techniques for determining the minimum rank of a small graph abstract the minimum rank of a simple graph g is defined to be the smallest possible rank over all symmetric real matrices whose ij th entry for i j is nonzero whenever i j is an edge in g and is zero otherwise minimum rank is a difficult parameter to compute however there are now a number of known reduction techniques and bounds that can be programmed on a computer we have developed a program using the open source mathematics software sage to implement several techniques we have also established several additional strategies for computation of minimum rank these techniques have been used to determine the minimum ranks of all graphs of order 7

the minimum rank of symmetric matrices described by a graph a survey abstract the minimum rank of a simple graph g is defined to be the smallest possible rank over all symmetric real matrices whose ijth entry for i j is nonzero whenever i j is an edge in g and is zero otherwise this paper surveys the current state of knowledge on the problem of determining the minimum rank of a graph and related issues

svd for multioffset linearized inversion resolution analysis in multicomponent acquisition acquisition of the full elastic response compressional and shear of the subsurface is an important technology in the seismic industry because of its potential to improve the quality of seismic data and to infer accurate information about rock properties fluid type and rock lithology in the framework of 3 d propagation in 1 d media we propose a computational tool to analyze the information about elastic parameters contained in the amplitudes of reflected waves with offset the approach is based on singular value decomposition svd analysis of the linearized elastic inversion problem and can be applied to any particular seismic data we applied this tool to examine the type of information in the model space that can be retrieved from sea bottom multicomponent measurements the results are compared with those obtained from conventional streamer acquisition techniques we also present multiparameter linearized inversion results obtained from synthetic data that illustrate the resolution of elastic parameters this approach allows us to investigate the reliability of the elastic parameters estimated for different offset ranges wave modes data types and noise levels involved in data space

the resolving power of seismic amplitude data an anisotropic inversion migration approach a description of the theory and numerical implementation of a 3 d linearized asymptotic anisotropic inversion method based on the generalized radon transform is given we discuss implementation aspects including 1 the use of various coordinate systems 2 regularization by both spectral and bayesian statistical techniques and 3 the effects of limited acquisition apertures on inversion we give applications of the theory in which well resolved parameter combinations are determined for particular experimental geometries and illustrate the interdependence of parameter and spatial resolutions procedures for evaluating uncertainties in the parameter estimates that result from the inversion are derived and demonstrated

flux normalized wavefield decomposition and migration of seismic data separation of wavefields into directional components can be accomplished by an eigenvalue decomposition of the accompanying system matrix in conventional pressurenormalized wavefield decomposition the resulting one way wave equations contain an interaction term which depends on the reflectivity function applying directional wavefield decomposition using flux normalized eigenvalue decomposition and disregarding interaction between up and downgoing wavefields these interaction terms were absent by also applying a correction term for transmission loss the result was an improved estimate of the up and downgoing wavefields in the wave equation angle transform a crosscorrelation function in local offset coordinates was fourier transformed to produce an estimate of reflectivity as a function of slowness or angle we normalized this wave equation angle transform with an estimate of the planewave reflection coefficient the flux normalized oneway wave propagation scheme was applied to imaging and to the normalized wave equation angle transform on synthetic and field data this proved the effectiveness of the new methods

a simultaneous inversion for background velocity and impedance maps full waveform inversion of seismic reflection data is highly nonlinear because of the irregular form of the function measuring the misfit between the observed and the synthetic data since the nonlinearity results mainly from the parameters describing seismic velocities an alternative to the full nonlinear inversion is to have an inversion method which remains nonlinear with respect to velocities but linear with respect to impedance contrasts the traditional approach is to decouple the nonlinear and linear parts by first estimating the background velocity from traveltimes using either traveltime inversion or velocity analysis and then estimating impedance contrasts from waveforms using either waveform inversion or conventional migration a more sophisticated strategy is to obtain both the subsurface background velocities and impedance contrasts simultaneously by using a single least squares norm waveform fit criterion the background velocity that adequately represents the gross features of the mediu

iterative multiparameter waveform inversion of precritical reflection data using prestack time kirchhoff approximation abstractwe have studied elastic wave scattering and iterative inversion in the context of the kirchhoff approximation the approach is more consistent with the weak contrast reflectivity functions of zoeppritz equations as compared to the born approximation to reduce the computational cost associated with inversion we demonstrated the use of amplitude variation with offset avo analysis prestack time migrations pstms and the corresponding forward modeling in an iterative scheme forward modeling and migration inversion operators are based on the double square root dsr equations of pstm and linearized reflectivity functions all operators involved in the inversion including the background model for dsr and avo are defined in p to p traveltime and are updated at each iteration our method is practical for real data applications because all operators of the inversion are known to be applicable for standard methods we have evaluated the inversion on synthetic and real data using the waveform chara

