high order finite difference simulations of marine csem surveys using a correspondence principle for wave and diffusion fields the computer time required to solve a typical 3d marine controlled source electromagnetic surveying csem simulation can be reduced by more than one order of magnitude by transforming low frequency maxwell equations in the quasi static or diffusive limit to a hyperbolic set of partial differential equations that give a representation of electromagnetic fields in a fictitious wave domain the dispersion and stability analysis can be made equivalent to that of other types of wave simulation problems such as seismic acoustic and elastic modeling second order to eighth order spatial derivative operators are implemented for flexibility fourth order and sixth order methods are the most numerically efficient implementations for this particular scheme an implementation with high order operators requires that both electric and magnetic fields are extrapolated simultaneously into the air layer the stability condition given for high order staggered derivative operators here should be equally valid for seismic wave simulation the bandwidth of recovered fields in the diffusive domain is independent of the bandwidth of the fields in the fictitious wave domain the fields in the fictitious wave domain do not represent observable fields propagation paths and interaction reflection amplitudes are not altered by the transform from the fictitious wave domain to the diffusive frequency domain however the transform contains an exponential decay factor that damps down late arrivals in the fictitious wave domain the propagation paths that contribute most to the diffusive domain fields are airwave shallow water plus typically postcritical events such as refracted and guided waves the transform from the diffusive frequency domain to the fictitious wave domain is an ill posed problem the transform is nonunique this gives a large degree of freedom in postulating temporal waveforms for boundary conditions in the fictitious wave domain that reproduce correct diffusive frequency domain fields

lipschitz metric for the periodic camassa holm equation we study stability of conservative solutions of the cauchy problem for the periodic camassa holm equation ut uxxt 3uux 2uxuxx uuxxx 0 with initial data u0 in particular we derive a new lipschitz metric dd with the property that for two solutions u and v of the equation we have dd u t v t e ct dd u0 v0 the relationship between this metric and usual norms in h 1 per and l 1 per is claried

type a images of galois representations and maximality given a compatible subsystem rho ell ell of n dimensional ell adic galois representations arising from etale cohomology of any complete non singular variety over a number field k we define gamma ell rho ell text gal bar k k and let mathbf g ell denote the zariski closure of gamma ell in mathrm gl n if mathbf g ell is of type a in the sense that all simple composition factors are of type a in the cartan killing classification then gamma ell is in a suitable sense maximal in mathbf g ell for all ell gg 0 as a corollary if rho ell is semisimple and ell is sufficiently large then mathbf g ell is unramified

residual shot profile migration we have developed a residual shot profile migration technique which consists of dip corrected residual normal moveout nmo and depth restretching the dip corrected residual nmo equation and the depth restretching equation were derived by generalizing al yahya s residual nmo equation using the dip correction residual nmo equation velocity errors can be estimated more accurately than without the dip correction term in the residual nmo equation residual shot profile migration was applied to migrated prestack data in a manner similar to conventional processing by cascading residual velocity analysis residual nmo and stack and depth stretching with residual migration we can either avoid remigration of the original prestack data or reduce the number of iterations required to produce a satisfactory image residual migration is efficient enough to be implemented on a workstation significant improvement in imaging to cases a pinchout and salt top and bottom is demonstrated using synthetic and field da

analysis and algorithms for a regularized cauchy problem arising from a non linear elliptic pde for seismic velocity estimation in the present work we derive and study a non linear elliptic pde coming from the problem of estimation of sound speed inside the earth the physical setting of the pde allows us to pose only a cauchy problem and hence is ill posed however we are still able to solve it numerically on a long enough time interval to be of practical use we used two approaches the first approach is a finite difference time marching numerical scheme inspired by the lax friedrichs method the key features of this scheme is the lax friedrichs averaging and the wide stencil in space the second approach is a spectral chebyshev method with truncated series we show that our schemes work because of i the special input corresponding to a positive finite seismic velocity ii special initial conditions corresponding to the image rays iii the fact that our finite difference scheme contains small error terms which damp the high harmonics truncation of the chebyshev series and iv the need to compute the solution only for a short interval of time we test our numerical schemes on a collection of analytic examples and demonstrate a dramatic improvement in accuracy in the estimation of the sound speed inside the earth in comparison with the conventional dix inversion our test on the marmousi example confirms the effectiveness of the proposed approach

