an abelian category of motivic sheaves abstract a category of motivic sheaves is constructed over a variety in characteristic 0 using nori s method although the relationship with many alternative constructions remains to be clarified it does have many of the properties one expects for example it is abelian and has betti hodge and l adic realizations and it has a tannakian subcategory of motivic local systems

target oriented wave equation least squares migration inversion with phase encoded hessian prestack depth migration produces blurred images resulting from limited acquisition apertures complexities in the velocity model and band limited characteristics of seismic waves this distortion can be partially corrected using the model space least squares migration inversion approach where a target oriented wave equation hessian operator is computed explicitly and then inverse filtering is applied iteratively to deblur or invert for the reflectivity however one difficulty is the cost of computing the explicit hessian operator which requires storing a large number of green s functions making it challenging for large scale applications a new method to compute the hessian operator for the wave equation based least squares migration inversion problem modifies the original explicit hessian formula enabling efficient computation of this operator an advantage is that the method eliminates disk storage of green s functions the modifications however also introduce undesired crosstalk artifacts two different phase encoding schemes plane wave phase encoding and random phase encoding suppress the crosstalk when the randomly phase encoded hessian operator is applied to the sigsbee2a synthetic data set an improved subsalt image with more balanced amplitudes is obtained

approximate linearized inversion by optimal scaling of prestack depth migration linearized inversion provides one possible sense of image amplitude correctness an image or bandlimited model perturbation has correct amplitudes if it is an approximate inversion that is if linearized modeling demigration with the image as input reproduces the data approximately the theory of linearized acoustic inverse scattering with slowly varying background or macromodel shows that an approximate inversion may be recovered from the output of prestack depth migration by a combination of scaling and filtering the necessary filter is completely specified by the theory and the scale factor may be estimated via filtering linearized modeling a second migration and the solution of a small auxiliary inverse problem

a discrepancy based penalty method for extended waveform inversion abstractextended waveform inversion globalizes the convergence of seismic waveform inversion by adding nonphysical degrees of freedom to the model thus permitting it to fit the data well throughout the inversion process these extra degrees of freedom must be curtailed at the solution for example by penalizing them as part of an optimization formulation for separable partly linear models a natural objective function combines a mean square data residual and a quadratic regularization term penalizing the nonphysical linear degrees of freedom the linear variables are eliminated in an inner optimization step leaving a function of the outer nonlinear variables to be optimized this variable projection method is convenient for computation but it requires that the penalty weight be increased as the estimated model tends to the physical solution we describe an algorithm based on discrepancy that is maintaining the data residual at the inner optimum within a prescribed range to control the pena

from migration to inversion velocity analysis abstractmigration velocity analysis mva is a technique defined in the image domain to determine the background velocity model controlling the kinematics of wave propagation in the presence of discontinuous interfaces the velocity gradient used to iteratively update the velocity model exhibits spurious oscillations for more stable results we replace the migration part by an inversion scheme by definition migration is the adjoint of the born modeling operator whereas inversion is its asymptotic inverse we have developed new expressions in 1d and 2d cases based on two way wave equation operators the objective function measures the quality of the images obtained by inversion in the extended domain depending on the subsurface offset in terms of implementation the new approach is very similar to classic mva a 1d analysis found that oscillatory terms around the interface positions can be removed by multiplying the inversion result with the velocity at a specific power before evaluating the objectiv

an alternative formula for approximate extended born inversion abstractvarious modifications of reverse time migration rtm provide asymptotic inverses to the subsurface offset extended born modeling operator for constant density acoustics these approximate inverses have the same quality asymptotic accuracy as do generalized radon transform pseudoinverses but they can be computed without any ray tracing whatsoever we have developed an approximate inverse of this type whose additional computational cost above that of subsurface offset extended rtm is negligible

discrete symbol calculus this paper deals with efficient numerical representation and manipulation of differential and integral operators as symbols in phase space i e functions of space x and frequency xi the symbol smoothness conditions obeyed by many operators in connection to smooth linear partial differential equations allow fast converging nonasymptotic expansions in adequate systems of rational chebyshev functions or hierarchical splines to be written the classical results of closedness of such symbol classes under multiplication inversion and taking the square root translate into practical iterative algorithms for realizing these operations directly in the proposed expansions because symbol based numerical methods handle operators and not functions their complexity depends on the desired resolution n very weakly typically only through log n factors we present three applications to computational problems related to wave propagation 1 preconditioning the helmholtz equation 2 decomposing wave fields into one way components and 3 depth extrapolation in reflection seismology the software is made available in the software sections of math mit edu sim laurent and www math utexas edu users lexing

computation of pseudo differential operators a simple algorithm is described for computing general pseudo differential operator actions our approach is based on the asymptotic expansion of the symbol together with the fast fourier transform fft the idea is motivated by the characterization of the pseudo differential operator algebra we show that the algorithm is efficient through analyzing its complexity some numerical experiments are also presented

the decomposition theorem perverse sheaves and the topology of algebraic maps we give a motivated introduction to the theory of perverse sheaves culminating in the decomposition theorem of beilinson bernstein deligne and gabber a goal of this survey is to show how the theory develops naturally from classical constructions used in the study of topological properties of algebraic varieties while most proofs are omitted we discuss several approaches to the decomposition theorem indicate some important applications and examples

