viewed as an inverse problem where the locations and amplitudes of a number of spikes reflectivity are estimated from noisy data seismic traces the main objective is to find the least number of spikes that when convolved with the available band limited seismicwaveletestimate fitthedatawithinagiventolerance errormisfit thedetectionofthespikes timelagsisahighly nonlinearoptimizationproblemthatcanbesolvedusingvery fastsimulatedannealingsa amplitudesareeasilyestimated using linear least squares at each sa iteration at this stage quadratic regularization is used to stabilize the solution to reduce its nonuniqueness and to provide meaningful reflectivitysequences thusavoidingtheneedtoconstrainthe spikes time lags and or amplitudes to force valid solutions impedanceconstraintsalsocanbeincludedatthisstage providing the low frequencies required to recover the acoustic impedance oneadvantageoftheproposedmethodoverother sparse spike deconvolution techniques is that the uncertaintyoftheobtainedsolutionscanbeestimatedstochastically further errorsinthephaseofthewaveletestimatearetolerated foranoptimumconstant phaseshiftisobtainedtocalibratetheeffectivewaveletthatispresentinthedata results using synthetic data including simulated data for the marmousi2 modelandfield 3d data show that physically meaningful high resolution sparse spike sections can be derived fromband limitednoisydata evenwhentheavailablewaveletestimateisinaccurate

multiple solutions of two point boundary value problems of ordinary differential equations in banach spaces soit p un cone normal de l espace de banach reel e on note par n la constante normale de p on considere le probleme aux deux points limites x f t x t i x 0 x 1 ou i 0 1 et f une application continue de i p dans p on etudie l existence de solutions positives multiples

nonlinear impedance inversion for attenuating media applications of seismic impedance inversion normally assume the data are free of multiples and transmission effects requiring knowledge of the seismic pulse that is assumed to be stationary an alternative formulation for impedance inversion is based on an exact frequency domain zero offset reflectivity function for a 1d medium analytical formulas for the frechet derivatives are derived for efficient implementation of an iterative nonlinear inversion the exact zero offset reflectivity accounts for internal multiples and transmission effects in the data absorption and dispersion are also conveniently handled if a reasonable estimate for the quality q factor of the medium is available a series of convenient features are included in the inversion algorithm an automatic estimation of the amplitude spectrum of the seismic pulse an impedance transform that makes the inversion independent from the initial smooth model and a practical approach to estimate the regularization weight numerical tests usin

wave propagation and sampling theory part ii sampling theory and complex waves morlet et al 1982 this issue showed the advantages of using complex values for both waves and characteristics of the media we simulated the theoretical tools we present here using the goupillaud kunetz algorithm now we present sampling methods for complex signals or traces corresponding to received waves and sampling methods for complex characterization of multilayered or heterogeneous media regarding the complex signals we present a twodimecsional 2 d method of sampling in the time frequency domain using a special or extended gabor expansion on a set of basic wavelets adapted to phase preservation such a 2 d expansion permits us to handle in a proper manner instantaneous frequency spectra we show the differences between wavelet resolution and sampling grid resolution we also show the importance of phase preservation in high resolution seismic regarding the media we show how analytical studies of wave propagation in periodic structured layers could help when trying to characterize the physical properties of the layers and their large scale granularity as a result of complex deconvolution analytical studies of wave propagation in periodic structures are well known in solid state physics and lead to the so called bloch waves the introduction of complex waves leads to replacing the classical wave equation by a schriidinger equation finally we show that complex wave equations gabor expansion and bloch waves are three different ways of introducing the tools of quantum mechanics in highresolution seismic gabor 1946 kittel 1976 morlet 1975 and conversely the goupillaud kunetz algorithm and an extended gabor expansion may be of some use in solid state physics

