rayleigh marchenko redatuming for target oriented true amplitude imaging abstractmarchenko redatuming is a revolutionary technique to estimate green s functions from virtual sources in the subsurface using only data measured at the earth s surface without having to place either sources or receivers in the subsurface this goal is achieved by crafting special wavefields so called focusing functions that can focus energy at a chosen point in the subsurface despite its great potential strict requirements on the reflection response such as knowledge and accurate deconvolution of the source wavelet including absolute scaling factor and co location of sources and receivers have so far challenged the application of marchenko redatuming to real world scenarios i combine the coupled marchenko equations with a one way version of the rayleigh integral representation to obtain a new redatuming scheme that handles internal as well as free surface multiples using dual sensor band limited seismic data with an unknown source signature from any acquisition system that presents arbit

reciprocity theorems for two way and one way wave vectors a comparison for acoustic applications in which there is a preferred direction of propagation the axial direction it is useful to arrange the two way and one way wave equations into the same matrix vector formalism in this formalism axial variations of the wave vector are expressed in terms of lateral variations of the same wave vector the two way wave vector contains the field quantities pressure and velocity axial component only whereas the one way wave vector contains waves propagating in the positive and negative axial direction by exploiting the equivalent form of the two way and one way matrix vector equations it appears to be possible to derive two way and one way reciprocity theorems that have an equivalent form but a different interpretation the main differences appear in the boundary integrals for unbounded media in the contrast terms and for the correlation type theorems in the handling of evanescent waves

an envelope for malcev algebras we prove that for every malcev algebra m there exist an algebra u m and a monomorphism m u m of m into the commutator algebra u m such that the image of m lies into the alternative center of u m and u m is a universal object with respect to such homomorphisms the algebra u m in general is not alternative but it has a basis of poincare birkhoff witt type over m and inherits some good properties of universal enveloping algebras of lie algebras in particular the elements of m can be characterized as the primitive elements of the algebra u m with respect to the diagonal homomorphism u m u m u m an extension of ado iwasawa theorem to malcev algebras is also proved

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

magic squares and matrix models of lie algebras this paper is concerned with the description of exceptional simple lie algebras as octonionic analogues of the classical matrix lie algebras we review the tits freudenthal construction of the magic square which includes the exceptional lie algebras as the octonionic case of a construction in terms of a jordan algebra of hermitian 3 3 matrices tits or various plane and other geometries freudenthal we present alternative constructions of the magic square which explain its symmetry and show explicitly how the use of split composition algebras leads to analogues of the matrix lie algebras su 3 sl 3 and sp 6 we adapt the magic square construction to include analogues of su 2 sl 2 and sp 4 for all real division algebras

reciprocity properties of one way propagators acoustic reciprocity is a fundamental property of the wave equation for the total acoustic wavefield rayleigh 1878 in its most elementary form the reciprocity principle states that an acoustic response remains the same when the source and receiver are interchanged in terms of green s functions this reciprocity principle reads g r2 r1 g r1 r2 with r x y z for more general expressions and their seismic applications see fokkema and van den berg 1993

migration of primaries and multiples using an imaging condition for amplitude normalized separated wavefields abstractmigration of primary and multiple reflections leads to enhanced subsurface illumination and to increased image resolution a joint migration approach using the complete wavefield requires properly imaged primaries and multiples of all orders in recent works primaries and multiples have therefore been imaged separately using the upgoing and downgoing pressure wavefields obtained by decomposing dual sensor streamer data the matches between the corresponding depth images are still not found to be sufficiently accurate so new and more appropriate imaging conditions need to be found we reviewed the classical imaging approach used in one way wave equation migration with the aim of extending it for simultaneous migration of primaries and all orders of multiples based on rayleigh s reciprocity theorem and well known theoretical developments we derived a new imaging condition described in terms of the upgoing pressure wavefield and a filtered version of the downgoing vertical velocity wavefield t

