least squares reverse time migration using controlled order multiple reflections abstractreverse time migration rtm using multiples generates inherent crosstalk artifacts due to the interference among multiples of different orders we have developed a method to remove such crosstalk this approach first separates the recorded seismic data into primary reflections and multiples using the surface related multiples elimination algorithm and then isolates the multiples into different orders we can take any specified say the nth order of multiples data as the incident wave and the next higher order multiples data n 1 th order as the corresponding primary reflection data for imaging we have applied the least squares migration scheme to these two successive orders of multiples our method is denoted as least squares rtm using controlled order multiples lsrtm cm our numerical tests demonstrated that lsrtm cm can significantly improve imaging quality compared with straightforward seismic imaging using multiples without multiples separation

adaptive surface related multiple elimination the major amount of multiple energy in seismic data is related to the large reflectivity of the surface a method is proposed for the elimination of all surface related multiples by means of a process that removes the influence of the surface reflectivity from the data an important property of the proposed multiple elimination process is that no knowledge of the subsurface is required on the other hand the source signature and the surface reflectivity do need to be provided as a consequence the proposed process has been implemented adaptively meaning that multiple elimination is designed as an inversion process where the source and surface reflectivity properties are estimated and where the multiple free data equals the inversion residue results on simulated data and field data show that the proposed multiple elimination process should be considered as one of the key inversion steps in stepwise seismic inversion

multiple subtraction using statistically estimated inverse wavelets surface related multiple elimination srme typically consists of two steps the first step is prediction and the second step is subtraction in subtraction it is important to effectively attenuate multiple events and preserve primary events when multiples cross with or overlap on primaries least square subtraction usually cannot subtract multiples effectively and may also damage the primaries when multiples overlap with primaries least square subtraction cannot always subtract multiples accurately and often damages the primaries to remedy this problem we propose to statistically estimate the inverse source wavelet correct for errors in the estimate of the inverse wavelet and then use the corrected inverse wavelets for multiple subtraction synthetic tests and real data examples show that the proposed method can effectively attenuate multiples while they also preserve the continuity of reflection events and successfully avoid amplitude distortion the proposed method is characterized by low computational costs and ease of implementation

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

an inverse scattering series method for attenuating multiples in seismic reflection data we present a multidimensional multiple attenuation method that does not require any subsurface information for either surface or internal multiples to derive these algorithms we start with a scattering theory description of seismic data we then introduce and develop several new theoretical concepts concerning the fundamental nature of and the relationship between forward and inverse scattering these include 1 the idea that the inversion process can be viewed as a series of steps each with a specific task 2 the realization that the inverse scattering series provides an opportunity for separating out subseries with specific and useful tasks 3 the recognition that these task specific subseries can have different and more favorable data requirements convergence and stability conditions than does the original complete inverse series and most importantly 4 the development of the first method for physically interpreting the contribution that individual terms and pieces of terms in the inv

time and offset domain internal multiple prediction with nonstationary parameters abstractpractical internal multiple prediction and removal is a high priority area of research in seismic processing technology its significance increases in plays in which data are complex and sophisticated quantitative interpretation methods are apt to be applied when the medium is unknown and or complex and moveout based primary multiple discrimination is not possible inverse scattering based internal multiple attenuation is the method of choice however challenges remain for its application in certain environments for instance when generators are widely distributed and are separated in space by a range of distances optimum prediction parameters such as e which limits the proximity of events combined in the prediction are difficult to determine in some cases we find that no stationary value of e can optimally predict all multiples without introducing damaging artifacts a reformulation and implementation in the time domain permits time nonstationarity to be enforced on e after which a rang

regularity of quasi linear equations in the heisenberg group we prove the c regularity of the gradient of weak solutions of a class of quasi linear equations in nilpotent stratified lie groups of step two as applications we prove higher regularity theorems and a liouville type theorem for 1 quasi conformal mappings between domains of the heisenberg group 1997 john wiley sons inc

adaptive multiple subtraction based on constrained independent component analysis adaptive multiple subtraction is a critical and challenging procedure for the widely used surface related multiple elimination techniques one of the problems encountered in this procedure is that a good result usually is hard to obtain when primaries and multiples have overlap or when the true and predicted multiples have mismatches such as wavelet difference time shift and scalar inconsistency we propose an adaptive multiple subtraction method based on constrained independent component analysis cica it combines the advantages of two current adaptive multiple subtraction methods the independent component analysis method and the multidimensional prediction error filters pef method in cica the prediction error obtained by the pefs which measures the lateral continuity of the primaries is adopted as a constraint term of its objective function

the mobius function and distal flows we prove that the mobius function is linearly disjoint from an analytic skew product on the 2 torus these flows are distal and can be irregular in the sense that their ergodic averages need not exist for all points the previous cases for which such disjointness has been proved are all regular we also establish the linear disjointness of the mobius function from various distal homogeneous flows

relative performance of moveout based multiple suppression methods for amplitude variation with offset avo analysis and common midpoint cmp stacking we apply three moveout based methods for multiple suppression to simple synthetic common midpoint cmp gathers and compare their performance in terms of level of multiple rejection and more importantly in terms of 1 suitability of processed output for use in analysis of amplitude variations with offset avo 2 primary to multiple amplitude p m ratio of the data after cmp stacking and 3 preservation of primary wavelet shape and amplitude on the stack the three approaches are filtering in the frequency wavenumber f k domain the hampson method of filtering in the parabolic radon transform domain and a hybrid method that improves upon hampson s approach by using a variation of harlan s statistical pattern recognition approach to separate primaries from multiples in the parabolic radon transform domain the f k approach is unsuitable prior to avo analysis moreover it is little better in suppressing the multiples than the cmp stack itself hampson s method performs considerably better and is su