incorporating geologic information into reflection tomography in areas of complex geology prestack depth migration is often necessary if we are to produce an accurate image of the subsurface prestack depth migration requires an accurate interval velocity model with few exceptions the subsurface velocities are not known beforehand and should be estimated when the velocity structure is complex with significant lateral variations reflection tomography methods are often an effective tool for improving the velocity estimate unfortunately reflection tomography often converges slowly to a model that is geologically unreasonable or it does not converge at all the large null space of reflection tomography problems often forces us to add a sparse parameterization of the model and or regularization criteria to the estimation standard tomography schemes tend to create isotropic features in velocity models that are inconsistent with geology these isotropic features result in large part from using symmetric regularization operators or from choosing a poor model parameterization if we replace the symmetric operators with nonstationary operators that tend to spread information along structural dips the tomography will produce velocity models that are geologically more reasonable in addition by forming the operators in helical 1d space and performing polynomial division we apply the inverse of these space varying anisotropic operators the inverse operators can be used as a preconditioner to a standard tomography problem thereby significantly improving the speed of convergence compared with the typical regularized inversion problem results from 2d synthetic and 2d field data are shown in each case the velocity obtained improves the focusing of the migrated image

maximality of galois actions for compatible systems in the sense of serre 19 of course the image of is not generally zariski dense in gln q so the maximality condition must be formulated relative to the zariski closure g of gal k k one might hope that the image of is a maximal compact subgroup of g q but this is too optimistic the center causes problems and we do not even know a priori that g is reductive to formulate a maximality conjecture which avoids such problems we introduce maps g g g

lipschitz metric for the novikov equation we consider the lipschitz continuous dependence of solutions for the novikov equation with respect to the initial data in particular we construct a finsler type optimal transport metric which renders the solution map lipschitz continuous on bounded sets of h 1 mathbb r cap w 1 4 mathbb r although it is not lipschitz continuous under the natural sobolev metric from an energy law due to the finite time gradient blowup by an application of thom s transversality theorem we also prove that when the initial data is in an open dense subset of h 1 mathbb r cap w 1 4 mathbb r the solution is piecewise smooth this generic regularity result helps us extend the lipschitz continuous metric to the general weak solutions our method of constructing the metric can be used to treat other kinds of quasi linear equations provided a good knowledge about the energy concentration

integrable peakon equations with cubic nonlinearity we present a new integrable partial differential equation found by vladimir novikov like the camassa holm and degasperis procesi equations this new equation admits peaked soliton peakon solutions but it has nonlinear terms that are cubic rather than quadratic we give a matrix lax pair for v novikov s equation and show how it is related by a reciprocal transformation to a negative flow in the sawada kotera hierarchy infinitely many conserved quantities are found as well as a bi hamiltonian structure the latter is used to obtain the hamiltonian form of the finite dimensional system for the interaction of n peakons and the two body dynamics n 2 is explicitly integrated finally all of this is compared with some analogous results for another cubic peakon equation derived by zhijun qiao

efficient calculation of a partial derivative wavefield using reciprocity for seismic imaging and inversion linearized inversion of surface seismic data for a model of the earth s subsurface requires estimating the sensitivity of the seismic response to perturbations in the earth s subsurface this sensitivity or jacobian matrix is usually quite expensive to estimate for all but the simplest model parameterizations we exploit the numerical structure of the finite element method modern sparse matrix technology and source receiver reciprocity to develop an algorithm that explicitly calculates the jacobian matrix at only the cost of a forward model solution furthermore we show that we can achieve improved subsurface images using only one inversion iteration through proper scaling of the image by a diagonal approximation of the hessian matrix as predicted by the classical gauss newton method our method is applicable to the full suite of wave scattering problems amenable to finite element forward modeling we demonstrate our method through some simple 2 d synthetic examples

