how singular functions define distributions following dirac schwartz and others distributions are well understood and widely used in physics as generalized functions however a function with a nonintegrable singularity does not define a distribution automatically or unambiguously we review a variety of ways in which such distributions can be defined sometimes with inequivalent results or results containing arbitrary constants we give special attention to the function cosech2 x and its semiclassical scaling limit which have recently attracted some attention in statistical mechanics

detection of signals by information theoretic criteria a new approach is presented to the problem of detecting the number of signals in a multichannel time series based on the application of the information theoretic criteria for model selection introduced by akaike aic and by schwartz and rissanen mdl unlike the conventional hypothesis testing based approach the new approach does not requite any subjective threshold settings the number of signals is obtained merely by minimizing the aic or the mdl criteria simulation results that illustrate the performance of the new method for the detection of the number of signals received by a sensor array are presented

a bootstrap procedure for high resolution velocity analysis a method is given for further improving velocity estimates derived from high resolution velocity analysis in conventional velocity analysis a set of tentative velocities is used to apply a normal moveout nmo correction to a set of spatio temporal windows the coherence measure is evaluated for each velocity and finally the velocity estimate is retrieved from the peak of the coherence measure because analytical expressions for velocity uncertainties are difficult to derive i propose an intensive statistical procedure the bootstrap method to assess the accuracy of the velocity estimate in the bootstrap method i create a data sample by randomly drawing seismic traces with replacement from a window of the common midpoint gather cmp next i calculate the velocity that maximizes the coherence measure for each bootstrap realization the variation of this velocity provides a means to compute standard errors i also use the bootstrap method to construct an average coherence measure and a kernel density estimator of the velocity that maximizes the coherence the average coherence exhibits an important attenuation of spurious events while retaining enough resolution to model reflections properly with similar moveout curves the latter is illustrated with synthetic and field data examples

regularization pseudofunction and hadamard finite part abstract first we discuss and correlate the various types of regularizations available in the literature for the singular function h x x k where k is an integer and h x is the heaviside function then we present the corresponding regularization for the function r k where r is the radial distance in r n thereafter we express the recently discovered distributional derivatives of this function in terms of pseudofunctional language commonly used in the coulomb gravitational and interparticle potentials where the function 1 r plays a fundamental role

a high resolution weighted ab semblance for dealing with amplitude variation with offset phenomenon abstractvelocity analysis is an essential step in seismic reflection data processing the conventional and fastest method to estimate how velocity changes with increasing depth is to calculate semblance coefficients traditional semblance has two problems low time and velocity resolution and an inability to handle amplitude variation with offset avo phenomenon although a method known as the ab semblance can arrive at peak velocities in the areas with an avo anomaly it has a lower velocity resolution than conventional semblance we have developed a weighted ab semblance method that can handle both problems simultaneously we have developed two new weighting functions to weight the ab semblance to enhance the resolution of velocity spectra in the time and velocity directions in this way we increase the time and velocity resolution while eliminating the avo problem the first weighting function is defined based on the ratio between the first and the second singular values of the time window to improve

homogenization and corrector for the wave equation with discontinuous coefficients in time abstract in this paper we analyze the homogenization of the wave equation with bounded variation coefficients in time generalizing the classical result which assumes lipschitz continuity we start showing a general existence and uniqueness result for a general sort of hyperbolic equations then we obtain our homogenization result comparing the solution of a sequence of wave equations to the solution of a sequence of elliptic ones we conclude the paper making an analysis of the corrector firstly we obtain a corrector result assuming that the derivative of the coefficients in the time variable is equicontinuous this result was known for non time dependent coefficients after we show with a counterexample that the regularity hypothesis for the corrector theorem is optimal in the sense that it does not hold if the time derivative of the coefficients is just bounded

window length selection for optimum slowness resolution of the local slant stack transform one of the most critical decisions in the design of a local slant stack transform lsst is the selection of its aperture or more precisely the selection of the appropriate number of traces and their weighting coefficients for each slant stack the challenge is to achieve a good compromise between the slowness and the spatial resolution conventionally the window size is chosen in a more intuitive manner by visual inspection and some limited tests we analyzed the lsst to establish rigorous criteria for the window selection to achieve the optimum slowness and spatial resolution in the transformed domain for a given data set for this purpose we estimated the slowness resolution in the lsst domain as a function of the spatial window bandwidth and of the spectral characteristics of the waves for a wave with a given bandpass spectrum the slowness resolution the stopband attenuation and the wavefront tracking capability are determined by the spatial window for narrowband signals the spatial window m