the relationship between born inversion and migration for common midpoint stacked data the relationship between born inversion and wave equation migration of common midpoint cmp stacked seismic reflection data is analytically determined the three dimensional 3 d velocity distribution obtained by born inversion is shown to be directly related to the 3 d reflectivity function obtained by wave equation migration for full bandwidth or band limited data the relationship is obtained by the reformulations of migration and born inversion methods as inverse source problems for the 3 d wave equation the reformulation leads to a definition of the reflectivity function as the source function for the wave equation it also leads to determination of the born inversion results by applying the algorithm for wave equation migration to modified surface data the modified data are simply related to the cmp stacked data alternatively born inversion results may be obtained directly from the migrated section results from synthetic and recorded data are presented and found to be consistent with the theo

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

on the imaging of reflectors in the earth in this paper i present a modification of the beylkin inversion operator this modification accounts for the band limited nature of the data and makes the role of discontinuities in the sound speed more precise the inversion presented here partially dispenses with the small parameter constraint of the born approximation this is shown by applying the proposed inversion operator to upward scattered data represented by the kirchhoff approximation using the angularly dependent geometrical optics reflection coefficient a fully nonlinear estimate of the jump in sound speed may be extracted from the output of this algorithm interpreted in the context of these kirchhoff approximate data for the forward problem the inversion of these data involves integration over the source receiver surface the reflecting surface and frequency the spatial integrals are computed by the method of stationary phase the output is asymptotically a scaled singular function of the reflecting surface the singular function of a

computation of minimal rank and path cover number for certain graphs for a given undirected graph g the minimum rank of g is defined to be the smallest possible rank over all real symmetric matrices a whose i j th entry is non zero whenever i j and i j is an edge in g the path cover number of g is the minimum number of vertexdisjoint paths occurring as induced subgraphs of g that cover all the vertices of g for trees the relationship between minimum rank and path cover number is completely understood however for non trees only sporadic results are known we derive formulae for the minimum rank and path cover number for graphs obtained from edge sums and formulae for minimum rank of vertex sums of graphs in addition we examine previously identified special types of vertices and attempt to unify the theory behind them

on the graph complement conjecture for minimum rank abstract the minimum rank of a graph has been an interesting and well studied parameter investigated by many researchers over the past decade or so one of the many unresolved questions on this topic is the so called graph complement conjecture which grew out of a workshop in 2006 this conjecture asks for an upper bound on the sum of the minimum rank of a graph and the minimum rank of its complement and may be classified as a nordhaus gaddum type problem involving the graph parameter minimum rank the conjectured bound is the order of the graph plus two other variants of the graph complement conjecture are introduced here for the minimum semidefinite rank and the colin de verdiere type parameter we show that if the graph complement conjecture is true for two graphs then it is true for the join of these graphs related results for the graph complement conjecture and the positive semidefinite version for joins of graphs are also established we also report on the use of recent results on partial k trees to establish the graph complement conjecture for graphs of low minimum rank

full waveform inversion of plane wave seismograms in stratified acoustic media applicability and limitations a nonlinear waveform inversion procedure derived from tarantola and valette s inversion formalism has been implemented in the horizontal slowness and intercept time p domain it infers simultaneously the velocity bulk modulus and density profiles of layered acoustic media an arbitrary partition of the full wavefield information i e an arbitrary selection of traces and time ranges of the whole p seismic section is used as input to the inversion procedure to explore its applicability and limitations the algorithm has been tested against a series of noise free and noisy minimum phase band limited p data sets synthesized from models of many microlayers which are meant to emulate the behavior of well logs for a given data covariance we confirmed that the specification of the starting model covariance determines the generalized damping factor which predicts the resolution of the final inverted model and the convergence rate of the inversion procedure we found three major obstacles in th

critical ideals minimum rank and zero forcing number abstract there are profound relations between the zero forcing number and minimum rank of a graph we study the relation of both parameters with a third one the algebraic co rank which is defined as the largest i such that the ith critical ideal is trivial this gives a new perspective for bounding and computing these three graph parameters