free surface multiple attenuation using inverse data processing in the coupled plane wave domain free surface multiples contain a large amount of energy in the seismic data because of large reflectivity of the free surface we propose a method for free surface multiple attenuation by a simple muting in the inverse coupled plane wave domain our method is based on inverse data processing and the well known 2d invariant embedding technique if the lateral variation in subsurface structure is smooth the data are well compressed in the 2d coupled plane wave domain reducing computation costs and stabilizing the inversion procedure surface multiples and primaries are well separated in the inverse coupled plane wave domain and multiples can be eliminated by simple muting which does not damage the primary energy to reduce artifacts wraparound and noise introduced by the frequency domain data matrix inversion horizontal and vertical tapers are applied a least squares matrix inversion method is chosen to stabilize the inversion synthetic data examples show that plane wave inverse data processing is stable and successful in attenuating free surface multiples

perturbation approach to depth extrapolation the born series solution of the lippmann schwinger equation does not necessarily converge properly for depth extrapolation problems even more seriously the lippmann schwinger equation is not necessarily self consistent for depth extrapolation unless it is interpreted correctly a solution is proposed by using the generalized lippmann schwinger equation that includes boundary value contributions that normally are ignored a modified version of the lippmann schwinger equation for depth extrapolation is obtained the corrections to the born series expansion for depth extrapolation are demonstrated the new system has the desired convergence and self consistency properties explicit depth extrapolation is discussed in the context of the lippmann schwinger equation a partial explanation for why explicit depth extrapolation works reasonably well in laterally inhomogeneous media may be that the contribution from the perturbation term in the lippmann schwinger equation for that problem is small for a wide var

data driven wavefield focusing and imaging with multidimensional deconvolution numerical examples for reflection data with internal multiples standard imaging techniques rely on the single scattering assumption this requires that the recorded data do not include internal multiples i e waves that have bounced multiple times between reflectors before reaching the receivers at the acquisition surface when multiple reflections are present in the data standard imaging algorithms incorrectly image them as ghost reflectors these artifacts can mislead interpreters in locating potential hydrocarbon reservoirs recently we introduced a new approach for retrieving the green s function recorded at the acquisition surface due to a virtual source located at depth we refer to this approach as data driven wavefield focusing additionally after applying source receiver reciprocity this approach allowed us to decompose the green s function at a virtual receiver at depth in its downgoing and upgoing components these wavefields were then used to create a ghost free image of the medium with either crosscorrelation or multidimensional deconvolution presenting an advantage over standard prestack migration we tested the robustness of our approach when an erroneous background velocity model is used to estimate the first arriving waves which are a required input for the datadriven wavefield focusing process we tested the new method with a numerical example based on a modification of the amoco model

weak solutions for the levi equation and envelope of holomorphy we consider the dirichlet problem for the levi equation in c2 and we prove an existence theorem of weak solutions i e in the sense of viscosity as an application we obtain the existence of the envelope of holomorphy for certain compact 2 manifolds

regularity for quasilinear second order subelliptic equations in this paper we study the regularity of solutions of the quasilinear equation where x x1 xm is a system of real smooth vector fields aij b c rm 1 assume that x satisfies the hormander condition and aij x z is positive definite we prove that if u s2 see section 2 is a solution of the above equation then u c

improving multiple removal using least squares dip filters and independent component analysis the suppression of multiple events is a crucial task in seismic data processing and the adaptive subtraction of the predicted multiples is recognized as one of the main challenges for the success of the surface related multiple elimination technique the traditional least squares matching approach can affect the primary events because the estimated multiples tend to adapt to the primaries under the minimum energy condition we investigate two filtering techniques for improving the multiple removal results in the first proposed method we combine the advantages of the least squares and pattern dip based subtraction methods doing so we exploit the separation of primaries and multiples in the dip domain and then we apply the least squares adaptive subtraction in each dip band before recomposing the data to obtain the final subtraction result as a result of the dip decomposition the primary multiple interferences are reduced allowing for a more reliable least squares filtering in the second method w

maximum principle and propagation for intrinsicly regular solutions of differential inequalities structured on vector fields abstract we prove suitable versions of the weak maximum principle and of the maximum propagation for solutions u of a differential inequality h u 0 here h i j a i j z z i z j z 0 is a differential operator structured on the vector fields z j s whereas u belongs to an appropriate intrinsic class of regularity modelled on the z j s

mobius disjointness for topological models of ergodic measure preserving systems with quasi discrete spectrum abstract by using a decomposition result for ergodic measure preserving system with quasi discrete spectrum we prove that a generic point of an ergodic quasi discrete spectrum measure in a topological dynamical system satisfies the required disjointness condition in sarnak s mobius disjointness conjecture as a direct application we have that sarnak s mobius disjointness conjecture holds for any topological model of an ergodic measure preserving system with quasi discrete spectrum

removal of internal multiples with the common focus point cfp approach part 1 explanation of the theory removal of surface and internal multiples can be formulated by removing the influence of downward scattering boundaries and downward scattering layers the involved algorithms can be applied in a model driven or a data driven way a unified description is proposed that relates both types of algorithms based on wave theory the algorithm for the removal of surface multiples shows that muted shot records play the role of multichannel prediction filters the algorithm for the removal of internal multiples shows that muted cfp gathers play the role of multichannel prediction filters the internal multiple removal algorithm is illustrated with numerical examples the conclusion is that the layer related version of the algorithm has significant practical advantages

free surface multiple attenuation for blended data abstractblended data sets are now being acquired because of improved efficiency and reduction in cost compared with conventional seismic data acquisition we have developed two methods for blended data free surface multiple attenuation the first method is based on an extension of surface related multiple elimination srme theory in which free surface multiples of the blended data can be predicted by a multidimensional convolution of the seismic data with the inverse of the blending operator a least squares inversion method is used which indicates that crosstalk noise existed in the prediction result due to the approximate inversion an adaptive subtraction procedure similar to that used in conventional srme is then applied to obtain the blended primary this can damage the energy of primaries the second method is based on inverse data processing idp theory adapted to blended data we derived a formula similar to that used in conventional idp and we attenuated free surface multiples by simple mut