inversion of seismic reflection data in the acoustic approximation the nonlinear inverse problem for seismic reflection data is solved in the acoustic approximation the method is based on the generalized least squares criterion and it can handle errors in the data set and a priori information on the model multiply reflected energy is naturally taken into account as well as refracted energy or surface waves the inverse problem can be solved using an iterative algorithm which gives at each iteration updated values of bulk modulus density and time source function each step of the iterative algorithm essentially consists of a forward propagation of the actual sources in the current model and a forward propagation backward in time of the data residuals the correlation at each point of the space of the two fields thus obtained yields the corrections of the bulk modulus and density models this shows in particular that the general solution of the inverse problem can be attained by methods strongly related to the methods of migration of unstacked data and commerc

automatic wave equation migration velocity inversion using multiobjective evolutionary algorithms to solve nonlinear seismic velocity inversion problems we have developed an automatic velocity estimation technique based on the stochastic method called multiobjective evolutionary algorithms moea semblance and differential semblance are used as objective functions to cope with the high computational cost we customized moea added domain knowledge velocity increases with depth slowly varies along layers and so forth which improves the conditioning of the problem and accelerates convergence this approach is robust because it can cope with large velocity errors computational cost of this algorithm is at least two orders of magnitude faster than other stochastic methods and comparable to that of direct gradient methods

elastic least squares reverse time migration abstractwe use elastic least squares reverse time migration lsrtm to invert for the reflectivity images of p and s wave impedances elastic lsrtm solves the linearized elastic wave equations for forward modeling and the adjoint equations for backpropagating the residual wavefield at each iteration numerical tests on synthetic data and field data reveal the advantages of elastic lsrtm over elastic reverse time migration rtm and acoustic lsrtm for our examples the elastic lsrtm images have better resolution and amplitude balancing fewer artifacts and less crosstalk compared with the elastic rtm images the images are also better focused and have better reflector continuity for steeply dipping events compared to the acoustic lsrtm images similar to conventional least squares migration elastic lsrtm also requires an accurate estimation of the p and s wave migration velocity models however the problem remains that when there are moderate errors in the velocity model and strong multiples lsrtm

on the shafarevich and tate conjectures for hyperkahler varieties 1 problems and results 2 polarized hyperk ihler varieties and cubic fourfoids 3 the period mapping 4 the kuga satake construction 5 the kuga satake construction in the relative setting 6 proof of theorem 1 5 1 7 proof of theorem 1 6 1 8 descent 9 proof of theorem 1 3 1 appendix 1 spinorial shimura varieties appendix 2 a short proof of the hodge conjecture for cubic fourfolds 205 210 212 216 220 224 229 233 240 244 245

multiscale adjoint waveform difference tomography using wavelets abstractfull waveform seismic inversions based on minimizing the distance between observed and predicted seismograms are in principle able to yield better resolved earth models than those minimizing misfits derived from traveltimes alone adjoint based methods provide an efficient way of calculating the gradient of the misfit function via a sequence of forward modeling steps which using spectral element codes can be carried out in realistically complex media convergence and stability of full waveform difference adjoint schemes are greatly improved when data and synthetics are progressively presented to the algorithms in a constructive multiscale approximation using a bi orthogonal wavelet transform wavelets provide the nonredundant spectral decomposition that paves the way for the inversion to proceed successively from long wavelength fitting to detailed exploration of the phases in the seismogram the choice of wavelet class and type the initial depth of the multiscale decomposition and the min

blow up phenomenon for the integrable novikov equation abstract in this paper we investigate a new integrable equation derived recently by v s novikov generalizations of the camassa holm equation j phys a 42 34 2009 342002 14 pp analogous to the camassa holm equation and the degasperis procesi equation this new equation possesses the blow up phenomenon under the special structure of this equation we establish sufficient conditions on the initial data to guarantee the formulation of singularities in finite time a global existence result is also found