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

fast least squares migration with a deblurring filter least squares migration lsm is a linearized waveform inversion for estimating a subsurface reflectivity model that relative to conventional migration improves spatial resolution of migration images the cost however is high because lsm typically requires 10 or more iterations which is at least 20 times more than the cpu cost of conventional migration to alleviate this expense we offer a deblurring filter that can be used in a regularization scheme or a preconditioning scheme to give acceptable lsm images with less than one third the cost of the standard lsm method our results in applying deblurred lsm to synthetic data and field data support this claim in particular a marmousi2 model test shows that the data residual for preconditioned deblurred lsm decreases rapidly in the first iteration which is equivalent to 10 or more iterations of lsm empirical results suggest that regularized deblurred lsm after three iterations is equivalent to about 10 iterations of lsm applying deblurred lsm to 2d marine data gives a higher resolution image compared to those from migration or lsm with three iterations these results suggest that lsm combined with a deblurring filter allows lsm to be a fast practical tool for improved imaging of complicated structures

q compensated least squares reverse time migration using low rank one step wave extrapolation abstractattenuation of seismic waves needs to be taken into account to improve the accuracy of seismic imaging in viscoacoustic media reverse time migration rtm can be performed with q compensation which is also known as q rtm least squares rtm lsrtm has also been shown to be able to compensate for attenuation through linearized inversion however seismic attenuation may significantly slow down the convergence rate of the least squares iterative inversion process without proper preconditioning we have found that incorporating attenuation compensation into lsrtm can improve the speed of convergence in attenuating media obtaining high quality images within the first few iterations based on the low rank one step seismic modeling operator in viscoacoustic media we have derived its adjoint operator using nonstationary filtering theory the proposed forward and adjoint operators can be efficiently applied to propagate viscoacoustic waves and to implement attenuation compensation recognizing that

an approximate inverse to the extended born modeling operator abstractgiven a correct data consistent velocity model reverse time migration rtm correctly positions reflectors but generally with incorrect amplitudes and wavelets iterative least squares migration lsm corrects the amplitude and wavelet by fitting data in the sense of born modeling that is replacing migration by born inversion however lsm also requires a correct velocity model and it may require many migration demigration cycles we modified rtm in the subsurface offset domain to create an asymptotic high frequency approximation to extended lsm this extended born inversion operator outputs extended reflectors depending on the subsurface offset and position in the earth with correct amplitude and phase in the sense that similarly extended born modeling reproduces the data to good accuracy although the theoretical justification of the inversion property relies on ray tracing and stationary phase application of the weight operators does not require any computational ray tracing the co

least squares wave equation migration for avp ava inversion we present an acoustic migration inversion algorithm that uses extended double square root wave equation migration and modeling operators to minimize a constrained least squares data misfit function least squares migration we employ an imaging principle that allows for the extraction of ray parameter domain common image gathers cigs from the propagated seismic wavefield the cigs exhibit amplitude variations as a function of half offset ray parameter avp closely related to the amplitude variation with reflection angle ava our least squares wave equation migration inversion is constrained by a smoothing regularization along the ray parameter this approach is based on the idea that rapid amplitude changes or discontinuities along the ray parameter axis result from noise incomplete wavefield sampling and numerical operator artifacts these discontinuities should therefore be penalized in the inversion the performance of the proposed algorithm is examined with two synthetic examples in the first

plane wave least squares reverse time migration with a preconditioned stochastic conjugate gradient method abstractthis study derives a preconditioned stochastic conjugate gradient cg method that combines stochastic optimization with singular spectrum analysis ssa denoising to improve the efficiency and image quality of plane wave least squares reverse time migration plsrtm this method reduces the computational costs of plsrtm by applying a controlled group sampling method to a sufficiently large number of plane wave sections and accelerates the convergence using a hybrid of stochastic descent sd iteration and cg iteration however the group sampling also produces aliasing artifacts in the migration results we use ssa denoising as a preconditioner to remove the artifacts moreover we implement the preconditioning on the take off angle domain common image gathers cigs for better results we conduct numerical tests using the marmousi model and sigsbee2a salt model and compare the results of this method with those of the sd method and the cg method the results demonstrate that our method efficientl