polynomial sequences in groups abstract given a groupgwith lower central seriesg g1 g2 g3 we say that a sequenceg z gispolynomialif for anykthere isdsuch that the sequence obtained fromgby applying the difference operatordg n g n 1g n 1 dtimes takes its values ingk we introduce the notion ofthe degree of a polynomial sequenceand we prove that polynomial sequences of degrees not exceeding a given one form a group as an application we obtain the following extension of the hall petresco theorem theorem letg g1 g2 g3 be the lower central series of a group g let x gk y gland letp qbe polynomials z z of degrees k and l respectively then there is a sequencez0 g zi gifori n such thatfor all n n

linear equations in primes consider a system of nonconstant affine linear forms 1 t d no two of which are linearly dependent let n be a large integer and let k n n d be convex a generalisation of a famous and difficult open conjecture of hardy and littlewood predicts an asymptotic as n for the number of integer points n d k for which the integers 1 n t n are simultaneously prime this implies many other well known conjectures such as the twin prime conjecture and the weak goldbach conjecture it also allows one to count the number of solutions in a convex range to any simultaneous linear system of equations in which all unknowns are required to be prime in this paper we conditionally verify this asymptotic under the assumption that no two of the affine linear forms 1 t are affinely related this excludes the important binary cases such as the twin prime or goldbach conjectures but does allow one to count nondegenerate configurations such as arithmetic progressions our result assumes two families of conjectures which we term the inverse gowers norm conjecture gi s and the mobius and nilsequences conjecture mn s where s 1 2 is the complexity of the system and measures the extent to which the forms i depend on each other the case s 0 is somewhat degenerate and follows from the prime number theorem in aps roughly speaking the inverse gowers norm conjecture gi s asserts the gowers u s 1 norm of a function f n 1 1 is large if and only if f correlates with an s step nilsequence while the mobius and nilsequences conjecture mn s asserts that the mobius function is strongly asymptotically orthogonal to s step nilsequences of a fixed complexity these conjectures have long been known to be true for s 1 essentially by work of hardy littlewood and vinogradov and were established for s 2 in two papers of the authors thus our results in the case of complexity s 2 are unconditional in particular we can obtain the expected asymptotics for the number of 4 term progressions p 1 p 2 p 3 p 4 n of primes and more generally for any nondegenerate problem involving two linear equations in four prime unknowns

3d prediction of surface related and interbed multiples multiple attenuation during data processing does not guarantee a multiple free final section multiple identification plays an important role in seismic interpretation a target oriented method for predicting 3d multiples on stacked or migrated cubes in the time domain is presented the method does not require detailed knowledge of the subsurface geological model or access to prestack data and is valid for both surface related and interbed multiples the computational procedure is based on kinematic properties of the data and uses fermat s principle to define the multiples since no prestack data are required the method can calculate 3d multiples even when only multi 2d survey data are available the accuracy and possible use of the method are demonstrated on synthetic and real data examples

fundamental solutions for non divergence form operators on stratified groups we construct the fundamental solutions and for the non divergence form operators i j a i j x t x i x j t and i j a i j x x i x j where the x i s are hormander vector fields generating a stratified group g and a i j i j is a positive definite matrix with holder continuous entries we also provide gaussian estimates of r and its derivatives and some results for the relevant cauchy problem suitable long time estimates of allow us to construct using both t saturation and approximation arguments

smoothness of lipschitz continuous graphs with nonvanishing levi curvature in 1 2 x y t denotes the point of r 3 ut is the first derivative of u with respect to t and analogous notations are used for the other firstand second order derivatives of u the notion of levi curvature for a real manifold was introduced by e e levi in 1909 in order to characterize the holomorphy domains of c 2 since then it has played a crucial role in the geometric theory of several complex variables in looking for the polynomial hull of a graph slodkowski and tomassini implicitly introduced in 1991 the following definition of levi curvature for lipschitz continuous graphs 16

multidimensional inverse scattering series internal multiple prediction in the coupled plane wave domain the inverse scattering series internal multiple prediction and attenuation algorithm predicts multiples using certain combinations of input seismic reflection data events which are computed in the wavenumber pseudo depth or plane wave vertical travel time i e p domains significant differences can arise in the algorithms output and computational expense depending on which domain is used many of these are traceable to the response of the algorithm to the users choice of the search limiting parameter the question of which domain is optimal can be addressed with benchmark synthetics the compactness of the input to the plane wave domain algorithm leads to the expectation that it will have a reduced computational expense also the lack of increase in the dominant period i e width of input events as the horizontal slowness increases leads to the expectation that it will respond well to a constant both of these expectations are borne out with a 1 5d benchmark example a 2d plane wave pred

dynamical approximation entropies and topological entropy in operator algebras dynamical entropy invariants based on a general approximation approach are introduced for c and w algebra automorphisms this includes a noncommutative extension of topological entropy

multiple recurrence and nilsequences aiming at a simultaneous extension of khintchine s and furstenberg s recurrence theorems we address the question if for a measure preserving system x mathcal x mu t and a set a in mathcal x of positive measure the set of integers n such that mu a cap t n a cap t 2n a cap ldots cap t kn a mu a k 1 epsilon is syndetic the size of this set surprisingly enough depends on the length k 1 of the arithmetic progression under consideration in an ergodic system for k 2 and k 3 this set is syndetic while for k 4 it is not the main tool is a decomposition result for the multicorrelation sequence int f x f t n x f t 2n x ldots f t kn x d mu x where k and n are positive integers and f is a bounded measurable function we also derive combinatorial consequences of these results for example showing that for a set of integers e with upper banach density d e 0 and for all e 0 the set big n in mathbb z colon d big e cap e n cap e 2n cap e 3n big d e 4 epsilon big is syndetic

