seismic data restoration via data driven tight frame restoration interpolation of missing traces plays a crucial role in the seismic data processing pipeline efficient restoration methods have been proposed based on sparse signal representation in a transform domain such as fourier wavelet curvelet and shearlet transforms most existing methods are based on transforms with a fixed basis we considered an adaptive sparse transform for restoration of data with complex structures in particular we evaluated a data driven tight frame based sparse regularization method for seismic data restoration the main idea of the data driven tight frame tf is to adaptively learn a set of framelet filters from the currently interpolated data under which the data can be more sparsely represented hence the sparsity promoting l1 norm spl1 minimization methods can produce better restoration results by using the learned filters a split inexact uzawa algorithm which can be viewed as a generalization of the alternating direction of multiplier method admm was applied to solve the presented spl1 model numerical tests were performed on synthetic and real seismic data for restoration of randomly missing traces over a regular data grid our experiments showed that our proposed method obtains the state of the art restoration results in comparison with the traditional fourier based projection onto convex sets the tight frame based method and the recent shearlet regularization admm method

slope inequality for families of curves over surfaces in this paper we investigate the general notion of the slope for families of curves f x rightarrow y the main result is an answer to the above question when dim y 2 and we prove a lower bound for this new slope in this case over fields of any characteristic both the notion and the slope inequality are compatible with the theory for dim y 0 1 in a very natural way and this gives a strong evidence that the slope for an n fold fibration of curves f x rightarrow y may be k x y n mathrm ch n 1 f omega x y rather than the usual stability methods the whole proof of the slope inequality here is based on a completely new method using characteristic p 0 geometry a simpler version of this method yields a new proof of the slope inequality when dim y 1

adaptive multiple subtraction with wavelet based complex unary wiener filters abstractadaptive subtraction is a key element in predictive multiple suppression methods it minimizes misalignments and amplitude differences between modeled and actual multiples and thus reduces multiple contamination in the data set after subtraction due to the high crosscorrelation between their waveforms the main challenge resides in attenuating multiples without distorting primaries as they overlap on a wide frequency range we split this wide band problem into a set of more tractable narrow band filter designs using a 1d complex wavelet frame this decomposition enables a single pass adaptive subtraction via complex single sample unary wiener filters consistently estimated on overlapping windows in a complex wavelet transformed domain each unary filter compensates for amplitude differences within its frequency support and can correct small and large misalignment errors through phase and integer delay corrections this approach greatly simplifies the matching filter estimation and despit

seislet transform and seislet frame we introduce a digital waveletlike transform which is tailored specifically for representing seismic data the transform provides a multiscale orthogonal basis with basis functions aligned along seismic events in the input data it is defined with the help of the wavelet lifting scheme combined with local plane wave destruction in the 1d case the seislet transform is designed to follow locally sinusoidal components in the 2d case it is designed to follow local plane wave components with smoothly variable slopes if more than one component is present the transform turns into an overcomplete representation or a tight frame in these terms the classic digital wavelet transform is simply a seislet transform for a zero frequency in one dimension or zero slope in two dimensions the main objective of the new transform is an effective seismic data compression for designing efficient data analysis algorithms traditional signal processing tasks such as noise attenuation and trace interpolation become simply defined in the seislet domain when applied in the offset direction on common midpoint or common image point gathers the seislet transform finds an additional application in optimal stacking of seismic records

boundedness of solutions for a class of reversible systems in this paper we are concerned with the boundedness of all solutions for the following differential systemx email protected p y e t y email protected q x f x f p y f x e t where f r s s r 2s r 1 f x is odd e t c 4 r z is even max 1 p 2 p 1

five dimensional interpolation recovering from acquisition constraints although 3d seismic data are being acquired in larger volumes than ever before the spatial sampling of these volumes is not always adequate for certain seismic processes this is especially true of marine and land wide azimuth acquisitions leading to the development of multidimensional data interpolation techniques simultaneous interpolation in all five seismic data dimensions inline crossline offset azimuth and frequency has great utility in predicting missing data with correct amplitude and phase variations although there are many techniques that can be implemented in five dimensions this study focused on sparse fourier reconstruction the success of fourier interpolation methods depends largely on two factors 1 having efficient fourier transform operators that permit the use of large multidimensional data windows and 2 constraining the spatial spectrum along dimensions where seismic amplitudes change slowly so that the sparseness and band limitation assumptions remain valid fourier reconstruction can be performed when enforcing a sparseness constraint on the 4d spatial spectrum obtained from frequency slices of five dimensional windows binning spatial positions into a fine 4d grid facilitates the use of the fft which helps on the convergence of the inversion algorithm this improves the results and computational efficiency the 5d interpolation can successfully interpolate sparse data improve avo analysis and reduce migration artifacts target geometries for optimal interpolation and regularization of land data can be classified in terms of whether they preserve the original data and whether they are designed to achieve surface or subsurface consistency

