computation theory of cellular automata self organizing behaviour in cellular automata is discussed as a computational process formal language theory is used to extend dynamical systems theory descriptions of cellular automata the sets of configurations generated after a finite number of time steps of cellular automaton evolution are shown to form regular languages many examples are given the sizes of the minimal grammars for these languages provide measures of the complexities of the sets this complexity is usually found to be non decreasing with time the limit sets generated by some classes of cellular automata correspond to regular languages for other classes of cellular automata they appear to correspond to more complicated languages many properties of these sets are then formally non computable it is suggested that such undecidability is common in these and other dynamical systems

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

shaping regularization in geophysical estimation problems regularization is a required component of geophysical estimation problems that operate with insufficient data the goal of regularization is to impose additional constraints on the estimated model i introduce shaping regularization a general method for imposing constraints by explicit mapping of the estimated model to the space of admissible models shaping regularization is integrated in a conjugate gradient algorithm for iterative least squares estimation it provides the advantage of better control on the estimated model in comparison with traditional regularization methods and in some cases leads to a faster iterative convergence simple data interpolation and seismic velocity estimation examples illustrate the concept

chaos and diffusion in deterministic cellular automata abstract it is shown that some of the deterministic one dimensional cellular automata studied recently by wolfram exhibit a kind of spontaneous symmetry breaking the associated kinks bloch walls perform annihilating diffusive walks for infinite lattices and large times in addition close analogies between these automata and strange attractors are exhibited by mapping the automata onto dynamical systems defined by iterations of the unit square

seismic modeling in the domain of intercept time and ray parameter in recent years seismic modeling has been based primarily on two methods generalized ray theory and the reflectivity method variations within these methods are characterized by the domains of computation and the approximations applied one common feature of both approaches is the straightforward translation of a mathematical solution of the wave equation into a computational algorithm which develops the seismic response for a given earth structure in the observation plane of seismic experiments source receiver offset and travel time x t we present an alternative approach in which we develop acoustic and seismic models in the domain of vertical delay or intercept time and horizontal ray parameter p this is accomplished by a modified version of the reflectivity method followed by a fourier transform over frequency in these models the p seismic response for compressional and shear waves is readily interpreted for all primary and multiple arrivals the p response can also be used to compute broad band synthetic x t seismograms by an appropriate integration through the p plane a fast method of computing seismograms is suggested based on a weighted integration designed to automatically evaluate the points of stationary phase

migration velocity analysis by wave field extrapolation velocity information is essential to both common midpoint cmp stacking and migration cmp stacking provides the basis for conventional velocity estimation techniques in that for a number of trial velocities the stack response of a cmp gather is computed and displayed in the form of a velocity table an alternative approach to velocity estimation makes use of the basic ingredients of migration downward extrapolation and imaging of seismic wave fields the procedure involves migration of a cmp gather with a number of trial velocities and collection of the zero offset information again in the form of a velocity table operating on a cmp gather the migration based approach produces results similar to those of the conventional method analyses of synthetic cmp gathers using both methods show essentially equivalent treatments of seismic signal and similar dependence of accuracy and resolving power on recording geometry we have extended the migration based approach to include more than one cmp gather in

aliasing in the tau p transform and the removal of spatially aliased coherent noise the tau p transform is a discretized radon transform the choice of discretization parameters is a very important part of performing the transform insufficient sampling in the tau direction leads to aliasing problems equivalent to those encountered in any one dimensional time series a simple graphical method illustrates that too coarse sampling in the p direction results in reconstructions containing data duplicated incorrectly at different spatial positions the spacing of these duplications is dependent on the temporal frequency of the data insufficient spatial sampling of the original seismic data causes events to plot at multiple p values in tau p domain again dependent on temporal frequency therefore to velocity filter spatially aliased noise efficiently multiple p values must be filtered the use of appropriate filters in the p f domain can substantially improve the noise rejection capabilities of velocity filters on spatially aliased noise while having little effect on desired reflection sig

selecting an optimal aperture in kirchhoff migration using dip angle images we present a method for selecting a migration aperture in kirchhoff migration we first split migrated data into constant dip angle partial images then in every partial image we estimate the consistency between each event and the constant dip of the analyzed section we filter out events whose slope is far from the corresponding dip stacking of the filtered partial images corresponds to migration having an optimal aperture synthetic and real data examples demonstrate that the proposed approach to migration aperture optimization is able to reduce migration noise while preserving diffraction energy which characterizes small geologic objects and brings additional resolution to the image

