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

on the liouville theorem for the stationary navier stokes equations in a critical space abstract in this paper we prove the liouville type theorem for the stationary navier stokes equations on r 3 more specifically if a solution u h 1 r 3 to the stationary navier stokes system satisfies additional conditions characterized by the decays near infinity and by the oscillation then we show that u 0

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

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

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

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

on the implementation of singly implicit runge kutta methods a modified newton method is often used to solve the algebraic equations that arise in the application of implicit runge kutta methods when the runge kutta method has a coefficient matrix a with a single point spectrum with eigenvalue a the efficiency of the modified newton method is much improved by using a similarity transformation of a each iteration involves vector transformations in this article an alternative iteration scheme is obtained which does not require vector transformations and which is simpler in other respects also both schemes converge in a finite number of iterations when applied to linear systems of differential equations but the new scheme uses the nilpotency of a ai to achieve this numerical results confirm the predicted convergence for nonlinear systems and indicate that the scheme may be a useful alternative to the modified newton method for low dimensional systems the scheme seems to become less effective as the dimension increases however it has clear advantages for parallel computation making it competitive for high dimensional systems

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

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

algebraically stable and implementable runge kutta methods of high order there are three interesting properties of methods for stiff ordinary differential equations order stability and efficiency of implementation this paper constructs runge kutta methods of orders 5 and 6 which possess these properties to a high extent we further classify all algebraically stable methods of an arbitrary order and give various relationships between contractivity and order of implicit methods

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

liouville type theorems for the 3d stationary navier stokes mhd and hall mhd equations abstract in this paper we consider the liouville type problem of the three dimensional stationary incompressible navier stokes equations and establish an improved liouville type result only under the condition that the l p norm of a velocity u is finite moreover these results are extended to the mhd and hall mhd equations

efficiently implementable algebraically stable runge kutta methods recent work by butcher bit 16 1976 pp 237 240 on the implementation of implicit runge kutta methods has shown that if his technique is used the most efficiently implementable methods are those whose runge kutta matrix has a single real eigenvalue in this paper therefore we analyze the algebraic stability of two families of methods with this property namely singly implicit and diagonally implicit methods

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

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

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

liouville type theorems for the forced euler equations and the navier stokes equations in this paper we study the liouville type properties for solutions to the steady incompressible euler equations with forces in mathbb r n if we assume single signedness condition on the force then we can show that a c 1 mathbb r n solution v p with v 2 p in l frac q 2 mathbb r n q in frac 3n n 1 infty is trivial v 0 for the solution of the steady navier stokes equations satisfying v x to 0 as x to infty the condition int mathbb r 3 delta v frac 6 5 dx infty which is stronger than the important d condition int mathbb r 3 nabla v 2 dx infty but both having the same scaling property implies that v 0 in the appendix we reprove theorem 1 1 chae commun math phys 273 203 215 2007 using the self similar euler equations directly

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

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

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

the potential for parallelism in runge kutta methods part 1 rk formulas in standard form the authors examine the potential for parallelism in runge kutta rk methods based on formulas in standard one step form both negative and positive results are presented many of the negative results are based on a theorem that bounds the order of an rk formula in terms of the minimal polynomial associated with its coefficient matrix the positive results are largely examples of prototypical formulas that offer a potential for effective coarse grain parallelism on machines with a few processors

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

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

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

analysis of field experiments by least squares smoothing summary assuming a smooth trend plus independent error model for the environmental effects in the yields of a field plot experiment least squares smoothing methods are developed to estimate both the treatment effects and the unknown trend treatment estimates are closely related to those resulting from a generalized least squares analysis in which the covariance structure for the environmental effects has a particular form however the main emphases are on the accuracy of treatment estimates under a fixed smooth trend plus error model and the exploratory power of the basic method to isolate trend effects of unknown form although the detailed development is for the one dimensional case generalizations of the smoothness concept and extensions to two dimensions are also discussed application of the basic method is illustrated on three data sets and the results compared with other analyses

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

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

a modification of cauchy s method for quadratic equations the plan of this paper is to obtain one point iterative methods with any r order of convergence when they are applied to approximate solutions of quadratic equations in banach spaces to do this we consider real cauchy s method and under certain natural modifications it is extended to banach spaces some applications are also provided

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

on liouville type theorems for the steady navier stokes equations in r3 abstract in this paper we prove three different liouville type theorems for the steady navier stokes equations in r 3 in the first theorem we improve logarithmically the well known l 9 2 r 3 result in the second theorem we present a sufficient condition for the trivially of the solution v 0 in terms of the head pressure q 1 2 v 2 p the imposed integrability condition here has the same scaling property as the dirichlet integral in the last theorem we present fubini type condition which guarantee v 0

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

liouville theorems for the navier stokes equations and applications we study bounded ancient solutions of the navier stokes equations these are solutions with bounded velocity defined in rn 1 0 in two space dimensions we prove that such solutions are either constant or of the form u x t b t depending on the exact definition of admissible solutions the general 3 dimensional problem seems to be out of reach of existing techniques but partial results can be obtained in the case of axisymmetric solutions we apply these results to some scenarios of potential singularity formation for axi symmetric solutions and obtain extensions of results in a recent paper by chen strain tsai and yau 4

a statistical model for positron emission tomography abstract positron emission tomography pet still in its research stages is a technique that promises to open new medical frontiers by enabling physicians to study the metabolic activity of the body in a pictorial manner much as in x ray transmission tomography and other modes of computerized tomography the quality of the reconstructed image in pet is very sensitive to the mathematical algorithm to be used for reconstruction in this article we tailor a mathematical model to the physics of positron emissions and we use the model to describe the basic image reconstruction problem of pet as a standard problem in statistical estimation from incomplete data we describe various estimation procedures such as the maximum likelihood ml method using the em algorithm the method of moments and the least squares method a computer simulation of a pet experiment is then used to demonstrate the ml and the least squares reconstructions the main purposes of this article are to report on what we believe is an

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