signal and traveltime parameter estimation using singular value decomposition abstractsignal detection and traveltime parameter estimation can be performed by computing a coherence function in a data window centered around a traveltime function defined by its parameters we used singular value decomposition of the data matrix not eigendecomposition of a covariance matrix to review the most commonly used coherence measures this resulted in a new reduced semblance coefficient defined from the first eigenimage assuming that the signal amplitude was the same on all data channels as in classical semblance in a second signal model the time signal was constant on each channel but the amplitude changed then the semblance coefficient is the square of the first singular value divided by the energy of the data two normalized crosscorrelation coefficients derived from the first eigenimage can also be used as a coherence measure the normalized crosscorrelation of the spatial singular vector with a vector with all elements equal to one and the normalized crosscorrelation of the tem

high resolution velocity gathers and offset space reconstruction we present a high resolution procedure to reconstruct common midpoint cmp gathers first we describe the forward and inverse transformations between offset and velocity space then we formulate an underdetermined linear inverse problem in which the target is the artifacts free aperture compensated velocity gather we show that a sparse inversion leads to a solution that resembles the infinite aperture velocity gather the latter is the velocity gather that should have been estimated with a simple conjugate operator designed from an infinite aperture seismic array this high resolution velocity gather is then used to reconstruct the offset space the algorithm is formally derived using two basic principles first we use the principle of maximum entropy to translate prior information about the unknown parameters into a probabilistic framework in other words to assign a probability density function to our model second we apply bayes s rule to relate the a priori probability density function pdf with the pdf corresponding to the experimental uncertainties likelihood function to construct the a posteriori distribution of the unknown parameters finally the model is evaluated by maximizing the a posteriori distribution when the problem is correctly regularized the algorithm converges to a solution characterized by different degrees of sparseness depending on the required resolution the solutions exhibit minimum entropy when the entropy is measured in terms of burg s definition we emphasize two crucial differences in our approach with the familiar burg method of maximum entropy spectral analysis first burg s entropy is minimized rather than maximized which is equivalent to inferring as much as possible about the model from the data second our approach uses the data as constraints in contrast with the classic maximum entropy spectral analysis approach where the autocorrelation function is the constraint this implies that we recover not only amplitude information but also phase information which serves to extrapolate the data outside the original aperture of the array the tradeoff is controlled by a single parameter that under asymptotic conditions reduces the method to a damped least squares solution finally the high resolution or aperture compensated velocity gather is used to extrapolate near and far offset traces

nonlocal problems for boussinesq equations abstract in this paper the existence and uniqueness of solution of the integral boundary value problem for abstract boussinesq equations are obtained

regularization and derivatives of multipole potentials abstract the only harmonic homogeneous functions defined in r n 0 are the harmonic polynomials and the so called multipole potentials namely functions of the type p x p x x 2 k n 2 for some harmonic polynomial p of degree k the first aim of this article is to study the distributional regularization of multipole potentials we show that even though the hadamard regularization p f p x x 2 k n 2 exists for any homogeneous polynomial of degree k the principal value p v p x x 2 k n 2 exists if and only if p is harmonic this means that if p is harmonic then for any test function the divergent integral r n p x x x 2 k n 2 d x can be computed by employing polar coordinates and performing the angular integral first we also find the first and second order distributional derivatives of these regularizations and more generally of the regularizations of functions of the form p l x p x x k l we find many interesting formulas that hold precisely when p is a harmonic polynomial of degree k in particular we prove that p v p x r 2 k n 2 1 k 1 n 2 2 k 2 n 2 k 1 p x generalizing the well known relation r 2 n 2 n c x where c is the area of a sphere of radius 1 actually formulas like this one hold for a homogeneous polynomial p of degree k if and only if p is harmonic