multiple prediction through inversion theoretical advancements and real data application wave equation based multiple attenuation seismic methods may be divided into the two distinct phases of multiple modeling and multiple subtraction these two are interrelated and must be optimized in order to produce an optimal final result the multiple prediction through inversion mpi scheme updates the multiple model iteratively as we usually do in a linearized inverse problem the scheme models the multiple wavefield without an explicit knowledge of surface and subsurface structures or of the source signature both are generally unknown in seismic surveys however compared to a conventional surface related multiple attenuation method the accuracy of the multiple model is improved both kinematically and dynamically it is because the mpi scheme implicitly takes account of the spatial variation of the surface reflectivity the source signature the detector patterns and receiver ghosts and other effects included in the so called surface operator when the mpi scheme is used in the first phase it also significantly reduces the nonlinearity of the problem in the second phase that involves attenuating multiples without removing or altering primaries the effectiveness of the mpi scheme is demonstrated by examples involving real marine seismic data

estimation of primaries by sparse inversion with scattering based multiple predictions for data with large gaps abstractwe have solved the estimation of primaries by sparse inversion problem for a seismic record with large near offset gaps and other contiguous holes in the acquisition grid without relying on explicit reconstruction of the missing data eliminating the unknown data as an explicit inversion variable is desirable because it sidesteps possible issues arising from overfitting the primary model to the estimated data instead we have simulated their multiple contributions by augmenting the forward prediction model for the total wavefield with a scattering series that mimics the action of the free surface reflector within the area of the unobserved trace locations each term in this scattering series involves convolution of the total predicted wavefield once more with the current estimated green s function for a medium without the free surface at these unobserved locations it is important to note that our method cannot by itself mitigate regular undersampling issues that result in significant aliases whe

dynamical entropy in banach spaces we introduce a version of voiculescu brown approximation entropy for isometric automorphisms of banach spaces and develop within this framework the connection between dynamics and the local theory of banach spaces as discovered by glasner and weiss our fundamental result concerning this contractive approximation entropy or ca entropy characterizes the occurrence of positive values both geometrically and topologically this leads to various applications for example we obtain a geometric description of the topological pinsker factor and show that a c algebra is type i if and only if every multiplier inner automorphism has zero ca entropy we also examine the behaviour of ca entropy under various product constructions and determine its value in many examples including isometric automorphisms of l p for 1 p and noncommutative tensor product shifts

sequences from zero entropy noncommutative toral automorphisms and sarnak conjecture abstract in this paper we study c algebra version of sarnak conjecture for noncommutative toral automorphisms let a be a noncommutative torus and be the noncommutative toral automorphism arising from a matrix s g l d z we show that if the voiculescu brown entropy of is zero then the sequence n u n z is a sum of a nilsequence and a zero density sequence where u a and is any state on a then by a result of green and tao 9 this sequence is linearly disjoint from the mobius function

higher order nonlinear diffraction tomography inversion of the rytov series nonlinear tomographic reconstruction algorithms are developed for inversion of data measured in scattering experiments in which the complex phase of the wavefields is modeled by an arbitrarily large possibly infinite number of terms in the rytov series the algorithms attain the form of a volterra series of nonlinear operators with the usual filtered backpropagation algorithm of diffraction tomography as the leading linear term a computer simulation study is included to illustrate the performance of the algorithms for the case of scattering objects with cylindrical symmetry

relating source receiver interferometry to an inverse scattering series to derive a new method to estimate internal multiples we have evaluated an explicit relationship between the representations of internal multiples by source receiver interferometry and an inverse scattering series this provides a new insight into the interaction of different terms in each of these internal multiple prediction equations and explains why amplitudes of estimated multiples are typically incorrect a downside of the existing representations is that their computational cost is extremely high which can be a precluding factor especially in 3d applications using our insight from source receiver interferometry we have developed an alternative computationally more efficient way to predict internal multiples the new formula is based on crosscorrelation and convolution two operations that are computationally cheap and routinely used in interferometric methods we have compared the results of the standard and the alternative formulas qualitatively in terms of the constructed wavefields and quantitatively in terms of the computational cost using examples from a synthetic data set

long time estimates for the heat kernel associated with a uniformly subelliptic symmetric second order operator abstract second order subelliptic operators have been the subject of a considerable amount of research in recent years starting with the paper rothschild and stein in which the sharp form of hormander s famous subellipticity theorem is proved and continuing through the work of fefferman and phong and sanchez calle it has become increasingly clear that precise regularity estimates for these operators depend intimately on the geometry associated with the operator under consideration the main purpose of this article is to obtain bounds from above and below on p t x y t epsilon 1 infinity in terms of standard heat kernels

potential analysis for a class of diffusion equations a gaussian bounds approach let h be a linear second order partial differential operator with non negative characteristic form in a strip s r n r we assume that h as a fundamental solution smooth out of its poles and bounded from above and from below by gaussian kernels modeled on subriemannian doubling distances in r n under these assumptions we show that h endows s with a structure of harmonic space this allows us to study boundary value problems for l with a perron wiener brelot bauer method and to obtain pointwise regularity estimates at the boundary in terms of wiener series modeled on the gaussian kernels our analysis includes the proof of a scale invariant harnack inequality for nonnegative solutions we also show an application to the real hypersurphaces of c n 1 with given levi curvature