wave propagation and sampling theory part i complex signal and scattering in multilayered media from experimental studies in digital processing of seismic reflection data geophysicists know that a seismic signal does vary in amplitude shape frequency and phase versus propagation time to enhance the resolution of the seismic reflection method we must investigate these variations in more detail we present quantitative results of theoretical studies on propagation of plane waves for normal incidence through perfectly elastic multilayered media as wavelet shapes we use zero phase cosine wavelets modulated by a gaussian envelope and the corresponding complex wavelets a finite set of such wavelets for an appropriate sampling of the frequency domain may be taken as the basic wavelets for a gabor expansion of any signal or trace in a two dimensional 2 d domain time and frequency we can then compute the wave propagation using complex functions and thereby obtain quantitative results including energy and phase of the propagating signals these results appear as complex 2 d functions of time and frequency i e as instantaneous frequency spectra choosing a constant sampling rate on the logarithmic scale in the frequency domain leads to an appropriate sampling method for phase preservation of the complex signals or traces for this purpose we developed a gabor expansion involving basic wavelets with a constant time duration mean period ratio for layered media as found in sedimentary basins

optimal simultaneous detection and estimation of filtered discrete semi markov chains an optimal algorithm for the detection of noisy filtered discrete semi markov chains is presented estimation of the underlying model parameters is also considered for a given path of the discrete semi markov chain the optimum estimates of the model parameters obtained by the maximum likelihood method are expressed as functions of the path these functions are then used to derive a single maximum a posteriori criterion for the optimal detection of the unknown single path the final optimization is carried out numerically by a combination of gradient divide and conquer and search techniques this set of techniques is referred to as the integer most likely search detector experimental results using synthetic data demonstrate the potential of the algorithm

positive solutions to semipositone k n k conjugate eigenvalue problems abstract in this paper we consider the existence of positive solutions to the following semipositone k n k conjugate eigenvalue problems scep 1 n k u n t f t u t 0 t 1 u i 0 0 0 i k 1 u j 1 0 0 j n k 1 where n 2 1 k n 1 and 0 is positive parameter the nonlinear function f may change sign for 0 t 1 i e we allow that the nonlinear term f is both semipositone and lower unbounded without making any monotone type assumptions by using the fixed point index theory we derive an explicit interval of such that for any in this interval the existence of at least one positive solution to the boundary value problem is guaranteed and the existence of at least two solutions for in an appropriate interval is also discussed

singular nonlinear k n k conjugate boundary value problems abstract for 1 k n 1 solutions are obtained for the boundary value problem 1 n k y n f x y y i 0 0 0 i k 1 and y j 1 0 0 j n k 1 where f x y is singular at y 0 an application is made of a fixed point theorem for operators that are decreasing with respect to a cone

positive solutions for p n p conjugate boundary value problems a new existence result is presented for the p n p conjugate boundary value problem 1 n py n h t y 0 t 1 withy i 0 0 0 i p 1 andy i 1 0 0 i n p 1 our nonlinearityhmay be singular aty 0 t 0 and ort 1

multiplicity results for singular conjugate focal and n p problems abstract a new technique is presented which enables us to discuss the existence of one or more solutions to singular conjugate focal and n p boundary value problems our nonlinearity f t y may be singular at y 0 t 0 and or t 1

viterbi sparse spike detection accurate interpretation of seismic traveltimes and amplitudes in the exploration and global scales is complicated by the band limited nature of seismic data we discovered a stochastic method to reduce a seismic waveform into a most probable constituent spike train model waveforms were constructed from a set of candidate spike trains convolved with a source wavelet estimate for each model waveform a profile hidden markov model hmm was constructed to represent the waveform as a stochastic generative model with a linear topology corresponding to a sequence of samples each match state in the hmm represented a sample in the model waveform in which the amplitude was represented by a gaussian distribution insert and delete states allowed the underlying source wavelet to dilate or contract accounting for nonstationarity in the seismic data and errors in the source wavelet estimate the gaussian distribution characterizing each sample s amplitude accounted for random noise the viterbi algorithm was employed to simultaneously find the optimal nonlinear alignment between a model waveform and the seismic data and to assign a score to each candidate spike train the most probable traveltimes and amplitudes were inferred from the alignments of the highest scoring models the method required no implicit assumptions regarding the distribution of traveltimes and amplitudes however in practice the solution set may be limited to mitigate the nonuniqueness of solutions and to reduce the computational effort our analyses found that the method can resolve closely spaced arrivals below traditional resolution limits and that traveltime estimates are robust in the presence of random noise and source wavelet errors the method was particularly well suited to fine scale interpretation problems such as thin bed interpretation tying seismic images to well logs and the analysis of anomalous waveforms in global seismology