boundedness of solutions of nonlinear differential equations abstract in this paper we are concerned with the boundedness of solutions of some nonlinear differential equations which are neither dissipative nor conservative having the form x x t x g x t 0 where f and g are odd polynomials in x with coefficients which are even and periodic in t using the kam theory for reversible systems we prove that all the solutions are bounded whenever a sharp condition on the degrees of f and g is satisfied we also obtain a boundedness result when f x t 0 i e the equation is conservative under some smoothness assumptions on g which improve the previously known ones in this case no symmetry conditions on g are needed

on the representation of operators in bases of compactly supported wavelets this paper describes exact and explicit representations of the differential operators d n dx n n 1 2 cdots in orthonormal bases of compactly supported wavelets as well as the representations of the hilbert transform and fractional derivatives the method of computing these representations is directly applicable to multidimensional convolution operators also sparse representations of shift operators in orthonormal bases of compactly supported wavelets are discussed and a fast algorithm requiring o n log n operations for computing the wavelet coefficients of all n circulant shifts of a vector of the length n 2 n is constructed as an example of an application of this algorithm it is shown that the storage requirements of the fast algorithm for applying the standard form of a pseudodifferential operator to a vector see g beylkin r r coifman and v rokhlin comm pure appl math 44 1991 pp 141 183 may be reduced from o n to o log 2 n significant entries

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

application of singular value decomposition to vertical seismic profiling an essential part of the interpretation of vertical seismic profiles vsp is the separation of the upgoing and downgoing waves this paper presents a new approach which is based on the decomposition of time shifted vsp sections into eigenimages using singular value decomposition svd the first few eigenimages of the time shifted vsp section contain the contributions of the horizontally aligned downgoing waves the last few eigenimages contain the contribution of uncorrelated noise components the separated upgoing waves are recovered as a partial sum of the eigenimages mportant aspects of this approach are that regular sampling of the recording levels is not required that the first break times need not be measured with extreme accuracy that noise rejection may be automatically included in the processing and that eigenimages or sums of eigenimages which may be computed as part of the approach can provide important additional information

boundedness of solutions for nonlinear reversible system at resonance abstract in this paper we are concerned with the problem of boundedness of solutions for the following nonlinear p laplacian p x p 1 n p p x f x p x g x e t where p u u p 2u p 2 for some n n f g c r are odd functions e t c r is odd and 2 p periodic

iterative deblending of simultaneous source seismic data using seislet domain shaping regularization abstractwe used a novel iterative estimation scheme for separation of blended seismic data from simultaneous sources the scheme is based on an augmented estimation problem that can be solved by iteratively constraining the deblended data using shaping regularization in the seislet domain we formulated the forward modeling operator in the common receiver domain in which two sources were assumed to be blended using a random time shift dithering approach the nonlinear shaping regularization framework offered some freedom in designing a shaping operator to constrain the model in an underdetermined inverse problem we designed the backward operator and the shaping operator for the shaping regularization framework the backward operator can be optimally chosen as half of the identity operator in the two source case and the shaping operator can be chosen as coherency promoting operator the high performance deblending effect of the iterative framework was tested on three numerically blended synthetic data s

severi inequality for varieties of maximal albanese dimension in this paper we prove the general severi inequality for varieties of maximal albanese dimension suppose that x is an n dimensional projective normal minimal and mathbb q gorenstein variety of general type in characteristic zero if x is of maximal albanese dimension then k n x ge 2 n chi omega x

boundedness of solutions for reversible systems in this paper we study the boundedness of all the solutions for the following second order nonlinear differential equation p x f x p x g x e t ddtwhere p u u p 2u p 1 f g c r 0 and odd e c r z is odd g is subquasilinear sgn x g x g x p x 0 as x

inversion of seismic reflection data in the acoustic approximation the nonlinear inverse problem for seismic reflection data is solved in the acoustic approximation the method is based on the generalized least squares criterion and it can handle errors in the data set and a priori information on the model multiply reflected energy is naturally taken into account as well as refracted energy or surface waves the inverse problem can be solved using an iterative algorithm which gives at each iteration updated values of bulk modulus density and time source function each step of the iterative algorithm essentially consists of a forward propagation of the actual sources in the current model and a forward propagation backward in time of the data residuals the correlation at each point of the space of the two fields thus obtained yields the corrections of the bulk modulus and density models this shows in particular that the general solution of the inverse problem can be attained by methods strongly related to the methods of migration of unstacked data and commerc