on the semisimplicity of low dimensional representations of semisimple groups in characteristic p there is an unfortunate conflict of terminology in the theory of algebraic groups in characteristic p unlike in characteristic zero the representations of a semisimple group need not be semisimple thanks to chevalley 15 over any algebraically closed field we can associate to each dominant weight a representation v satisfying weyl s character formula but it is no longer in general irreducible springer 13 has shown that in one sense representation theory for large p stabilizes to the usual characteristic zero theory for a fixed root system a fixed dominant weight and p 0 v is irreducible unfortunately there exist representations of low dimension which are not of the form v no matter how large p is taken a typical example is the absolute frobenius representation of sl 2

finite difference time dispersion transforms for wave propagation the finite difference fd wave equation is widely implemented in seismic imaging for oil exploration but the numerical dispersion due to discretization of time and space derivatives can introduce severe numerical errors we have investigated time dispersion which might distort the phase and introduce severe artifacts to the data and images especially for long time propagation we first studied precisely how the time dispersion was produced and then we developed an efficient approach the time dispersion transforms to cope with time dispersion errors in wave propagation the proposed forward time dispersion transform can predict or simulate the time dispersion errors and it can be applied to remove time dispersion errors in reverse time migration the inverse time dispersion transform can be used to eliminate time dispersion errors from synthetic data after modeling the proposed method is general and works for low and high order time fd schemes by using this method large time steps were allowed in wave propagation thus we can remove the time dispersion errors at a negligible numerical cost and the efficiency is not affected to test the effectiveness and robustness of the proposed time dispersion transforms we used isotropic and anisotropic examples as well as modeling and imaging examples

waveform inversion by one way wavefield extrapolation one way riemannian wavefield extrapolation rwe on computational meshes conforming to the direction of turning wave propagation is an alternative forward modeling procedure for waveform inversion forward modeling tests demonstrate that the rwe approach may be a sufficiently accurate approximation for calculating the wavefield phases important for early arrival waveform inversion initial results indicate that rwe waveforms are well matched at wide offsets to finite difference data and can be used in a waveform inversion scheme to invert synthetic data for moderate 1d velocity perturbations

finite subgroups of algebraic groups generalizing a classical theorem of jordan to arbitrary characteristic we prove that every finite subgroup of gln over a field of any characteristic p possesses a subgroup of bounded index which is composed of finite simple groups of lie type in characteristic p a commutative group of order prime to p and a p group while this statement can be deduced from the classification of finite simple groups our proof is self contained and uses methods only from algebraic geometry and the theory of linear algebraic groups we believe that our results can serve as a viable substitute for classification in a range of applications in various areas of mathematics

constrained full waveform inversion by model reparameterization the waveform inversion problem is inherently ill posed traditionally regularization schemes are used to address this issue for waveform inversion where the model is expected to have many details reflecting the physical properties of the earth regularization and data fitting can work in opposite directions the former smoothing and the latter adding details to the model we propose constraining estimated velocity fields by reparameterizing the model this technique also called model space preconditioning is based on directional laplacian filters it preserves most of the details of the velocity model while smoothing the solution along known geological dips preconditioning also yields faster convergence at early iterations the laplacian filters have the property to smooth or kill local planar events according to a local dip field by construction these filters can be inverted and used in a preconditioned waveform inversion strategy to yield geologically meaningful models we illustrate with 2d syn

cascaded wavefield tomography and inversion using extended common image point gathers a case study abstractimage domain wavefield tomography wt exploits focusing characteristics of extended images for updating the velocity field to make good use of this information one must understand how such images behave if the migration velocity is accurate this is not trivial because focusing depends on not only the model error but also on the acquisition setup the data bandwidth and illumination variation caused by the overburden we address this problem by constructing penalty functions based on the point spread functions of the imaging operator that characterize focusing in extended images moreover instead of sampling the extended images at preset distances along the surface we sample the image by constructing common image point gathers which are more economical from a computational point of view and also measure the spatial and temporal focusing of the wavefield coupled with image residuals exploiting illumination based penalty functions we construct robust wavefield tomographic updates from spar