globally optimized finite difference extrapolator for strongly vti media implicit finite difference fd migration is unconditionally stable and is popular in handling strong velocity variations but its extension to strongly transversely anisotropic media with vertical symmetric axis media is difficult traditional local optimizations generate the optimized coefficients for each pair of thomsen anisotropy parameters independently which can degrade results substantially for large anisotropy variations and lead to a huge table we developed an implicit fd method using the analytic taylor series expansion and used a global optimization scheme to improve its accuracy for wide phase angles we first extended the number of the constant coefficients then we relaxed the coefficient of the time delay extrapolation term by tuning a small factor such that error is less than 0 1 finally we optimized the constant coefficients using a simulated annealing algorithm by constraining that all the error functions on a fine grid of the whole anisotropic region did not exceed 0 5 simultaneously the extended number of the constant coefficients and the relaxed coefficient greatly enhanced the flexibility of matching the dispersion relation and significantly improved the ability of handling strong anisotropy over a much wider range compared with traditional local optimization our scheme does not need any table and table lookup for each order of the fd method only one group of optimized coefficients is enough to handle strong variations in velocity and anisotropy more importantly our global optimization scheme guarantees the accuracy for various possible ranges of anisotropy parameters no matter how strong the anisotropy is for the globally optimized second order fd method the accurate phase angle is up to 58 and the increase is about 18 22 for the globally optimized fourth order fd method the accurate phase angle is up to 77 and the increase is about 22 27

fourier finite difference migration many existing migration schemes cannot simultaneously handle the two most important problems of migration imaging of steep dips and imaging in media with arbitrary velocity variations in all directions for example phase shift k migration is accurate for nearly all dips but it is limited to very simple velocity functions on the other hand finite difference schemes based on one way wave equations consider arbitrary velocity functions but they attenuate steeply dipping events we propose a new hybrid migration method named fourier finite difference migration wherein the downward continuation operator is split into two downward continuation operators one operator is a phase shift operator for a chosen constant background velocity and the other operator is an optimized finite difference operator for the varying component of the velocity function if there is no variation of velocity then only a phase shift operator will be applied automatically on the other hand if there is a strong variatio

representation of seismic sources in the one way wave equations one way extrapolation of downgoing and upgoing acoustic waves plays an essential role in the current practice of seismic migration berkhout 1985 stolt and benson 1986 claerbout 1985 gardner 1985 generally one way wave equations are derived for the source free situation sources are then represented as boundary conditions for the one way extrapolation problem this approach is valid provided the source representation is done with utmost care for instance it is not correct to represent a monopole source by a spatial delta function and to use this as input data for a standard one way extrapolation scheme this yields an erroneous directivity pattern as illustrated below

a structure theory of freudenthal kantor triple systems abstract the notion of the radical of a freudenthal kantor triple system is introduced using this it is shown that a semisimple freudenthal kantor triple system is decomposed into a direct sum of simple ideals also this paper defines and characterizes the structure groups for freudenthal kantor triple systems

compensation for transmission losses based on one way propagators in the mixed domain most true amplitude migration algorithms based on oneway wave equations involve corrections of geometric spreading and seismic q attenuation however few papers discuss the compensation of transmission losses ctl based on one way wave equations here we present a method to compensate for transmission losses using one way wave propagators for a 2d case the scheme is derived from the lippmann schwinger integral equation the ctl scheme is composed of a transmission term and a phase shift term the transmission term compensates amplitudes while the wave propagates through subsurfaces the transmission term is a function of the vertical wavenumbers of two adjacent heterogeneous screens the phase shift term is a fourier finite difference ffd propagator implemented in a mixed domain via fourier transform the transmission term can be flexibly incorporated into the conventional phaseshift migration algorithm i e ffd at every depth step we analyze the effects of frequency lateral velocity contrast and vertical velocity ratio on the accuracy of the presented formulae numerical examples from a flat model and a fault model with lateral velocity variations are presented to demonstrate the ability of the proposed scheme for compensation of transmission losses

theory of true amplitude one way wave equations and true amplitude common shot migration one way wave operators are powerful tools for forward modeling and migration here we describe a recently developed true amplitude implementation of modified one way operators and present some numerical examples by true amplitude one way forward modeling we mean that the solutions are dynamically correct as well as kinematically correct that is ray theory applied to these equations yields the upward and downward traveling eikonal equations of the full wave equation and the amplitude satisfies the transport equation of the full wave equation the solutions of these equations are used in the standard wave equation migration imaging condition the boundary data for the downgoing wave is also modified from the one used in the classic theory because the latter data is not consistent with a point source for the full wave equation when the full wave form solutions are replaced by their ray theoretic approximations the imaging formula reduces to the common shot kirchhoff inversion formula in this sense