scattering theory for temporally inhomogeneous hamilton equations in hilbert space in the scattering theory for the equation y p t y 0 in banach space x it is known that for a perturbation which is summable in time the scattering operator is everywhere defined in the space of the generalized solutions and onto we prove here that a stronger notion of scattering operator defined in the space of the true solutions is everywhere defined and onto provided that x is reflexive and that p satisfies the following property if p denotes the average operator p t tt 1p s ds then the cauchy problem for the equation y p t y 0 is uniformly well posed on the real line this result is applied to the hamilton equation in hilbert space jx h t x generalizing those of the author arch rational mech anal 86 1984 147 180 for the abstract wave equation u a t u 0

linear second order differential equations in hilbert spaces the cauchy problem and asymptotic behaviour for large time on considere des equations d evolutions lineaires d ordre 2 dans un espace de hilbert ceci fournit un schema abstrait pour l etude des equations aux derivees partielles hyperboliques a coefficients qui peuvent etre discontinus par rapport aux variables de temps et d espace on trouve l existence d une solution unique continue pour toutes les valeurs positives du temps

distributions in spaces with thick points abstract we present a theory of distributions in a space with a thick point in dimensions n 2 generalizing the theory of thick distributions in one variable given in estrada and fulling 2007 8 the higher dimensional situation is quite different from the one dimensional case we construct topological vector spaces of thick test functions and by duality spaces of thick distributions we study several operations on these distributions both algebraic and analytic particularly partial differentiation we introduce the notion of thick delta functions at the special point not only of order 0 but of any integral order we also consider the thick distributions constructed by the hadamard finite part procedure we give formulas for the derivatives of important thick distributions including the finite part of power functions we obtain the new formula 2 p f r 1 x i x j 3 x i x j i j r 2 p f r 5 4 i j 4 n i n j for the second order thick derivatives of the finite part of r 1 in r 3 where is a thick delta of order 0

the nonexistence of regularization operators we show that there are no continuous regularization procedures for the extension of distributions we also show that there are no continuous projection operators from the spaces of distributions onto subspaces of distributions with support on a given closed set

new approach to seismic reflection event detection and velocity determination recently a number of high resolution techniques for event detection and parameter estimation velocity have emerged in the field of multichannel sonar array processing we find that a practical and computationally efficient implementation of these methods is possible for seismic reflection data this implementation results in development of the covariance measure for event detection and velocity estimation in seismic reflection data the measure is based on the eigenstructure of the sampled data covariance matrix this decomposition is carried out within hyperbolic windows that are moved through common depth point cdp data eigenvalues of the data covariance matrix allow simultaneous estimation of the noise and signal energy present within each window resulting in the development of a coherency measure computer implementation of the covariance measure provides a high resolution method for determination of velocity spectra relative to currently used techniques the method applied to synthetic data res

abstract schroedinger type differential equations with variable domain abstract the cauchy problem is studied for a class of linear abstract differential equations of schroedinger type with variable domain existence and uniqueness results are proved for suitably defined weak solutions some applications are given they concern initial boundary value problems for linear schroedinger type p d e either with mixed variable lateral conditions or in non cylindrical regions

high resolution bootstrapped differential semblance abstractwe formulated two coherency measures based on the bootstrapped differential semblance bds estimator that offered higher resolution in parameter tracking than did standard normalized differential semblance bootstrapping is a statistical resampling procedure used to infer estimates of standard errors and confidence intervals from data samples for which the statistical properties are unattainable via simple means or when the probability density function is unkown or difficult to estimate the first proposed estimator was based on a deterministic sorting of original offset traces by alternating near and far offsets to achieve maximized time shifts between adjacent traces the near offsets were indexed with odd integers while the even integers were used to index far offsets that were located at a constant index increment from the previous trace the second was the product of several bds terms with the first term being the deterministic bds defined above the other terms were generated by random

high resolution velocity spectra using eigenstructure methods stacking spectra provide maximum likelihood estimates for the stacking velocity or for the ray parameter of well separated reflections in additive white noise however the resolution of stacking spectra is limited by the aperture of the array and the frequency of the data despite these limitations parametric spectral estimation methods achieve better resolution than does stacking to improve resolution the parametric methods introduce a parsimonious model for the spectrum of the data in particular when the data are modeled as the superposition of wavefronts the properties of the eigenstructure of the data covariance matrix can be used to obtain high resolution spectra the traditional stacking spectra can also be expressed as a function of the data covariance matrix and directly compared to the eigenstructure spectra the superiority of the latter in separating closely interfering reflections is then apparent from a simple geometric interpretation eigenstructure methods were originally developed