e adic and x adic representations associated to abelian varieties defined over number fields 0 introduction let a be an abelian variety over a number field k of dimension n 1 and let g gal kik for each prime number let vi a be the qe adic tate module of a one general problem is to study the image of the e adic representation vr a of g let ge be the e adic lie algebra of the image of this f adic representation fix an embedding cr k c c and denote by a c the abelian variety a x c in 12 mumford and tate conjectured that ge m 0q qe where mn is the lie algebra of the mumford tate group mt a c cf 9 section 3 as is now well known in general g is contained in m 0q qe cf 9 16 various results toward this general problem have been obtained for some classes of abelian varieties cf 22 section 4 more recently some important general results on e adic representations attached to abelian varieties have been established especially one has the following results i the rank of g is independent of c serre 24 zarhin 34 ii geis algebraic and contains the homotheties bogomolov 1 iii g is reductive and end ve a endk a 0 qe faltings 10 these make the determination of g possible for some other classes of abelian varieties for example when d dim a is odd and endt a z j p serre has proved that ge m 0q qe sp 2d qe 3 qe id where id is the 2d x 2d identity matrix cf 25 the purpose of this article is to extend serre s method to some other cases as a result ge can be determined for the following types of simple abelian varieties

seismic envelope inversion and modulation signal model abstractwe recognized that the envelope fluctuation and decay of seismic records carries ultra low frequency ulf i e the frequency below the lowest frequency in the source spectrum signals that can be used to estimate the long wavelength velocity structure we then developed envelope inversion for the recovery of low wavenumber components of media smooth background so that the initial model dependence of waveform inversion can be reduced we derived the misfit function and the corresponding gradient operator for envelope inversion to understand the long wavelength recovery by the envelope inversion we developed a nonlinear seismic signal model the modulation signal model as the basis for retrieving the ulf data and studied the nonlinear scale separation by the envelope operator to separate the envelope data from the wavefield data envelope extraction a demodulation operator envelope operator was applied to the waveform data numerical tests using synthetic data for the marmousi model pro

correlation based reflection full waveform inversion because modeling for full waveform inversion fwi cannot produce reflections unless the velocity model has the scattering potential high wavenumbers using a migration demigration process to generate modeling data which is a key step in what is now known as reflection fwi rfwi is a credible alternative to tackle the reflection nonlinearity associated with fwi however because rfwi depends on a conventional data residual or zero lag correlation objective function high nonlinearity can still exist when the true amplitude migration is not used as well as at far offsets due to cycle skipping to avoid the cycle skipping and the need for a true amplitude migration we have developed a correlation based reflection full waveform inversion method to update the low wavenumber components of the velocity model the success of this method relies on a sensitivity kernel decomposition and a correlation based objective function the sensitivity kernel decomposition makes it possible to separate out the contributions of different subkernels and to smear the reflected wave residuals along the rabbit ear wavepath to obtain middle and deep background model estimates the correlation based objective function measures differences in kinematic information and behaves in a more linear way than the traditional waveform residual misfit moreover our approach is less sensitive to the frequency content and amplitude information of the seismic data enabling reliable background velocity estimates to be obtained without the need for low frequencies and full physics modeling because the kinematic features of reflected waves are described correctly the inversion result of the proposed method can be used as a migration model or an initial model for conventional fwi to achieve a correct high wavenumber model update

on the rationality of certain type a galois representations let x be a complete smooth variety defined over a number field k and let i be an integer the absolute galois group galk of k acts on the ith etale cohomology group forumala presented for all primes l producing a system of l adic representations l l the conjectures of grothendieck tate and mumford tate predict that the identity component of the algebraic monodromy group of l admits a reductive q form that is independent of l if x is projective denote by l and gl respectively the monodromy group and the algebraic monodromy group of ss l the semisimplification of l assuming that gl0 satisfies some group theoretic conditions for some prime l0 we construct a connected quasi split q reductive group gq which is a common q form of g l for all sufficiently large l let gscq be the universal cover of the derived group of gq as an application we prove that the monodromy group l is big in the sense that forumala presented for all sufficiently large l