the radon transform and its properties this paper presents a summary of the fundamental properties of the radon transform including delay effects data shifting rotation scaling windowing bowtie events energy conservation etc and includes representative examples on standard data sets such as 2 d delta functions boxcar events gaussian bell and conic sections which reinforce the basic concepts of the radon transform

thermodynamics of systems of finite sequences it is proposed that the hamming distance between bit strings plays the role of interaction energy the statistical sum and thermodynamic functions are calculated for the canonical ensemble of bit sequences the method is illustrated with an ensemble of two sequence systems which evolve according to cellular automata rules

a note on weak type l1 boundedness of czos with rough kernels abstract for f s r 2 and l 1 s 1 s 1 x d x 0 define t f x lim 0 x y y y y 2 f x y d y in this paper we shall prove that there is a class of functions in h 1 s 1 l ln l s 1 such that t is weak type l 1 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

velocity independent time domain seismic imaging using local event slopes by estimating local event slopes in prestack seismic reflection data it is possible to accomplish all common time domain imaging tasks from normal moveout to prestack time migration without the need to estimate seismic velocities or any other attributes local slopes contain complete information about the reflection geometry once they are estimated seismic velocities and all other moveout parameters turn into data attributes and are directly mappable from the prestack data domain into the time migrated image domain i develop an analytical theory which defines the transformation from data and local slopes to the image space for different time domain imaging operators computational experiments with synthetic and field data examples confirm theoretical expectations and demonstrate practical effectiveness of the proposed method

multicomponent wave separation using hosvd unimodal ica subspace method multicomponent sensor arrays now are commonly used in seismic acquisition to record polarized waves in this article we use a three mode model polarization mode distance mode and temporal mode to take into account the specific structure of signals that are recorded with these arrays providing a data structure preserving processing with the suggested model we propose a multilinear decomposition named higher order singular value decomposition and unimodal independent component analysis hosvd unimodal ica to split the recorded threemode data into two orthogonal subspaces the signal and noise subspaces this decomposition allows the separation and identification of polarized waves with infinite apparent horizontal propagation velocity the hosvd leads to a definition of a subspace method that is the counterpart of the well known subspace method for matrices that is driven by singular value decompositionsvd aclassictoolinmonocomponentarrayprocessing the proposed three mode subspace decomposition provides a multicomponentwave separationalgorithm toenhancetheseparation result when the signal to noise ratio is low or when orthogonality constraints are not well adapted to the recorded waves a unimodal ica step is included on the temporal mode doingthisreplacestheclassicorthogonalityconstraintsbetween estimated waves with independence constraints that might allow better recovery of recorded seismic waves asimulation on realistic two component 2c geophysical data shows qualitative and quantitative improvements for the wavefield separation results the relative mean square errors between the original and estimated signal subspaces are respectively 52 for svd applied on each component separately 27 4 for hosvd based technique applied to the whole three mode dataset and 7 3 for hosvd unimodal ica technique the efficiency of the threemodesubspacedecompositionsalsoisshownonrealthree component 3c geophysical data these results emphasize the potentialofthehosvd unimodal icasubspacemethodformulticomponentseismic waveseparation

singular integral operators with rough convolution kernels in all dimensions again under the assumption q e l log l it is conceivable that a variant of the arguments in 3 for the maximal operator could also work for the singular integral operator in fact in unpublished work the authors of 3 obtained a weak type 1 1 inequality in dimension d 7 however their arguments if applied to the singular integral operator lead to substantial technical difficulties and no

a pulsed beam summation formulation for short pulse radiation based on windowed radon transform wrt frames a new time domain td representation for radiation from apertures wherein the field is expanded in a discrete lattice of shifted tilted and retarded pulsed beam pb propagators is proposed the pb representation provides a uniformly accurate systematic framework for ray based tracking of ultrawide band transient fields in complex configurations e g slowly or abruptly varying inhomogeneous media the excitation coefficients of the pbs are extracted from the aperture source distribution via a new discretized local slant stack transform the formulation is structured upon a new class of frames termed windowed radon transform frames the sets of expansion and analysis functions are shown to be dual frames thus facilitating representation of arbitrary bandlimited source fields explicit expressions for the processing windows the analysis frame set and for the pb propagators the expansion set are provided for the class of isodiffracting pbs accuracy of the pb expansion technique is demonstrated for a focusing aperture case that tends to fail the conventional ray shooting schemes