spiking band limited traces with a relative entropy algorithm a novel method for the inversion of band limited seismic traces to full bandwidth reflectivity traces is based on a probabilistic spiky model of the reflectivity trace in which position indicators and amplitudes of the spikes occur as random variables and relies on relative entropy inference from information theory first an a priori model for general reflectivity traces in the prospect is derived from nearby wells second the a priori distribution is updated into an a posteriori distribution for the specific trace being studied by the addition of the fourier data of the seismic trace within a passband uncertainty about the fourier coefficients can be accounted for by specification of a noise variance which implicitly is infinite outside the passband the update with relative entropy inference is justified because of its relationship with bayesian inference application of maximum a posteriori map estimation to the a posteriori distribution results in the most likely spiky reflectivity trace of f

infinite boundary value problems for nth order nonlinear impulsive integro differential equations in banach spaces abstract in this paper the existence of unbounded nonnegative solutions of a boundary value problem for n th order impulsive integro differential equations on an infinite interval in banach spaces is obtained by means of the monch fixed point theory in a banach space

positive solutions of nonlinear singular boundary value problems in abstract spaces by constructing a particular closed convex set and applying the monch fixed point theorem we study the existence of positive solutions of the singular boundary value problem x t f t x t t 0 1 x 0 x 1 in banach space singularities occurring at t 0 1 and x our method is completely distinct from what the former literature used even if we discuss the above problem in scalar space

a note on rearrangements spectral concentration and the zero order prolate spheroidal wavefunction if the measure of the support of a function f is small its symmetric decreasing rearrangement f is more nearly bandlimited to low frequencies than f while their norms are equal an immediate corollary is that the time limited zero order prolate spheroidal wavefunction is the extremal function for a new optimization problem involving time and bandlimiting the result has an application in exploration seismology

multichannel deconvolution of seismic signals using statistical mcmc methods in this paper we propose two multichannel blind deconvolution algorithms for the restoration of two dimensional 2d seismic data both algorithms are based on a 2d reflectivity prior model and use iterative multichannel deconvolution procedures which deconvolve the seismic data while taking into account the spatial dependency between neighboring traces the first algorithm employs in each step a modified maximum posterior mode mpm algorithm which estimates a reflectivity column from the corresponding observed trace using the estimate of the preceding reflectivity column the second algorithm takes into account estimates of both the preceding and subsequent columns in the estimation process both algorithms are applied to synthetic and real data and demonstrate better results compared to those obtained by a single channel deconvolution method expectedly the second algorithm which utilizes more information in the estimation process of each reflectivity column is shown to produce better results than the first algorithm

global structure instability of riemann solutions for general quasilinear hyperbolic systems of conservation laws in the presence of a boundary abstract this work is a continuation of our previous work z q shao d x kong y c li shock reflection for general quasilinear hyperbolic systems of conservation laws nonlinear anal tma 66 1 2007 93 124 in this paper we study the global structure instability of the riemann solution u u x t containing shocks at least one rarefaction wave for general n n quasilinear hyperbolic systems of conservation laws in the presence of a boundary we prove the nonexistence of global piecewise c 1 solution to a class of the mixed initial boundary value problem for general n n quasilinear hyperbolic systems of conservation laws on the quarter plane our result indicates that this kind of riemann solution u u x t mentioned above for general n n quasilinear hyperbolic systems of conservation laws in the presence of a boundary is globally structurally unstable some applications to quasilinear hyperbolic systems of conservation laws arising from physics and mechanics are also given