a procedure for multidimensional inversion of seismic data a new space time approach to inverting multidimensional seismic data is formulated this work constitutes a natural but assuredly nontrivial extension of a previously reported one dimensional 1 d bremmer inversion procedure in its most general format the scheme applies to the three dimensional 3 d inversion problem however appropriately reduced forms applicable to two dimensional 2 d as well as 1 d configurations are also presented the basic scheme and all its variants utilize exclusively zero offset data although interpretable in terms of a distorted wave born model the inversion procedure and subsequent algorithms differ substantially from their born based predecessors here the background is not only inhomogeneous but is not known a priori the choice of a uniquely defined phase corrected background is cardinal to the proposed reconstruction scheme the improvement in the inversion accuracy is considerable comparatively large discontinuities can be handled and the phenomenon of erro

primaries the only events that can be migrated and for which migration has meaning abstract without exception primaries are the only events that are ever migrated arranging for a more complete set of primaries will enhance imaging the approximate image of an unrecorded primary can be realized by isolating and migrating a subevent primary of a multiple a more complete set of images of primaries consisting of recorded primaries and approximated unrecorded primaries can enhance imaging multiples are not and cannot be migrated and care must be taken for creating inevitably false events in this method called cross talk

zero forcing sets and bipartite circulants abstract in this paper we introduce a class of regular bipartite graphs whose biadjacency matrices are circulant matrices a generalization of circulant graphs which happen to be bipartite and we describe some of their properties notably we compute upper and lower bounds for the zero forcing number for such a graph based only on the parameters that describe its biadjacency matrix the main results of the paper characterize the bipartite circulant graphs that achieve equality in the lower bound and compute their minimum ranks

some applications of fractional calculus operators to certain subclasses of prestarlike functions with negative coefficients abstract the object of the present paper is to prove various distortion theorems for the fractional calculus of functions in the subclasses r and c consisting of prestarlike and normalized analytic functions with negative coefficients furthermore distortion theorems involving a generalized fractional integral operator for functions in the subclasses r and c are given

fast 2 d ray born migration inversion in complex media in this paper we evaluate the capacity of a fast 2 d ray born migration inversion algorithm to recover the true amplitude of the model parameters in 2 d complex media the method is based on a quasi newtonian linearized inversion of the scattered wavefield asymptotic green s functions are computed in a smooth reference model with a dynamic ray tracing based on the wavefront construction method the model is described by velocity perturbations associated with diffractor points both the first traveltime and the strongest arrivals can be inverted the algorithm is implemented with several numerical approximations such as interpolations and aperture limitation around common midpoints to speed the algorithm both theoritical and numerical aspects of the algorithm are assessed with three synthetic and real data examples including the 2 d marmousi example comparison between logs extracted from the exact marmousi perturbation model and the computed images shows that the amplitude of the velocity perturbations

full waveform inversion of plane wave seismograms in stratified acoustic media theory and feasibility we have implemented an inversion procedure for obtaining velocity and density profiles from multioffset data in a layered acoustic medium using an iterative modeling technique and the p representation the procedure is derived from the nonlinear least squares formalism of tarantola and valette the feasibility of this method depends upon obtaining frechet derivatives during the modeling process and on vectorizing the thompson haskell reflectivity algorithm test data sets for this study are gathers of two to four plane wave synthetic seismograms which may contain both precritical and postcritical arrivals generated by a seismic wavelet these examples demonstrate convergence to the correct velocity and density profiles with reasonable accuracy

vertex and edge spread of zero forcing number maximum nullity and minimum rank of a graph abstract the minimum rank of a simple graph g is defined to be the smallest possible rank over all symmetric real matrices whose ijth entry for i j is nonzero whenever i j is an edge in g and is zero otherwise maximum nullity is taken over the same set of matrices the zero forcing number is the minimum size of a zero forcing set of vertices and bounds the maximum nullity from above the spread of a graph parameter at a vertex v or edge e of g is the difference between the value of the parameter on g and on g v or g e rank spread at a vertex was introduced in 4 this paper introduces vertex spread of the zero forcing number and edge spreads for minimum rank maximum nullity and zero forcing number properties of the spreads are established and used to determine values of the minimum rank maximum nullity and zero forcing number for various types of grids with a vertex or edge deleted

zero forcing sets and the minimum rank of graphs the minimum rank of a simple graph g is defined to be the smallest possible rank over all symmetric real matrices whose ijth entry for i 6 j is nonzero whenever i j is an edge in g and is zero otherwise this paper introduces a new graph parameter z g that is the minimum size of a zero forcing set of vertices and uses it to bound the minimum rank for numerous families of graphs often enabling computation of the minimum rank