spatial antialias filtering in the slowness frequency domain a rigorous explicit spatial antialias filter is designed and applied to spatially coarsely sampled seismic data by removing all energy above the first nyquist wavenumber and aliased energy that is folded back across the nyquist in the horizontal slowness frequency domain the spatial filtering in the slowness frequency domain is explicit free from any event linearity assumption and does not require any interpolation the spatially aliased energy is dispersive and present at small and large slownesses comparison of the output data after antialias spatial filtering with output data after conventional antialias frequency filtering shows that the filter removes the spatially aliased frequencies selectively at each slowness antialias low pass frequency filtering under or overcorrects for spatial aliasing at all slownesses a seismic gather can be spatially dealiased only at the expense of wavelet spectral changes dealiasing and preservation of amplitude variations with offset are not simultaneously possible

direct mapping of seismic data to the domain of intercept time and ray parameter a plane wave decomposition marine seismic data recorded as a function of source receiver offset and traveltime are mapped directly to the domain of intercept or vertical delay time and horizontal ray parameter this is a plane wave decomposition based on beam forming of wide aperture seismic array data to determine automatically the loci of coherent seismic reflection and refraction events in this computation semblance in addition to the required slowness or horizontal ray parameter stack is found for linear x t trajectories across subarrays subsequently semblance is used to derive a windowing filter that is applied to the slowness stack to determine the points of stationary phase and eliminate aliasing the resulting filtered slowness stacks for multiple subarrays can then be linearly transformed and combined according to ray parameter range and time the resulting function of intercept time and horizontal ray parameter offers significant computational and interpretational advantages for the case of horizontal homogeneou

on p dependent boundedness of singular integral operators we study the classical calderon zygmund singular integral operator with homogeneous kernel suppose that is an integrable function with mean value 0 on s1 we study the singular integral operator t omega f rm p v f frac omega x x x 2 we show that for 0 the condition bigg int limits i omega theta d theta bigg leq c log i 1 alpha quad quad quad quad 0 1 for all intervals i frac 3 alpha 1 6 alpha 1

the weak type l1 convergence of eigenfunction expansions for pseudodifferential operators soit m une variete compacte connexe lisse sans bord de dimension n 2 soit p d un operateur pseudodifferentiel elliptique d ordre 1 sur m qui est auto adjoint par rapport a une densite positive lisse dx on demontre la convergence faible de type l 1 pour les developpements en fonctions propres

3 d traveltime computation using huygens wavefront tracing traveltime computation is widely used in seismic modeling imaging and velocity analysis the two most commonly used methods are ray tracing and numerical solutions to the eikonal equation eikonal solvers are fast and robust but are limited to computing only the first arrival traveltimes ray tracing can compute multiple arrivals but lacks the robustness of eikonal solvers we propose a robust and complete method of traveltime computation it is based on a system of partial differential equations which is equivalent to the eikonal equation but formulated in the ray coordinates system we use a first order discretization scheme that is interpreted very simply in terms of the huygens s principle our explicit finite difference solution to the eikonal equation solved in the ray coordinates system delivers both computational speed and stability since we use more than one point on the current wavefront at every time step the finite difference method has proven to be a robust alternative to conventional ray tracing while being faster and having a better ability to handle rough velocity media and penetrate shadow zones

diffraction imaging and time migration velocity analysis using oriented velocity continuation abstractwe perform seismic diffraction imaging and time migration velocity analysis by separating diffractions from specular reflections and decomposing them into slope components we image the slope components using migration velocity extrapolation in time space slope coordinates the extrapolation is described by a convection type partial differential equation and implemented in a highly parallel manner in the fourier domain synthetic and field data experiments show that the proposed algorithms are able to detect accurate time migration velocities by measuring the flatness of diffraction events in slope gathers for single and multiple offset data

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

prestack migration by split step dsr the double square root dsr prestack migration equation though defined for depth variable velocity can be used to image media with strong velocity variations using a phase shift plus interpolation pspi or split step correction the split step method is based on applying a phase shift correction to the extrapolated wavefield a correction that attempts to compensate for the lateral velocity variations i show how to extend dsr prestack migration to lateral velocity media and exemplify the method by applying the new algorithm to the marmousi data set the split step dsr migration is very fast and offers excellent imaging results

adaptive multiple subtraction using regularized nonstationary regression stationary regression is the backbone of seismic data processing algorithms including match filtering which is commonly applied for adaptive multiple subtraction however the assumption of stationarity is not always adequate for describing seismic signals i have developed a general method of nonstationary regression and that applies to nonstationary match filtering the key idea is the use of shaping regularization to constrain the variability of nonstationary regression coefficients simple computational experiments demonstrate advantages of shaping regularization over classic tikhonov s regularization including a more intuitive selection of parameters and a faster iterative convergence using benchmark synthetic data examples i have successfully applied this method to the problem of adaptive subtraction of multiple reflections