well ties for seismic with severe stratigraphic filtering abstractstratigraphic filtering sf or short period multiples is prominent in cyclically stratified sedimentation with large impedance contrasts that result in normal incident reflection magnitudes greater than 0 5 because sf attenuates and delays the propagating wavelet similar to the effects of q attenuation the integrity of well ties is often jeopardized a method is proposed to obtain better well ties in areas with severe sf starting with a well log acoustic impedance curve two way transmitted wavefields and their equivalent inverse filters are generated at each time sample because a time varying convolution of the transmitted wavefields with the primary only reflectivity yields the multiple reflectivity a time varying deconvolution of the multiple synthetic with the inverse filters yields the primary only reflectivity in essence when the multiple synthetic matches the near angle stack at a well location the near angle stack is deconvolved in a time varying fashion to match the primary on

multimodel adaptive subtraction with regularized parameter selection via generalized information criterion abstractadaptive subtraction remains a critical part of today s seismic data processing many existing adaptive subtraction methods can be formulated as a parameter estimation problem and all of them although fundamentally different have a common restriction that the dimension of the unknown parameters must be determined in advance we have developed an adaptive subtraction framework called multimodel adaptive subtraction mmas that aims to relax this restriction as well as regularize the estimation of the parameters through a generalized information criterion combined with a nonconcave penalized likelihood function mmas is a fully general framework that can be applied to many existing adaptive subtraction methods as an example we determined how mmas can be applied to the popular least squares adaptive subtraction lsas method and we call the resulting algorithm the multimodel lsas mmasls furthermore we studied its computational complexity and developed an efficient implementation for mmasls

a nilpotent roth theorem let t and s be invertible measure preserving transformations of a probability measure space x we prove that if the group generated by t and s is nilpotent then lim n ras infty frac 1 n sum n 1 n u t n x v s n x exists in l 2 norm for any u v l x we also show that for a with a 0 one has lim n ras infty frac 1 n sum n 1 n mu a cap t n a cap s n a 0 by the way of contrast we bring examples showing that if measure preserving transformations t s generate a solvable group then i the above limits do not have to exist ii the double recurrence property fails that is for some a a 0 one may have a t n a s n a 0 for all n finally we show that when t and s generate a nilpotent group of class c lim n ras infty frac 1 n sum n 1 n u t n x v s n x int ud mu int vd mu in l 2 x for all u v l x if and only if t s is ergodic on x x and the group generated by t 1 s t 2 s 2 t c s c acts ergodically on x

harnack inequality for non divergence form operators on stratified groups we prove lower bounds for the fundamental solutions of the non divergence form operators j i j x t x i x j t and i j i j x x i x j where the x i s are hormander vector fields generating a stratified group g and a i j i j is a positive definite matrix with holder continuous entries we then prove an invariant harnack inequality for such operators as a byproduct we also study some relevant properties of the green functions on bounded domains

generalized linear inversion and the first born theory for acoustic media a procedure is derived which incorporates a generalized linear inverse viewpoint within a multidimensional born inversion method the method we present is a more general born theory which can accommodate insufficient and inaccurate data this general method reduces to the ordinary born procedure when the data requirements of the latter technique are satisfied

the mobius function is strongly orthogonal to nilsequences we show that the mobius function n is strongly asymptotically or thogonal to any polynomial nilsequence f g n n2n here g is a sim ply connected nilpotent lie group with a discrete and cocompact subgroup so g is a nilmanifold g z g is a polynomial sequence and f g r is a lipschitz function more precisely we show that j 1 n p n n 1 n f g n j f g a log a n for all a 0 in particu lar this implies the mobius and nilsequence conjecture mn s from our earlier paper for every positive integer s this is one of two major in gredients in our programme to establish a large number of cases of the generalised hardy littlewood conjecture which predicts how often a collec tion 1 t z d z of linear forms all take prime values the proof is a relatively quick application of the results in our recent companion paper we give some applications of our main theorem we show for exam ple that the mobius function is uncorrelated with any bracket polynomial such as n p 3bn p 2c we also obtain a result about the distribution of nilse quences a n x n2n as n ranges only over the primes

internal multiple prediction and removal using marchenko autofocusing and seismic interferometry standard seismic processing steps such as velocity analysis and reverse time migration imaging usually assume that all reflections are primaries multiples represent a source of coherent noise and must be suppressed to avoid imaging artifacts many suppression methods are relatively ineffective for internal multiples we show how to predict and remove internal multiples using marchenko autofocusing and seismic interferometry we first show how internal multiples can theoretically be reconstructed in convolutional interferometry by combining purely reflected up and downgoing green s functions from virtual sources in the subsurface we then generate the relevant up and downgoing wavefields at virtual sources along discrete subsurface boundaries using autofocusing then we convolve purely scattered components of up and downgoing green s functions to reconstruct only the internal multiple field which is adaptively subtracted from the measured data crucially this is all possible without detailed modeled information about the earth s subsurface the method only requires surface reflection data and estimates of direct nonreflected arrivals between subsurface virtual sources and the acquisition surface the method is demostrated on a stratified synclinal model and shown to be particularly robust against errors in the reference velocity model used

measure complexity and mobius disjointness abstract in this paper the notion of measure complexity is introduced for a topological dynamical system and it is shown that sarnak s mobius disjointness conjecture holds for any system for which every invariant borel probability measure has sub polynomial measure complexity we then apply this result to a number of situations including certain systems whose invariant measure don t all have discrete spectrum

a wiener test a la landis for evolutive hormander operators abstract in this paper we prove a wiener type characterization of boundary regularity in the spirit of a classical result by landis for a class of evolutive hormander operators we actually show the validity of our criterion for a larger class of degenerate parabolic operators with a fundamental solution satisfying suitable two sided gaussian bounds our condition is expressed in terms of a series of balayages or as it turns out to be equivalently riesz potentials