using a new zero forcing process to guarantee the strong arnold property abstract the maximum nullity m g and the colin de verdiere type parameter g both consider the largest possible nullity over matrices in s g which is the family of real symmetric matrices whose i j entry i j is nonzero if i is adjacent to j and zero otherwise however g restricts to those matrices a in s g with the strong arnold property which means x o is the only symmetric matrix that satisfies a x o i x o and a x o this paper introduces zero forcing parameters z sap g and z vc g and proves that z sap g 0 implies every matrix a s g has the strong arnold property and that the inequality m g z vc g g holds for every graph g finally the values of g are computed for all graphs up to 7 vertices establishing g z g for these graphs

smoothing operators for waveform tomographic imaging the analytic kernel in the space time domain for the frechet derivative of acoustic waveform data with respect to changes in the slowness model is given by the born approximation solution to the integral equation of waveform scattering preconditioning operators in the solution of this forward problem which may incorporate a priori information and approximate solutions are smoothing operators in the imaging problem the first iteration of a nonlinear inversion for the slowness model some preconditioning operators are determined for solutions to the parabolic wave equation and then used to create new sensitivity functions that retain appropriate characteristics of the true frechet kernel in forward calculations the new sensitivity functions define near source near receiver and far field kernels as well as kernels that exhibit an amplitude decay off the ray yielding ray perpendicular sensitivity that scales with the fresnel zone size a sample calculation from a synthetic cross well imaging experimen

inversion versus migration a new perspective to an old discussion seismic imaging techniques can be subdivided into inversion and migration the object functions for inversion and migration are respectively the medium contrast parameters and reflectivity in this paper the relationship between inversion and migration is approached by analyzing the underlying representations the forward models it appears that the two way representation which underlies inversion as well as the one way representation which underlies migration can both be expressed in terms of a volume integral over the appropriate object function in their linearized form these representations account for primaries only in this case the one way representation in terms of reflectivity is the most accurate of the two which implies that proper migration is more accurate than linearized inversion internal multiples can be taken into account by the nonlinear representations as an alternative however the generalized primary representation is introduced in its explicit form this one way

on the minimum rank of the third power of a starlike tree abstract for a simple graph g of order n let a be a real n n symmetric matrix whose i j th entry for i j is nonzero whenever i j is an edge in g and is zero otherwise the minimum rank of g is the smallest possible rank over all such symmetric matrices the jth power of a graph g is the graph g j v f where u v f if and only if there is a walk of length j from u to v in 2007 brualdi hogben and shader reported a conjecture that if t is not the star k 1 n 1 then mr t 3 mr t 2 1 in this paper we construct a class of starlike trees such that mr t 3 mr t 2 which give a negative answer to this conjecture

a differential semblance algorithm for the inverse problem of reflection seismology abstract this paper presents an analysis of stability and convergence for a special case of differential semblance optimization dso this approach to model estimation for reflection seismology is a variant of the output least squares inversion of seismograms enjoying analytical and numerical properties superior to those of more straightforward versions we study a specialization of dso appropriate to the inversion of convolutional approximation planewave seismograms over layered constant density acoustic media we prove that the differential semblance variational principle is locally convex in suitable model classes for a range of data noise moreover the structure of the convexity estimates suggest a family of quasi newton algorithms we describe an implementation of one of these algorithms and present some numerical results

three dimensional born inversion with an arbitrary reference recent work of g beylkin helped set the stage for very general seismic inversions we have combined these broad concepts for inversion with classical high frequency asymptotics and perturbation methods to bring them closer to practically implementable algorithms applications include inversion schemes for both stacked and unstacked seismic data basic assumptions are that the data have relative true amplitude and that a reasonably accurate background velocity c x y z is available the perturbation from this background is then sought since high frequency approximations are used throughout the resulting algorithms essentially locate discontinuities in velocity an expression for a full 3 d velocity inversion can be derived for a general data surface in this degree of generality the formula does not represent a computationally feasible algorithm primarily because a key jacobian determinant is not expressed in practical terms in several important cases however this shortcoming can be overcome and