inversion gradients for acoustic vti wavefield tomography abstractwavefield tomography can handle complex subsurface geology better than ray based techniques and ultimately provide a higher resolution here we implement forward and adjoint wavefield extrapolation for vti transversely isotropic with a vertical symmetry axis media using a generalized pseudospectral operator based on a separable approximation for the p wave dispersion relation this operator is employed to derive the gradients of the differential semblance optimization dso and modified image power objective functions we also obtain the gradient expressions for a data domain objective function that can more easily incorporate borehole information necessary for stable vti velocity analysis these gradients are similar to the ones obtained with a space time finite difference fd scheme for a system of coupled wave equations but the pseudospectral method is not hampered by the imprint of the shear wave artifact numerical examples also show the potential advantages of the modified image power

maximality of galois actions for abelian and hyper kahler varieties let l l be the system of l adic representations arising from the ith l adic cohomology of a proper smooth variety x defined over a number field k let l and gl be respectively the image and the algebraic monodromy group of l we prove that the reductive quotient of gl is unramified over every degree 12 totally ramified extension of ql for all sufficiently large l we give a necessary and sufficient condition on l l such that for all sufficiently large l the subgroup l is in some sense maximal compact in gl ql this is used to deduce galois maximality results for l adic representations arising from abelian varieties for all i and hyper kahler varieties i 2 defined over finitely generated fields over q

global conservative solutions of the camassa holm equation this paper develops a new approach in the analysis of the camassa holm equation by introducing a new set of independent and dependent variables the equation is transformed into a semilinear system whose solutions are obtained as fixed points of a contractive transformation these new variables resolve all singularities due to possible wave breaking returning to the original variables we obtain a semigroup of global solutions depending continuously on the initial data our solutions are conservative in the sense that the total energy equals a constant for almost every time

lipschitz metrics for a class of nonlinear wave equations the nonlinear wave equation u tt c u c u u x x 0 determines a flow of conservative solutions taking values in the space h 1 mathbb r however this flow is not continuous with respect to the natural h 1 distance the aim of this paper is to construct a new metric which renders the flow uniformly lipschitz continuous on bounded subsets of h 1 mathbb r for this purpose h 1 is given the structure of a finsler manifold where the norm of tangent vectors is defined in terms of an optimal transportation problem for paths of piecewise smooth solutions one can carefully estimate how the weighted length grows in time by the generic regularity result proved in 7 these piecewise regular paths are dense and can be used to construct a geodesic distance with the desired lipschitz property

seismic waveform inversion in the frequency domain part 1 theory and verification in a physical scale model seismic waveforms contain much information that is ignored under standard processing schemes seismic waveform inversion seeks to use the full information content of the recorded wavefield in this paper i present apply and evaluate a frequency space domain approach to waveform inversion the method is a local descent algorithm that proceeds from a starting model to refine the model in order to reduce the waveform misfit between observed and model data the model data are computed using a full wave equation viscoacoustic frequency domain finite difference method ray asymptotics are avoided and higher order effects such as diffractions and multiple scattering are accounted for automatically the theory of frequency domain waveform wavefield inversion can be expressed compactly using a matrix formalism that uses finite difference finite element frequency domain modeling equations expressions for fast local descent inversion using back propagation techniques then follow naturally implementation of