fibered algebraic surfaces with low slope the background of this paper is the following conjecture on complex surfaces of general type let s be a smooth projective minimal surface of general type over ti with c 2 throughout the paper

interpolation and extrapolation using a high resolution discrete fourier transform we present an iterative nonparametric approach to spectral estimation that is particularly suitable for estimation of line spectra this approach minimizes a cost function derived from bayes theorem the method is suitable for line spectra since a long tailed distribution is used to model the prior distribution of spectral amplitudes since the data themselves are used as constraints phase information can also be recovered and used to extend the data outside the original window the objective function is formulated in terms of hyperparameters that control the degree of fit and spectral resolution noise rejection can also be achieved by truncating the number of iterations spectral resolution and extrapolation length are controlled by a single parameter when this parameter is large compared with the spectral powers the algorithm leads to zero extrapolation of the data and the estimated fourier transform yields the periodogram when the data are sampled at a constant rate the algorithm uses one levinson recursion per iteration for irregular sampling the algorithm uses one cholesky decomposition per iteration the performance of the algorithm is illustrated with three different problems that arise in geophysical data 1 harmonic retrieval from a time series contaminated with noise 2 linear event detection from a finite aperture array of receivers 3 interpolation extrapolation of gapped data the performance of the algorithm as a spectral estimator is tested with the kay and marple 1981 data set

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

the severi inequality k2 4χ for surfaces of maximal albanese dimension we prove the so called severi inequality stating that the invariants of a minimal smooth complex projective surface of maximal albanese dimension satisfy k 2 s ge4 chi s

relative noether inequality on fibered surfaces abstract we prove effective upper bounds on the global sections of nef line bundles of small generic degree over a fibered surface over a field of any characteristic it can be viewed as a relative version of the classical noether inequality for surfaces as a consequence we give a new proof of the slope inequality for fibered surface without using any stability method the treatment is essentially different from those of xiao cornalba harris and moriwaki we also study the geography problem of surfaces in positive characteristics and show that the severi inequality is true for surfaces of general type in positive characteristic whose albanese map is generically finite moreover the geography of surfaces with albanese fibrations is studied

invariant tori and lagrange stability of pendulum type equations abstract in this paper we prove that the pendulum type equation x g t x 0 possesses infinitely many invariant tori whenever g t x has zero mean value on the torus t2 where g t x belongs to c t2 this yields the boundedness for solutions of the considered pendulum type equation and thus leads to an answer to j moser s boundedness problem 1973 ann of math stud 77

boundedness problem for solutions of semilinear asymmetric equations in this paper we consider the boundedness problem for the solutions of the following semilinear differential equation f p x p 1 email protected p x email protected f p x g x e t where f p u u p 2u p 2 a b are positive constants satisfying a 1 p email protected 1 p email protected 1 g x c r is semilinear and e t c r z

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

computing traveltime and amplitude sensitivity kernels in finite frequency tomography the efficient computation of finite frequency traveltime and amplitude sensitivity kernels for velocity and attenuation perturbations in global seismic tomography poses problems both of numerical precision and of validity of the paraxial approximation used we investigate these aspects using a local model parameterization in the form of a tetrahedral grid with linear interpolation in between grid nodes the matrix coefficients of the linear inverse problem involve a volume integral of the product of the finite frequency kernel with the basis functions that represent the linear interpolation we use local and global tests as well as analytical expressions to test the numerical precision of the frequency and spatial quadrature there is a trade off between narrowing the bandpass filter and quadrature accuracy and efficiency using a minimum step size of 10km for s waves and 30km for ss waves relative errors in the quadrature are of the order of 1 for direct waves such as s and a few percent for ss waves which are below data uncertainties in delay time or amplitude anomaly observations in global seismology larger errors may occur wherever the sensitivity extends over a large volume and the paraxial approximation breaks down at large distance from the ray this is especially noticeable for minimax phases such as ss waves with periods 20s when kernels become hyperbolic near the reflection point and appreciable sensitivity extends over thousands of km errors becomes intolerable at epicentral distance near the antipode when sensitivity extends over all azimuths in the mantle effects of such errors may become noticeable at epicentral distances140 we conclude that the paraxial approximation offers an efficient method for computing the matrix system for finite frequency inversions in global tomography though care should be taken near reflection points and alternative methods are needed to compute sensitivity near the antipode