derivation of forward and adjoint operators for least squares shot profile split step migration the forward and adjoint operators for shot profile least squares migration are derived the forward operator is demigration and the adjoint operator is migration the demigration operator is derived from the born approximation the process begins with a green s function that allows for a laterally varying migration velocity model using the split step approximation next the earth is divided into horizontal layers and within each layer the migration velocity model is made to be constant with respect to depth for a given layer 1 the source side wavefield is propagated down to its top using the background wavefield this gives a background wavefield incident at the layer s upper boundary 2 the layer s contribution to the scattered wavefield is computed using the born approximation to the scattered wavefield and the background wavefield 3 next its scattered wavefield is propagated back up to the measurement surface using again the background wavefield the measured wavefield is approximated by the sum of scattered wavefields from each layer in the derivation of the measured wavefield the shot profile migration geometry is used for each shot the resulting wavefield modeling operator takes the form of a fredholm integral equation of the first kind and this is used to write down its adjoint the shot profile migration operator this forward adjoint pair is used for shot profile least squares migration shot profile least squares migration is illustrated with two synthetic examples the first uses data collected over a four layer acoustic model and the second uses data from the sigsbee 2a model

inverse problem of the wave equation and the schwinger approximation a new method of profile inversion for acoustic waves propagating in a medium with spherical inhomogeneity based on the schwinger variational method is presented the wave equation of interest is transformed into a schrodinger equation so that the born approximation and the new method could also be applied at high frequencies it is shown that the new method is stable and is more accurate than the born approximation to illustrate the method an exactly solvable analytical example is presented also numerical examples using synthetic data with and without additive noise are given and the corresponding inversion results and the stability of the method are studied

a born wkbj inversion method for acoustic reflection data density and bulk modulus variations in an acoustic earth are separately recoverable from standard reflection surveys by utilizing the amplitude versus offset information present in the observed wave fields both earth structure and a variable background velocity can be accounted for by combining the born and wkbj approximations in a before stack migration with two output sections one for density variations and the other for bulk modulus variations for the inversion the medium is considered to be composed of a known low spatial frequency variation the background plus an unknown high spatial frequency variation in bulk modulus and density the reflectivity the division between the background and the reflectivity depends upon the frequency content of the source for constant background parameters computations are done in the fourier domain where the first part of the algorithm includes a frequency shift identical to that in an f k migration the modulus and density variations are then determined by observing in a least squares sense amplitude versus offset wavenumber for a spatially variable background wkbj green s operators that model the direct wave in a medium with a smoothly varying background are used a downward continuation with these operators removes the effects of variable velocity from the problem and consequently the remainder of the inversion essentially proceeds as if the background were constant if the background is strictly depth dependent the inversion can be expressed in closed form the method neglects multiples and surface waves and it is restricted to precritical reflections density is distinguishable from bulk modulus only if a sufficient range of precritical incident angles is present in the data

forbidden minors for the class of graphs g with ξ g 2 abstract for a given simple graph g s g is defined to be the set of real symmetric matrices a whose i j th entry is nonzero whenever i j and ij is an edge in g in f barioli s fallat l hogben a variant on the graph parameters of colin de verdiere implications to the minimum rank of graphs electron j linear algebra 13 2005 387 404 g is defined to be the maximum corank i e nullity among a s g having the strong arnold property is used to study the minimum rank maximum eigenvalue multiplicity problem for g since is minor monotone the graphs g such that g k can be described by a finite set of forbidden minors we determine the forbidden minors for g 2 and present an application of this characterization to computation of minimum rank among matrices in s g

source receiver reverse time imaging of dual source vector acoustic seismic data abstractnovel technologies in seismic data acquisition allow for recording full vector acoustic va data pointwise recordings of pressure and its multicomponent gradient excited by pressure only as well as dipole gradient sources building on recent connections between imaging and seismic interferometry we present a wave equation based nonlinear reverse time imaging approach that takes full advantage of dual source multicomponent data the method s formulation relies on source receiver scattering reciprocity thus making proper use of va fields in the wavefield extrapolation and imaging condition steps in a self consistent manner the va imaging method is capable of simultaneously focusing energy from all in and outgoing waves the receiver side up and downgoing receiver ghosts fields are handled by the va receiver extrapolation whereas source side in and outgoing source ghosts arrivals are accounted for when combining dual source data at the imaging condition additionally va imaging handl

an iterative inversion of back scattered acoustic waves the application of the born approximation to the scattered wave field followed by a wkbj and far field approximation on the propagation green s function for a slowly varying background medium leads to a simple integral relation between the density and bulk modulus anomalies superimposed on the background medium and the scattered wave field an iterative inversion scheme based on successive back projections of the wave field is used to reconstruct the two acoustic parameters the scheme when applied to data generated using the direct integral relation shows that the variations of the parameters can be reconstructed the procedure is readily applicable to actual data since every iterative step is essentially a prestack kirchhoff migration followed by the application of the direct born approximation and far field operator