analysis on the blow up of solutions to a class of integrable peakon equations abstract we investigate the blow up mechanism of solutions to a class of quasilinear integrable equations which could possess peakons the dynamics of the blow up quantity along the characteristics is established by the riccati type differential inequality which involves the interaction among three parts a local nonlinearity a nonlocal term and a term stemming from the weak linear dispersion to analyse the interplay among these quantities we provide two different approaches the first one is designed for the case when the equations do not exhibit a weak linear dispersion and hence focuses on the interplay between the first two parts the method is based on a refined analysis on either evolution of the solution u and its gradient u x that is c u u x or the growth rate of the relative ratio u x u the second one handles the general situation when all of three parts are present the idea is to extract the truly blow up component from the riccati type differential inequality and utilizes the morawetz type identity or higher order conservation laws to show that such a component blows up in finite time before the other component degenerates

wave field reciprocity and optimization in remote sensing a unified approach to local optimization techniques and wave field reciprocity as applied to constructing solutions to remote sensing imaging and inversion problems in acoustic elastic and electromagnetic wave theory is presented the starting point is a system of linear first order partial differential equations in space time of the class to which the indicated wave phenomena gives rise for this system three types of remote sensing problems the inverse source the inverse scattering and the inverse transducted wave field problems are formulated and the construction of their solutions via local optimization techniques is discussed emphasis is placed on iterative algorithms that are based on a guaranteed decrease in the mismatch between modelled and observed data at each update of the medium subsequently the wave field reciprocity theorems of the time convolution and the time correlation types are derived and their occurrence in the optimization procedures is discussed also attention is paid to approximate methods in particular to the rayleigh gans born approximation approximations of this sort provide the means to invoke the method of preconditioning in the process of inverting the operator equations exotic media for example chiral media in electromagnetics are included in the analysis

geometric monodromy semisimplicity and maximality let x be a connected scheme smooth and separated over an alge braically closed field k of characteristic p 0 let f y x be a smooth proper morphism and x a geometric point on x we prove that the tensor invariants of bounded length d of 1 x x acting on the etale cohomology groups h yx fl are the reduction modulo l of those of 1 x x acting on h yx l for l greater than a constant depending only on f y x d we apply this result to show that the geometric variant with fl coefficients of the grothendieck serre semisimplicity conjecture namely that 1 x x acts semisimply on h yx fl for l 0 is equivalent to the condition that the image of 1 x x acting on h yx ql is almost maximal in a precise sense what we call almost hyperspecial with respect to the group of ql points of its zariski closure ultimately we prove the geometric variant with fl coefficients of the grothendieck serre semisimplicity conjecture

wave equation tomography the relation between ray trace and diffraction tomography is usually obscured by formulation of the two methods in different domains the former in space the latter in wavenumber here diffraction tomography is reformulated in the space domain under the title of wave equation tomography with this transformation wave equation tomography projects monochromatic scattered wavefields back over source receiver wavepaths just as ray trace tomography projects traveltime delays back over source receiver raypaths derived under the born approximation these wavepaths are wave theoretic back projection patterns for reflected energy derived under the rytov approximation they are wave theoretic back projection patterns for transmitted energy differences between ray trace and wave equation tomography are examined through comparison of wavepaths and raypaths followed by their application to a transmission geometry synthetic data set rytov wave equation tomography proves superior to ray trace tomography in de

linearized wave equation migration velocity analysis by image warping seismic imaging produces images of contrasts in physical parameters in the subsurface e g velocity or impedance to build such images a background model describing the wave kinematics in the earth is necessary in practice the structural image and background velocity model are unknown and have to be estimated from the acquired data migration velocity analysis deals with estimation of the background model in the framework of seismic migration and relies on two main elements data redundancy and invariance of the structures with respect to different seismic experiments because all the experiments probe the same model the reflectors must be invariant in suitable domains e g shots or reflection angle the semblance principle is the tool used to measure the invariance of a set of multiple images we measure the similarity of the structural features between pairs of single shot migrated images obtained from adjacent experiments by using the estimated warping vector field between two migrated images we construct an image perturbation which describes the difference in reflectivity observed by two shots we derive an expression for the image perturbation that drives a migration velocity analysis procedure based on a linearization of the wave equation with respect to the model parameters synthetic 2d examples show promising results in retrieving errors in the velocity model this methodology can be directly applied to 3d