global bifurcation of steady state solutions in this paper we study the bifurcation of steady state solutions of a reaction diffusion equation in one space variable the steady state solutions satisfy the equation a f u 0 on the interval l 0 or 0 for some nonzero function c c a for the neumann problem we use entirely different techniques to prove that s is never critical for any cubic polynomial j this implies at once that the neumann problem can have at most one nonconstant solution having a given number of maxima or minima this solution is necessarily strongly nondegenerate in the sense that zero is not contained in the spectrum of the linearized operator see 2 for the dirichlet problem the situation is far more complicated and the bifurcation diagrams undergo qualitative changes depending on the positions of the roots off for example fig 1 shows the bifurcation diagram for the positive solutions for two different cubic functions of the form f u a u u b u c it is interesting to note however that in every case we study there are at most three solutions to the dirichlet problem for each

optimized chebyshev fourier migration a wide angle dual domain method for media with strong velocity contrasts a wide angle propagator is essential when imaging complex media with strong lateral velocity contrasts in one way waveequation migration we have developed a dual domain one way propagator using truncated chebyshev polynomials and a globally optimized scheme our method increases the accuracy of the expanded square root operator by adding two high order terms to the traditional split step fourier propagator first we approximate the square root operator using taylor expansion around the reference background velocity then we apply the first kindchebyshevpolynomialstoeconomizetheresultsofthe taylor expansion finally we optimize the constant coefficients using the globally optimized scheme which are fixed and feasible for arbitrary velocity models theoretical analysis and numericalexperimentshavedemonstratedthatthemethodhasvery highaccuracyandexceedstheunoptimizedfourierfinite difference propagator for the entire range of practical velocity contrasts the accurate propagation angle of the method is always about 60 under the relative error of 1 for complex media with weak moderate and even strong lateral velocity contrasts the method allows us to handle wide angle propagations and strong lateral velocity contrast simultaneously by using fourier transform alone only four 2d fourier transforms are required for each step of 3d wavefield extrapolation and the computing cost is similar to that of the fourier finite difference method compared with the finite difference method our method has no twowaysplittingerrori e azimuthal anisotropyerrorfor3dcases and almost no numerical dispersion for coarse grids in addition it has strong potential to be accelerated when an enhanced fast fouriertransformalgorithmemerges

anisotropic complex pade hybrid finite difference depth migration standard real valued finite difference fd and fourier finite difference ffd migrations cannot handle evanescent waves correctly which can lead to numerical instabilities in the presence of strong velocity variations a possible solution to these problems is the complex pade approximation which avoids problems with evanescent waves by rotating the branch cut of the complex square root we have applied this approximation to the acoustic wave equation for vertical transversely isotropic media to derive more stable fd and hybrid fd ffd migrations for such media our analysis of the dispersion relation of the new method indicates that it should provide more stable migration results with fewer artifacts and higher accuracy at steep dips our studies lead to the conclusion that the rotation angle of the branch cut that should yield the most stable image is 60 for fd migration as confirmed by numerical impulse responses and work with synthetic data

symmetric triality relations and structurable algebras in this paper some properties of algebras satisfying local symmetric triality relations have been studied including both generalized symmetric composition algebras and the conjugate algebras of any structurable algebras we also discuss a general method of constructing lie algebras from such a system which can permit a construction of the magic square

directional illumination analysis using the local exponential frame we have developed an efficient method of directional illuminationanalysisinthelocalangledomainusinglocalexponential frame beamlets the space domain wavefields with different shot receiver geometries are decomposed into the local angle domain by using the local exponential beamlets which form a tight frame with the redundancy ratio two and are implemented by a linear combination of local cosine and localsinetransforms becauseofthefastalgorithmsofthelocal cosine sine transforms this method is much more efficient than the previously used decomposition methods in directional illumination analysis such as the local slant stackingmethodandthegabor daubechiesframemethod theresults of directional illumination di maps and the acquisitiondipresponsesadrforthe2dseg eagesaltmodel and the 45 shot 3d seg eage model demonstrated the validity and feasibility of our method compared with the illumination results using local slant stacking decomposition thenewmethodproducesilluminationmapsofsimilarquality but it does so a few times faster furthermore because of its high computational efficiency and saving in memory usage the new method makes the 3d directional illumination analysisreadilyapplicableintheindustry