kirchhoff scattering series insight into the multiple attenuation method there are two basic integral equations used to represent wavefields in theoretical seismology the lippmann schwinger integral equation and the representation theorem the born scattering series currently used for attenuating free surface multiples has been derived from the lippmann schwinger integral equation similarly we have used the representation theorem here to derive a kirchhoff scattering series for attenuating free surface multiples in towed streamer data the kirchhoff series for attenuating free surface multiples is in theory equivalent to the born series most important like the born series it does not require any knowledge of the subsurface however it still provides useful insight into the multiple attenuation methods because the form of some quantities involved in the kirchhoff series is different from the form in the born series for example in towed streamer seismic data the kirchhoff series requires measurements of the vertical derivative of the pressure field in addition to the

integrated receiver deghosting and closed loop surface multiple elimination abstractaccurate surface related multiple removal is an important step in conventional seismic processing and more recently primaries and surface multiples are separated such that each of them is available for imaging algorithms current developments in the field of surface multiple removal aim at estimating primaries in a large scale inversion process using such a so called closed loop process in each iteration primaries and surface multiples will be updated until they fit the measured data the advantage of redefining surface multiple removal as a closed loop process is that certain preprocessing steps can be included which can lead to an improved multiple removal in principle the surface related multiple elimination process requires deghosted data as input thus the source and receiver ghost must be removed we have focused on the receiver ghost effect and assume that the source is towed close to the sea surface such that the source ghost effect is well represented by a dipole source the rece

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

nilsequences and a structure theorem for topological dynamical systems abstract we characterize inverse limits of nilsystems in topological dynamics via a structure theorem for topological dynamical systems that is an analog of the structure theorem for measure preserving systems we provide two applications of the structure the first is to nilsequences which have played an important role in recent developments in ergodic theory and additive combinatorics we give a characterization that detects if a given sequence is a nilsequence by only testing properties locally meaning on finite intervals the second application is the construction of the maximal nilfactor of any order in a distal minimal topological dynamical system we show that this factor can be defined via a certain generalization of the regionally proximal relation that is used to produce the maximal equicontinuous factor and corresponds to the case of order 1

accounting for free surface multiples in marchenko imaging imagine placing a receiver at any location in the earth and recording the response at that location to sources on the surface in such a world we could place receivers around our reservoir to better image the reservoir and understand its properties realistically this is not a feasible approach for understanding the subsurface we have developed an alternative and realizable approach to obtain the response of a buried virtual receiver for sources at the surface our method is capable of retrieving the green s function for a virtual point in the subsurface to the acquisition surface in our case a physical receiver is not required at the subsurface point instead we require the reflection measurements for sources and receivers at the surface of the earth and a macromodel of the velocity no small scale details of the model are necessary we can interpret the retrieved green s function as the response to sources at the surface for a virtual receiver in the subsurface we obtain this green s function by solving the marchenko equation an integral equation pertinent to inverse scattering problems our derivation of the marchenko equation for the green s function retrieval takes into account the free surface reflections present in the reflection response previous work considered a response without free surface multiples we decompose the marchenko equation into up and downgoing fields and solve for these fields iteratively the retrieved green s function not only includes primaries and internal multiples as do previous methods but it also includes freesurface multiples we use these up and downgoing fields to obtain a 2d image of our area of interest in this case below a synclinal structure

removal of water layer multiples from multicomponent sea bottom data a method for suppressing water layer multiples in multicomponent sea floor measurements is presented the multiple suppression technique utilizes the concept of wavefield separation into upgoing and downgoing modes just below the sea floor for eliminating the sea floor ghost the sea surface ghost and the accompanying water layer reverberations the theory applies to each of the recorded components pressure vertical velocity and horizontal velocities the fundamental physical principle for the multiple suppression technique rests on identifying these multiples as downgoing waves just below the sea floor while the primaries of interest arriving from the subsurface are upgoing waves white presented this realization for the pressure component three decades ago hence the theory for the velocity field is an extension of the theory in this paper the theory is derived for an experiment with a marine source in the water layer above a locally flat elastic sea floor with known elastic parameters the method is otherwise multidimensional and operates on a shot to shot basis hence it is computationally fast aside from this we show that this demultiple method removes the strongest multiples in sea floor data without knowledge of the source wavelet synthetic and real data examples are provided to illustrate the application of the algorithms to the pressure in line velocity and vertical velocity components the numerical tests show that strong multiples have been attenuated on the pressure and the velocity recordings producing promising results

a new proof of moser s parabolic harnack inequality using the old ideas of nash in 1958 nash published his fundamental work on the local holder continuity of solutions of second order parabolic equations with coefficients which need not be smooth 8 the primary purpose of that work was to study the properties of the fundamental solution corresponding to the parabolic operator and from these properties to derive regularity for a general solution though the work is often cited in the literature about weak solutions of elliptic and parabolic equations one feels that nash s ideas were never fully understood and maybe still are not and that because of this the more understandable and seemingly more fruitful ideas of de giorgi 4 and moser 6 7 were subsequently adopted

direct nonlinear inversion of multiparameter 1d elastic media using the inverse scattering series in the direct nonlinear inversion method and in algorithms for 1d elastic media p wave velocity s wave velocity and density are depth dependent direct nonlinear means that the method uses explicit formulas that 1 input data and directly output changes in material properties without the need for indirect procedures such as model matching searching optimization or other assumed aligned objectives or proxies and that 2 the algorithms recognize and directly invert the intrinsic nonlinear relationship between changes in material properties and the recorded reflection wavefields to achieve full elastic inversion all components of data such as pp sp and ss data are needed the method assumes that only data and reference medium propertiesare input and terms in the inverse series for moving mislocated reflectors resulting from the linear inverse term are separated from amplitude correction terms although in principle this direct inversion approach requires all components of elastic data synth