generalizations of the camassa holm equation we classify generalized camassa holm type equations which possess infinite hierarchies of higher symmetries we show that the obtained equations can be treated as negative flows of integrable quasi linear scalar evolution equations of orders 2 3 and 5 we present the corresponding lax representations or linearization transformations for these equations some of the obtained equations seem to be new

full waveform inversion with extrapolated low frequency data abstractthe availability of low frequency data is an important factor in the success of full waveform inversion fwi in the acoustic regime the low frequencies help determine the kinematically relevant low wavenumber components of the velocity model which are in turn needed to avoid convergence of fwi to spurious local minima however acquiring data less than 2 or 3 hz from the field is a challenging and expensive task we have explored the possibility of synthesizing the low frequencies computationally from high frequency data and used the resulting prediction of the missing data to seed the frequency sweep of fwi as a signal processing problem bandwidth extension is a very nonlinear and delicate operation in all but the simplest of scenarios it can only be expected to lead to plausible recovery of the low frequencies rather than their accurate reconstruction even so it still requires a high level interpretation of band limited seismic records into individual events each of which can be extr

symplectic structures their backlund transformations and hereditary symmetries abstract it is shown that compatible symplectic structures lead in a natural way to hereditary symmetries we recall that a hereditary symmetry is an operator valued function which immediately yields a hierarchy of evolution equations each having infinitely many commuting symmetries all generated by this hereditary symmetry furthermore this hereditary symmetry usually describes completely the soliton structure and the conservation laws of these equations this result then provide us with a method for constructing hereditary symmetries and hence exactly solvable evolution equations in addition we show how symplectic structures transform under backlund transformations this leads to a method for generating a whole class of symplectic structures from a given one several examples and applications are given illustrating the above results also the connection of our results with those of gelfand and dikii and of magri is briefly pointed out

migration velocity analysis theory and an iterative algorithm imaging complex structures inside the earth requires reasonable velocities that can be provided by applying prestack depth migration to multichannel seismic data migration velocity analysis is based on the principle that the images in the migrated data will be distorted when an erroneous velocity is used and the difference of the imaged depths residual moveout at a common image gather is a measure of the error in the velocity the imaging equations that we derive from snell s law describe a general quantitative relationship between migration images and migration velocity based on the imaging equations we analyze properties of common image gathers and derive analytical formulas to represent residual moveout in some cases these formulas show what factors affect the sensitivity of velocity analysis which is useful to assess errors involved in velocity estimates in addition we develop a simple iteration algorithm to correct the layer velocities from residual moveout the algorithm presented here is applicable to a medium that consists of constant velocity layers separated by arbitrary smooth interfaces some computer implementations are presented for both synthetic data and physical tank data they demonstrate the effectiveness of our velocity analysis approach

hodge classes on certain types of abelian varieties the hodge conjecture for a states that such classes are algebraic in the sense that they are q linear combinations of cohomology classes of algebraic subvarieties of codimension p on a this statement is well known for p 1 lefschetz and it is trivially true for p 0 hence for a given a it holds for allp if the ring formed by the hodge classes under cup product is generated by the hodge classes of codimension 1 we thus have a simple criterion for the hodge conjecture to be true for a which we call the 1 1 criterion responding to a question posed by tate 14 p 107 mumford found that the 1 1 criterion while true for all a such that dim a c 3 is not necessarily satisfied by abelian varieties of dimension 4 although mumford s counterexample involved an abelian variety of cm type 51 weil 15 has stressed that its most important feature is the action of an imaginary quadratic field k on a in such a way that lie a becomes a free k x c module abelian varieties of precisely this type play an important role in deligne s proof 2 that hodge cycles on an arbitrary abelian variety are absolutely hodge there is speculation that the hodge conjecture is false at least in general for such abelian varieties but no method is available at present for settling this question either way indeed one knows no example of an algebraic variety for which the hodge conjecture is false and only scattered examples shioda 12 of

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