fourier finite difference wave propagation we introduce a novel technique for seismic wave extrapolation in time the technique involves cascading a fourier transform operator and a finite difference operator to form a chain operator fourier finite differences ffd we derive the ffd operator from a pseudoanalytical solution of the acoustic wave equation two dimensional synthetic examples demonstrate that the ffd operator can have high accuracy and stability in complex velocity media applying the ffd method to the anisotropic case overcomes some disadvantages of other methods such as the coupling of qp waves and qsv waves the ffd method can be applied to enhance accuracy and stability of seismic imaging by reverse time migration

generalization of the bremmer coupling series an operator formalism is developed to expand the acoustic wave field in a multi dimensionally smoothly varying medium generated by a source localized in space and time into a sum of constituents each of which can be interpreted as a wave that has traveled up and down with respect to a direction of preference a definite number of times this expansion is a generalization of the bremmer coupling series the condition of smoothness of the medium relates to the width of the signature of the source in the configuration both the existence and the convergence in the weak sense of the expansion are discussed the operator calculus involved leads to a natural generalization of the concept of slowness surface to multi dimensionally smoothly varying media the operator associated with the corresponding generalized vertical slowness induces the full one way wave operator in the type of media under consideration in addition a wavefield decomposition operator as well as an interaction operator that couples the decomposed constituents are derived

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

triality of structurable and pre structurable algebras abstract we introduce a notion of pre structurable algebras based on the structurable algebra of allison as well as lie algebras that satisfy triality

wide angle one way wave equations a one way wave equation also known as a paraxial or parabolic wave equation is a differential equation that permits wave propagation in certain directions only such equations are used regularly in underwater acoustics in geophysics and as energy absorbing numerical boundary conditions the design of a one way wave equation is connected with the approximation of 1 s2 1 2 on 1 1 by a rational function which has usually been carried out by pade approximation this article presents coefficients for l2 l and other alternative classes of approximants that have better wide angle behavior for theoretical results establishing the well posedness of these wide angle equations see the work of trefethen and halpern well posedness of one way wave equations and absorbing boundary conditions math comput 47 421 435 1986

recursive integral time extrapolation methods for waves a comparative review abstractwe compared several families of algorithms for recursive integral time extrapolation rite algorithms for waves in isotropic and anisotropic media these methods allow simulating accurate wave extrapolation with little numerical dispersion even when using larger time steps than are usually possible for conventional finite difference methods these various rite algorithms all share the use of mixed space wavenumber domain operators derived from fourier integral solutions of single mode wave equations we evaluated a taxonomy for rite methods based on how they approximated the influence of medium heterogeneity one family of methods uses mixed domain series expansions to provide accurate approximations to heterogeneous extrapolators even for large time steps we compared several methods for deriving coefficients for such series approximations another family of methods uses interpolation between different homogeneous extrapolations to approximate heterogeneous time extrapolation such methods can b

target oriented wave equation inversion a target oriented strategy can be applied to estimate a wave equation least squares inverse lsi image by explicitly computing the wave equation hessian the lsi image is obtained as the solution of a nonstationary least squares inverse filtering problem the rows of the hessian are the nonstationary filters containing information about the acquisition geometry the velocity model and the band limited characteristics of the seismic data by exploiting the sparsity and the structure of the hessian matrix a large number of iterations necessary to achieve convergence can be computed cheaply the results on a structurally complex model show the improvements of the lsi image versus the migrated image

lie algebras with s4 action and structurable algebras abstract the normal symmetric triality algebras sta s and the normal lie related triple algebras lrta s have been recently introduced by the second author in connection with the principle of triality it turns out that the unital normal lrta s are precisely the structurable algebras extensively studied by allison it will be shown that the normal sta s respectively lrta s are the algebras that coordinatize those lie algebras whose automorphism group contains a copy of the alternating respectively symmetric group of degree 4

source receiver marchenko redatuming obtaining virtual receivers and virtual sources in the subsurface abstractby solving the marchenko equations one can retrieve the green s function marchenko green s function between a virtual receiver in the subsurface and points at the surface no physical receiver is required at the virtual location we extend the idea behind these equations to retrieve the green s function between any two points in the subsurface i e between a virtual source and a virtual receiver no physical source or physical receiver is required at either of these locations this green s function is called the virtual green s function and it includes all primary internal and free surface multiples similar to the marchenko green s function this virtual green s function requires the reflection response at the surface single sided illumination and an estimate of the first arrival traveltime from the virtual locations to the surface these green s functions can be used to image the interfaces from above and below