generalized linear inversion of reflection seismic data generalized linear inversion sometimes known as model perturbation nonlinear regression or inverse modeling is applied to synthetic and real seismic data sets with the objective of obtaining an impedance profile as a function of time the impedances solved for are parameterized in a manner that describes the unknown earth using fewer variables than previous seismic generalized linear inversion techniques in this application only single traces of common midpoint cmp processed data will be inverted the method of generalized linear inversion gli presented here is designed to improve on the shortcomings of recursive inversion with respect to relative and absolute scale of the impedance results resolution of impedance boundaries and distortion from residual wavelet effects in obtaining these goals other advantageous aspects of gli were discovered for example it is insensitive to noise in many cases and it will allow an interpreter to fix the impedance of any number of known lithologies in an in

wavelet estimation revisited this article deals with the ubiquitous problem of wavelet estimation in essence the problem consists of estimating at best two unknowns from one equation in the seismic case as it is well known the model of the recorded seismic trace xt is generally formulated as a convolution xt wt rt where wt is the seismic source signature or wavelet and rt is the reflectivity series which is comprised of all primary and multiple reflections since in general we can assume that rt is white or nearly so the amplitude signature of wt may be estimated with some degree of confidence it is of course the phase of wt that presents the difficulty it turns out perhaps somewhat counterintuitively that except for rather special circumstances the phase of a signal uniquely determines the signal to within a scale factor the amplitude spectrum plays no part whatsoever we explore this further below no wonder then that the phase is the all important factor

autoregressive recovery of the acoustic impedance this paper presents a method of recovering the acoustic impedance from reflection seismograms using autoregressive ar modeling an approach originally applied to deconvolution by lines and clayton 1977 the algorithm which we describe is novel both in the manner in which the missing low and high frequency information is predicted and in the fact that the prediction may be constrained if acoustic impedance information is available the prediction of the low frequencies treats the missing data as a gap which extends from the low frequency cut off in the negative frequency band to the corresponding frequency in the positive frequency band the conjugate symmetry which governs the behavior of the spectrum in the band is taken into account in the prediction the missing high frequencies are predicted using a modified minimum entropy norm in the frequency domain both synthetic and field examples are presented and illustrate the robustness of the new ar algorithm under a variety of conditions the field e

low frequency reflection data augmentation by an inpainting method 1d acoustic media abstractthe augmentation of low frequencies is an important problem for inverting seismic reflection data through an acoustic scattering approach we have characterized how the data down to zero frequency grows in importance with increasing depth and the size of the velocity contrast we have developed a hermite distributed approximation functional hdaf inpainting method that provides a least squares fitting of the acoustic reflection data down to zero frequency the hdaf data fitting is applied to the first born approximation for compact velocity contrasts the top and bottom depth velocities are equal several fitting constraints are derived using a taylor expansion of the reflection data near zero frequency they provide a coupling with the first born approximation of the velocity contrast the method can be extended to the general geophysical case in which the final velocity i e is different from the starting velocity by use of the distorted wave born approximation numerical results for 1d t

automatic phase correction of common midpoint stacked data the residual wavelet on a processed seismic section is often not zero phase despite all efforts to make it so in this paper we adopt the convolutional model for the processed seismogram assume that the residual phase shift can be approximated by a frequency independent constant and use the varimax norm to generate an algorithm to estimate the residual phase directly application of our algorithm to reflectivities from well logs suggests that it should work in the majority of cases so long as the reflectivity is non gaussian an application of our algorithm to stacked data enhances the interpretability of the seismic section and leads to an improved match between the recovered relative acoustic impedance and a measured velocity log

mixed phase deconvolution we describe a new algorithm for mixed phase deconvolution it is valid only for pulses whose z transform has no zeros on the unit circle that is the amplitude spectrum cannot be zero for any frequency using the z transform of a discrete time signal and assuming that the signal has zeros inside the unit circle the inverse of its minimum delay component may be estimated by solving the extended yule walker eyw system of equations with the lag of the autocorrelation function acf on diagonal of the coefficient matrix this property of the solution of eyw equations is exploited to derive mixed phase inverse filters and their corresponding mixed phase pulses for different values of a suite of inverse filters is generated using the same acf to choose the best decomposition and its corresponding mixed phase inverse filter we have used the value of which gives the maximum value of the lp norm of the filtered signal the optimal value of does not seem to be very sensitive to the choice of norm