operateurs uniformement sous elliptiques sur les groupes de lie given a connected unimodular lie group g having polynomial volume growth and left invariant vector fields x1 xk which generate the lie algebra of g consider the operator la i j 1kxiaijxj where a g rk rk is a smooth symmetric matrix valued function which satisfies a 2 i j 1k aij x i j 1 2 x g rk for some 0 1 what we show is that the corresponding heat flow semigroup etla admits a kernel which satisfies two sided gaussian estimates in terms of the control distance determined by the xi s moreover the estimates can be made to depend on a only through we also prove harnack inequalities and holder regularity of solutions we end with a discussion of some other sub elliptic situations

clarifying the underlying and fundamental meaning of the approximate linear inversion of seismic data linearinversionisdefinedasthelinearapproximationofa direct inverse solution this definition leads to data requirements and specific direct inverse algorithms which differ with all current linear and nonlinear approaches and is immediately relevant for target identification and inversion in an elastic earth common practice typically starts with a directforwardormodelingexpressionandseekstosolveaforward equation in an inverse sense attempting to solve a direct forward problem in an inverse sense is not the same as solvinganinverseproblemdirectly distinctionsincludedifferences in algorithms in the need for a priori information and in data requirements the simplest and most accessible examples are the direct inversion tasks derived from the inverse scattering series iss for the removal of free surface and internal multiples the iss multiple removal algorithms require no subsurface information and they are independent ofearthmodeltype adirectforwardmethodsolvedinaninverse sense for modeling and subtracting multiples would require accurate knowledge of every detail of the subsurface the multiple has experienced in addition it requires a different modeling and subtraction algorithm for each different earth model type the iss methods for direct removal of multiples are not a forward problem solved in an inverse sense similarly the direct elastic inversion provided by the issisnotamodelingformulaforppdatasolvedinaninverse sense direct elastic inversion calls for pp ps ss data for direct linear and nonlinear estimates of changes in mechanical properties in practice a judicious combination of directandindirectmethodsarecalleduponforeffectivefield dataapplication

signal and noise separation in prestack seismic data using velocity dependent seislet transform abstractthe seislet transform is a waveletlike transform that analyzes seismic data by following varying slopes of seismic events across different scales and provides a multiscale orthogonal basis for seismic data it generalizes the discrete wavelet transform dwt in the sense that the dwt in the lateral direction is simply the seislet transform with a zero slope our earlier work used plane wave destruction pwd to estimate smoothly varying slopes however the pwd operator can be sensitive to strong noise interference which makes the seislet transform based on pwd pwd seislet transform occasionally fail in providing a sparse multiscale representation for seismic field data we adopted a new velocity dependent vd formulation of the seislet transform in which the normal moveout equation served as a bridge between local slope patterns and conventional moveout parameters in the common midpoint domain the vd slope has better resistance to strong random noise which indicated the potential of vd sei

removal of internal multiples with the common focus point cfp approach part 2 application strategies and data examples in the past the surface multiple removal method based on the feedback model has been successfully applied to many different field data sets the extension of surface to internal multiples can be made by replacing shot records with common focus point cfp gathers a cfp gather representing focused data with one source in the subsurface and all receivers at the surface or vice versa for a receiver gather the internal multiple removal algorithm can be formulated in terms of boundary related and layer related versions in the boundary related version the internal multiples are removed for one downward scattering reflector at a time in the layer related version the internal multiples are removed for a sequence of downward scattering reflectors at a time an exact velocity model is not required but proper muting is critical muting becomes straightforward in the cfp domain the strategy for applying the two versions of the multiple removal algorithm is demonstrated on physical model and field data one can conclude that the layer related version is the most appropriate in most situations because it requires less user action and does not need exact knowledge of the multiple generating boundary

antileakage least squares spectral analysis for seismic data regularization and random noise attenuation spatial transformation of irregularly sampled data series to regularly sampled data series is a challenging problem in many areas such as seismology the classical fourier analysis is limited to regularly sampled data series on the other hand the least squares spectral analysis lssa can analyze irregularly sampled data series although the lssa method takes into account the correlation among the sinusoidal basis functions of irregularly spaced series it still suffers from the problem of spectral leakage energy leaks from one spectral peak into another we propose an iterative method called antileakage lssa to attenuate the spectral leakage and consequently regularize irregular data series in this method we first search for a spectral peak with the highest energy and then we remove suppress it from the original data series in the next step we search for a new peak with the highest energy in the residual data series and remove the new and the old components simultaneously from the original data

seismic processing in the inverse data space until now seismic processing has been carried out by applying inverse filters in the forward data space because the acquired data of a seismic survey is always discrete seismic measurements in the forward data space can be arranged conveniently in a data matrix p each column in the data matrix represents one shot record if we represent seismic data in the temporal frequency domain then each matrix element consists of a complex valued number considering the dominant role of multiple scattering in seismic data it is proposed to replace data matrix p by its inverse p 1 before starting seismic processing making use of the feedback model for seismic data multiple scattered energy is mapped onto the zero time axis of the inverse data space the practical consequence of this remarkable property may be significant multiple elimination in the inverse data space simplifies to removing data at zero time only moving to the inverse data space may cause a fundamental change in the way we preprocess and image seismic data

a method for calculating synthetic seismograms which include the effects of absorption and dispersion a method is outlined for the calculation of synthetic seismograms which include the effects of absorption and dispersion the absorption model used is the usual model of exponential decay of amplitude with distance given by a a0e z where is a linear function of frequency this attenuation is accounted for mathematically by allowing the elastic modulus to be a complex function of frequency this results in a complex velocity and wavenumber and the reflection and transmission coefficients also become complex functions of frequency the method is based upon the communication theory approach and is applicable to plane waves in a flat layered model the source can be placed at an arbitrary depth the equations are outlined in detail for a particular absorption dispersion pair taken from futterman 1962 an example with a surface synthetic seismogram and synthetic traces at several depths is presented