true amplitude one way wave equation migration in the mixed domain one way wave equation migration is a powerful imaging tool for locating accurately reflectors in complex geologic structures however the classical formulation of one way wave equations does not provide accurate amplitudes for the reflectors when dynamic information is required after migration such as studies for amplitude variation with angle or when the correct amplitudes of the reflectors in the zero offset images are needed some modifications to the one way wave equations are required the new equations which are called true amplitude one way wave equations provide amplitudes that are equivalent to those provided by the leading order of the ray theoretical approximation through the modification of the transverse laplacian operator with dependence of lateral velocity variations the introduction of a new term associated with the amplitudes and the modification of the source representation in a smoothly varying vertical medium the extrapolation of the wavefields with the true amplitude one way

a complete classification of bifurcation diagrams of classes of multiparameter p laplacian boundary value problems we study the bifurcation diagrams of positive solutions of the multiparameter p laplacian problem p u x f u x 0 1 x 1 u 1 u 1 0 where p 1 p y y p 2y p u is the one dimensional p laplacian f u g u h u and 0 and 0 are two bifurcation parameters 0 and 0 are two given real numbers assuming that functions g and h satisfy hypotheses h1 h3 and h4a resp h1 h3 and h4b for fixed 0 resp 0 we give a classification of totally eight qualitatively different bifurcation diagrams we prove that on the u plane resp u plane each bifurcation diagram consists of exactly one curve which is either a monotone curve or has exactly one turning point where the curve turns to the right hence the problem has at most two positive solutions for each 0 resp 0 more precisely we prove the exact multiplicity of positive solutions in addition for all p 1 we give interesting examples which show complete evolution of bifurcation diagrams as resp varies

migration of seismic data by phase shift plus interpolation under the horizontally layered velocity assumption migration is defined by a set of independent ordinary differential equations in the wavenumber frequency domain the wave components are extrapolated downward by rotating their phases this paper shows that one can generalize the concepts of the phase shift method to media having lateral velocity variations the wave extrapolation procedure consists of two steps in the first step the wave field is extrapolated by the phase shift method using l laterally uniform velocity fields the intermediate result is l reference wave fields in the second step the actual wave field is computed by interpolation from the reference wave fields the phase shift plus interpolation pspi method is unconditionally stable and lends itself conveniently to migration of three dimensional data the performance of the methods is demonstrated on synthetic examples the pspi migration results are then compared with those obtained from a finite difference method

review of elastic and electromagnetic wave propagation in horizontally layered media the objective of this paper is to provide a unified treatment of elastic and electromagnetic em wave propagation in horizontally layered media for which the parameters in the partial differential equations are piece wise continuous functions of only one spatial variable by applying a combination of fourier laplace and bessel transforms to the partial differential equations describing the elastic or em wave propagation i obtain a system of 2n linear ordinary differential equations the 2n 2n coefficient matrix is partitioned into 4n n submatrices by a proper choice of variables the diagonal submatrices are zero and the off diagonal submatrices are symmetric all the results in the paper are derived from the symmetry properties of this general equation in the appendices it is shown that three dimensional elastic waves cylindrical p sv waves acoustic waves and electromagnetic waves in isotropic layered media can all be represented by an equation with the same properties the symmetry properties of

on algebras satisfying x 2x 2 n x x the commutative algebras satisfying the adjoint identity x 2x 2 n x x where n is a cubic form are shown to be related to a class of generically algebraic jordan algebras of degree at most 4 and to the pseudo composition algebras they are classified under a nondegeneracy condition

a complete classification of bifurcation diagrams of a p laplacian dirichlet problem abstract we study the bifurcation diagrams of classical positive solutions u with u 0 of the p laplacian dirichlet problem p u x f q u x 0 1 x 1 u 1 0 u 1 where p 1 p y y p 2 y p u is the one dimensional p laplacian 0 is a bifurcation parameter and f q u 1 u q is defined on 0 with q 0 more precisely for different p q we give a complete classification of bifurcation diagrams of classical positive solutions on the u plane hence we are able to determine the exact multiplicity of classical positive solutions for each p q