minimum entropy deconvolution with frequency domain constraints a method for reconstructing the reflectivity spectrum using the minimum entropy criterion is presented the algorithm fmed described is compared with the classical minimum entropy deconvolution med as well as with the linear programming lp and autoregressive ar approaches the med is performed by maximizing an entropy norm with respect to the coefficients of a linear operator that deconvolves the seismic trace by comparison the approach presented here maximizes the norm with respect to the missing frequencies of the reflectivity series spectrum this procedure reduces to a nonlinear algorithm that is able to carry out the deconvolution of band limited data avoiding the inherent limitations of linear operators the proposed method is illustrated under a variety of synthetic examples field data are also used to test the algorithm the results show that the proposed method is an effective way to process band limited data the fmed and the lp arise from similar conceptions both methods seek an extremum of a particular norm subjected to frequency constraints in the lp approach the linear programming problem is solved using an adaptation of the simplex method which is a very expensive procedure the fmed uses only two fast fourier transforms ffts per iteration hence the computational cost of the inversion is reduced

a computationally fast approach to maximum likelihood deconvolution in this paper we derive and implement a maximum likelihood deconvolution mld algorithm based on the same channel and statistical models used by kormylo and mendel 1983a that leads to many fewer computations than their mld algorithm both algorithms can simultaneously estimate a nonminimum phase wavelet and statistical parameters detect locations of significant reflectors and deconvolve the data our mld algorithm is implemented by a two phase block component method bcm the phase 1 block functions like a coarse adjustment of unknown quantities and provides a set of good initial conditions for the phase 2 block which functions like a fine adjustment of unknown quantities we demonstrate good performance of our algorithm for both synthetic and real data

evaluation and applications of the iterated window maximization method for sparse deconvolution estimating a sparse signal from a linearly degraded and noisy data record is often desirable in seismic and ultrasonic applications bernoulli gaussian modeling and maximum a posteriori estimation has proven successful but entails computationally difficult optimization problems that must be solved by suboptimal methods the iterated window maximization iwm algorithm was proposed for such optimization by kaaresen see ibid vol 45 p 1173 83 1997 the purpose of this paper is twofold first the iwm is evaluated against several established alternatives for bernoulli gaussian deconvolution the restoration quality is quantified by various loss functions and the average performance is studied through simulation in all cases examined the iwm combined better restoration and significantly faster execution than the other algorithms this motivates extension of iwm to other models which is the second objective of this paper promising real data results are obtained from such diverse applications as robust modeling of ultrasonic nondestructive evaluation data deblurring of two dimensional 2 d astronomical star fields and segmentation of seismic well logs it is also argued that iwm can be used for other deconvolution problems as long as the function to be reconstructed is in some sense sparse

2 d bayesian deconvolution inverse problems can be solved in different ways one way is to define natural criteria of good recovery and build an objective function to be minimized if instead we prefer a bayesian approach inversion can be formulated as an estimation problem where a priori information is introduced and the a posteriori distribution of the unobserved variables is maximized when this distribution is a gibbs distribution these two methods are equivalent furthermore global optimization of the objective function can be performed with a monte carlo technique in spite of the presence of numerous local minima application to multitrace deconvolution is proposed in traditional 1 d deconvolution a set of unidimensional processes models the seismic data while a markov random field is used for 2 d deconvolution in fact the introduction of a neighborhood system permits one to model the layer structure that exists in the earth and to obtain solutions that present lateral coherency moreover optimization of an appropriated objective function by simulated annealing allows one to control the fit with the input data as well as the spatial distribution of the reflectors extension to 3 d deconvolution is straightforward