rough bilinear singular integrals abstract we study the rough bilinear singular integral introduced by coifman and meyer 8 t f g x p v r n r n y z 2 n y z y z f x y g x z d y d z when is a function in l q s 2 n 1 with vanishing integral and 2 q when q we obtain boundedness for t from l p 1 r n l p 2 r n to l p r n when 1 p 1 p 2 and 1 p 1 p 1 1 p 2 for q 2 we obtain that t is bounded from l 2 r n l 2 r n to l 1 r n for q between 2 and infinity we obtain the analogous boundedness on a set of indices around the point 1 2 1 2 1 to obtain our results we introduce a new bilinear technique based on tensor type wavelet decompositions

velocity continuation and the anatomy of residual prestack time migration velocity continuation is an imaginary continuous process of seismic image transformation in the postmigration domain it generalizes the concepts of residual and cascaded migrations understanding the laws of velocity continuation is crucially important for a successful application of time migration velocity analysis these laws predict the changes in the geometry and intensity of reflection events on migrated images with the change of the migration velocity in this paper i derive kinematic and dynamic laws for the case of prestack residual migration from simple geometric principles the main theoretical result is a decomposition of prestack velocity continuation into three different components corresponding to residual normal moveout residual dip moveout and residual zero offset migration i analyze the contribution and properties of each of the three components separately this theory forms the basis for constructing efficient finite difference and spectral algorithms for time migration velocity ana

migrating around in circles given the identified and picked primary reflection arrivals on a cmp stack section or a zero offset section it is well known assuming a constant macro velocity model that the depth migrated subsurface image of each point on the picked reflection time curve is a half circle in the depth domain as a consequence the poststack or zero offset depth migrated subsurface image of the reflection time curve upper half of figure 1 is the envelope of all half circles lower half of figure 1 these half circles also known as isochrons or aplanats are explained in all texts about seismic migration and are quite familiar to geophysicists

imaging and velocity estimation with depth focusing analysis prestack depth migration uses two imaging conditions zero time and zero offset during downward continuation to form a migrated depth section when the migration velocities are exact the two imaging conditions act in a complementary fashion to yield a focused image when the migration velocities are in error reflected energy collapses to zero offset at depths that are inconsistent with the zero time imaging condition the result is a deteriorated seismic image however by interpreting the nonzero times at which focusing actually occurs the migration velocities can be updated iteratively in a process called depth focusing analysis to produce a well focused seismic image the goal of depth focusing analysis must be the elimination of focusing errors however practical considerations can prevent this goal from being achieved therefore to relax the sensitivity of the migrated image to focusing errors we introduce a nonzero time imaging condition by extracting the data along the interpreted surface o

weak type 1 1 bounds for rough operators ii in this paper we consider four operators arising in harmonic analysis in r n 1 they share one feature each is known to be bounded on lp for all 1 p o but for each it has been an open question whether a weak type 1 1 bound also holds in two cases we show that the weak 1 1 bound does hold and for the others we establish weaker estimates which are still sharper than the lp boundedness although the four operators are quite distinct they all involve convolution kernels which are related to the singular integrals of calderon and zygmund but which lack the smoothness required for the usual analysis in each case this defect is mitigated by the presence of some form of curvature consider the bochner riesz means

estimates for singular integrals on homogeneous groups we consider singular integral operators and maximal singular integral operators with rough kernels on homogeneous groups we prove certain estimates for the operators that imply lp boundedness of them by an extrapolation argument under a sharp condition for the kernels also we prove some weighted lp inequalities for the operators

time migration velocity analysis by velocity continuation time migration velocity analysis can be performed by velocity continuation an incremental process that transforms migrated seismic sections according to changes in the migration velocity velocity continuation enhances residual normal moveout correction by properly taking into account both vertical and lateral movements of events on seismic images finite difference and spectral algorithms provide efficient practical implementations for velocity continuation synthetic and field data examples demonstrate the performance of the method and confirm theoretical expectations

sharp lorentz space estimates for rough operators we demonstrate the h 1 l 1 2 or l p l p 2 mapping properties of several rough operators in all cases these estimates are sharp in the sense that the lorentz exponent 2 cannot be replaced by any lower number