pointwise local estimates and gaussian upper bounds for a class of uniformly subelliptic ultraparabolic operators abstract we consider a class of second order ultraparabolic differential equations in the form t u i j 1 m x i a i j x j u x 0 u where a a i j is a bounded symmetric and uniformly positive matrix with measurable coefficients under the assumption that the operator i 1 m x i 2 x 0 t is hypoelliptic and the vector fields x 1 x m and x 0 t are invariant with respect to a suitable homogeneous lie group we adapt the moser s iterative methods to the non euclidean geometry of the lie groups and we prove an l loc bound of the solution u in terms of its l loc p norm we then use a technique going back to aronson to prove a pointwise upper bound of the fundamental solution of the operator i j 1 m x i a i j x j x 0 t the bound is given in terms of the value function of an optimal control problem related to the vector fields x 1 x m and x 0 t finally by using the upper bound the existence of a fundamental solution is then established for smooth coefficients a i j

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

direct nonlinear q compensation of seismic primaries reflecting from a stratified two parameter absorptive medium q compensation of seismic primaries that have reflected from a stratified absorptive dispersive medium may be posed as a direct nonlinear inverse scattering problem if the reference medium is chosen to be nonattenuating and homogeneous an inverse scattering q compensation procedure may be derived that is highly nonlinear in the data but which operates in the absence of prior knowledge of the properties of the subsurface including its q structure it is arrived at by 1 performing an order by order inversion of a subset of the born series 2 isolating and extracting a component of the resulting nonlinear inversion equations argued to enact q compensation and 3 mapping the result back to data space once derived the procedure can be understood in terms of nonlinear interaction of the input primary reflection data the attenuation of deeper primaries is corrected by an operator built automatically using the angle and frequency variations of all shallower primaries a simple synthetic exam

adaptive overburden elimination with the multidimensional marchenko equation abstractiterative substitution of the multidimensional marchenko equation has been introduced recently to integrate internal multiple reflections in the seismic imaging process in so called marchenko imaging a macro velocity model of the subsurface is required to meet this objective the model is used to back propagate the data during the first iteration and to truncate integrals in time during all successive iterations in case of an erroneous model the image will be blurred akin to conventional imaging and artifacts may arise from inaccurate integral truncations however the scheme is still successful in removing artifacts from internal multiple reflections inspired by these observations we rewrote the marchenko equation such that it can be applied early in a processing flow without the need of a macro velocity model instead we have required an estimate of the two way traveltime surface of a selected horizon in the subsurface we have introduced an approximation such that adaptive subtracti

closed loop surface related multiple elimination and its application to simultaneous data reconstruction surface related multiple elimination srme is one of the most commonly used methods for suppressing surface multiples however to obtain an accurate surface multiple estimation dense source and receiver sampling is required the traditional approach to this problem is performing data interpolation prior to multiple estimation although appropriate in many cases this methodology fails when big data gaps are present or when relevant information is not recovered e g near offset data in shallow water environments we have developed a solution in which multiple estimation was performed simultaneously with data reconstruction such that data reconstruction helped obtain better multiple estimates and in which the physical primary multiple relationship helped constrain the data interpolation to accomplish this we proposed to extend the recently introduced closed loop srme cl srme algorithm to account for primary estimation in the case of coarsely sampled data we achieved this by introducing a focal domain parameterization of the primaries in a sparsity promoting cl srme method results proved that the method was capable of reliably estimating primaries data in case of shallow water and with large undersampling factors

fredholm integral equation for shape reconstruction of an underwater object using backscattered ramp response signatures inverse scattering problems for identifying underwater objects are of importance for all kinds of applications in seas that occupy more than two thirds of the surface of the earth the physical optics approximation has been applied to the acoustic scattering integral representation yielding an inverse scattering equation which relates the profile function of a submerged object to the fourier transform of the complex backscattered acoustic field this technique is the so called ramp response technique and usually requires the full scattering information for all frequencies to obtain accurate profile function of an object in order to retrieve the shape information of an object from the band limited backscattered frequency data a fredholm integral equation is formulated in this paper for calculating the ramp response of the submerged object this fredholm integral equation is for the unknown ramp response of an object in terms of the known incomplete scattering information the kernel of this integral equ

multidimensional signature deconvolution and free surface multiple elimination of marine multicomponent ocean bottom seismic data this paper presents a wave equation method for multidimensional signature deconvolution designature and elimination of free surface related multiples demultiple in four component 4c ocean bottom seismic data the designature demultiple method has the following characteristics it preserves primary amplitudes while attenuating free surface related multiples it requires no knowledge of the sea floor parameters and the subsurface it requires information only of the local density and acoustic wave propagation velocity just above the sea floor it accommodates source arrays and no information except location of the physical source array its volume and its radiation characteristics wavelet is required designature is an implicit part of the demultiple process hence the method is capable of transforming recorded reflection data excited by any source array below the sea surface into free surface demultipled data that would be recorded from a point source with any desired signature in addition the incident wavefield is not subtracted from the data prior to free surface demultiple hence separation of incident and scattered fields is not an issue as it is for most other free surface demultiple schemes the designature demultiple algorithm can be divided into two major computational steps first a multidimensional deconvolution operator inversely proportional to the time derivative of the downgoing part of the normal component of the particle velocity just above the sea floor is computed second an integral equation is solved to find any component of the designatured free surface demultipled multicomponent field when the geology is horizontally layered the designature and free surface demultiple scheme greatly simplifies and lends itself toward implementation in the p domain or frequency wavenumber domain as deterministic deconvolution of common shot gathers or common receiver gathers when source array variations are negligible