minimum weighted norm interpolation of seismic records in seismic data processing we often need to interpolate and extrapolate data at missing spatial locations the reconstruction problem can be posed as an inverse problem where from inadequate and incomplete data we attempt to reconstruct the seismic wavefield at locations where measurements were not acquired we propose a wavefield reconstruction scheme for spatially band limited signals the method entails solving an inverse problem where a wavenumber domain regularization term is included the regularization term constrains the solution to be spatially band limited and imposes a prior spectral shape the numerical algorithm is quite efficient since the method of conjugate gradients in conjunction with fast matrix vector multiplications implemented via the fast fourier transform fft is adopted the algorithm can be used to perform multidimensional reconstruction in any spatial domain

wavelet transform inversion with prior scale information the concept of scale is of central importance in all problems related to the interpretation of earth data inversion of geophysical data does not at present however allow the incorporation of scale dependent information directly this article presents an application of the wavelet transform to traveltime inversion where we are specifically concerned with the selection and application of a priori scale information in the wavelet domain to reconstruct tomographic models because of the nature of the problem the wavelet basis chosen for this example is the haar basis the results are compared with the classic damped least squares dls method which is a special case of the wavelet transform approach when all the scales are treated equally we show that with scale dependent prior information most of the artifacts created by the dls technique are diminished considerably

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

asymmetric chirplet transform for sparse representation of seismic data abstractfor a fast and accurate extraction of important information in seismic signals a sparse representation based on the physical parameters of the given data is crucial we have used the asymmetric gaussian chirplet model agcm and established a dictionary free variant of the orthogonal matching pursuit a greedy algorithm for sparse approximation the atoms of agcm so called chirplets display asymmetric oscillation attenuation properties which make the agcm very suitable for sparse representation of absorption decay seismic signals unlike the fourier transform which assumes that the seismic signals consist of plane waves the agcm assumes that the seismic signal consists of nonstationary compressed plane waves i e symmetric or asymmetric chirplets thus agcm is a general model for seismic wave simulation and its model parameters include an envelope part and a phase part we have mainly concentrated on the parameters of the envelope part such as the envelope amplitude and arrival time we

seismic data decomposition into spectral components using regularized nonstationary autoregression summary seismic data can be decomposed into nonstationary spectral components with smoothly variable frequencies and smoothly variable amplitudes to estimate local frequencies i use a nonstationary version of prony s spectral analysis method defined with the help of regularized nonstationary autoregression rnar to estimate local amplitudes of different components i fit their sum to the data using regularized nonstationary regression rnr shaping regularization ensures stability of the estimation process and provides controls on smoothness potential applications of the proposed technique include noise attenuation seismic data compression and seismic data regularization

boundedness for solutions of nonlinear hill s equations with periodic forcing terms via moser s twist theorem we consider the nonlinear hill s equation with periodic forcing term x x2n 1 a1 ea t x p t n 1 where a t and p t are continuous and 1 periodic a1 and are positive constants and e is a small parameter the purpose of this paper is to prove that all solutions of the above mentioned equation are bounded for t r and that there are infinitely many quasi periodic solutions and an infinity of periodic solutions of minimal period m for each positive integer m

quasiperiodic motions in superquadratic time periodic potentials it is shown that for a large class of potentials on the line with superquadratic growth at infinity and with the additional time periodic dependence all possible motions under the influence of such potentials are bounded for all time and that most in a precise sense motions are in fact quasiperiodic the class of potentials includes as very particular examples the exponential polynomial and much more this extends earlier results and gives an answer to a problem posed by littlewood in the mid 1960 s along the way machinery is developed for estimating the action angle transformation directly in terms of the potential and also some apparently new identities involving singular integrals are derived

on the arakelov inequality in positive characteristic in this note we generalize the arakelov inequality in positive characteristic for non isotrivial semistable families of curves of g ge 2 which are liftable to w 2 k resp w k as a consequence we give an analogue of beauville s conjecture in positive characteristic there are at least 5 singular fibers for non isotrivial semistable families of curves of g ge 2 over mathbb p 1 which are liftable to w k

boundedness of solutions of some nonlinear differential equations in this paper the boundedness of solutions for the nonlinear differential equations is studied x f x t x g x t 0 where f x t k 0 ma k t x 2 k 1 g x t j 0 n 1b j t x 2 j 1 a k t b j t are continuous periodic functions with period 1 0

boundedness of solutions for reversible system via moser s twist theorem abstract in this paper we consider the problem of the boundedness of all solutions for the reversible system x j 0 l b j t x 2 j 1 x x 2 n 1 i 0 n 1 a i t x 2 i 1 0 it is shown that all the solutions are bounded provided that the a i t 0 i n 1 2 are of bounded variation in 0 1 and the derivatives of b j t and a i t n 1 2 1 i n 1 0 j l are lipschitzian it is also shown that there exist a i s being discontinuous everywhere such that all solutions of the equation are bounded this implies that the continuity of a i s is not necessary for the boundedness of solutions of the equation