observer design for unknown input nonlinear descriptor systems via convex optimization this paper treats the design problem of full order observers for nonlinear descriptor systems with unknown input ui depending on the available knowledge on the ui dynamics two cases are considered first a ui proportional observer uipo is proposed when the spectral domain of the ui is unknown second a pio is proposed when the spectral domain of the ui is in the low frequency range sufficient conditions for the existence and stability of such observers are given and proved based on the linear matrix inequality lmi approach an algorithm is presented to compute the observer gain matrix that achieves the asymptotic stability objective an example is included to illustrate the method

an optimal true amplitude least squares prestack depth migration operator in order to define an optimal true amplitude prestack depth migration for multishot and multitrace data we develop a general methodology based on the least squares data misfit function associated with a forward model the amplitude of the migrated events are restored at best for any given geometry and any given preliminary filtering and amplitude correction of the data the migrated section is then the gradient of the cost function multiplied by a weight matrix a study of the hessian associated with this data misfit shows how efficiently to find a good weight matrix via the computation of only few elements of this hessian thanks to this matrix the resulting migration operator is optimal in the sense that it ensures the best possible restoration of the amplitudes among the large class of least squares migrations applied to a forward model based on born ray tracing and diffracting points approximation this optimal migration outperforms or at least equals the classic kirchhoff formula since the latter belongs to the class of least squares migrations and is only optimal for one shot and an infinite aperture numerical results illustrate this construction and confirm the above expectations

imaging of discontinuities in the inverse scattering problem by inversion of a causal generalized radon transform this paper treats the linearized inverse scattering problem for the case of variable background velocity and for an arbitrary configuration of sources and receivers the linearized inverse scattering problem is formulated in terms of an integral equation in a form which covers wave propagation in fluids with constant and variable densities and in elastic solids this integral equation is connected with the causal generalized radon transform grt and an asymptotic expansion of the solution of the integral equation is obtained using an inversion procedure for the grt the first term of this asymptotic expansion is interpreted as a migration algorithm as a result this paper contains a rigorous derivation of migration as a technique for imaging discontinuities of parameters describing a medium also a partial reconstruction operator is explicitly derived for a limited aperture when specialized to a constant background velocity and specific source receiver geometries our results are directly related to some known migration algorithms

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

elastic least squares reverse time migration via linearized elastic full waveform inversion with pseudo hessian preconditioning abstracttime domain elastic least squares reverse time migration lsrtm is formulated as a linearized elastic full waveform inversion problem the elastic born approximation and elastic reverse time migration rtm operators are derived from the time domain continuous adjoint state method our approach defines p and s wave impedance perturbations as unknown elastic images our algorithm is obtained using continuous functional analysis in which the problem is discretized at the final stage optimize before discretize approach the discretized numerical versions of the elastic born operator and its adjoint elastic rtm operator pass the dot product test the conjugate gradient least squares method is used to solve the least squares migration quadratic optimization problem in other words the hessian operator for elastic lsrtm is implicitly inverted via a matrix free algorithm that only requires the action of forward and adjoint operators on vectors the diagonal of the pseudo hessian operator is used t

an adaptive multiscale algorithm for efficient extended waveform inversion abstractsubsurface offset extended full waveform inversion fwi may converge to kinematically accurate velocity models without the low frequency data accuracy required for standard data domain fwi however this robust alternative approach to waveform inversion suffers from a very high computational cost resulting from its use of nonlocal wave physics the computation of strain from stress involves an integral over the subsurface offset axis which must be performed at every space time grid point we found that a combination of data fit driven offset limits grid coarsening and low pass data filtering can reduce the cost of extended inversion by one to two orders of magnitude

frobenius semisimplicity for convolution morphisms this article concerns properties of mixed ell adic complexes on varieties over finite fields related to the action of the frobenius automorphism we establish a fiberwise criterion for the semisimplicity and frobenius semisimplicity of the direct image complex under a proper morphism of varieties over a finite field we conjecture that the direct image of the intersection complex on the domain is always semisimple and frobenius semisimple this conjecture would imply that a strong form of the decomposition theorem of beilinson bernstein deligne gabber is valid over finite fields we prove our conjecture for generalized convolution morphisms associated with partial affine flag varieties for split connected reductive groups over finite fields as a crucial tool we develop a new schematic theory of big cells for loop groups with suitable reformulations the main results are valid over any algebraically closed ground field

preserving the discontinuities in least squares reverse time migration of simultaneous source data abstractthe simultaneous source shooting technique can accelerate field acquisition and improve spatial sampling but it will cause strong interferences in the recorded data and artifacts in the final image the previously proposed structural smoothing operator can effectively attenuate artifacts for relatively simple reflection structures during least squares inversion but it will cause damage to complicated reflection events such as discontinuities to preserve discontinuities in a seismic image we apply the singular spectrum analysis ssa operator to attenuate artifacts during least squares inversion considering that global ssa cannot deal with overcomplicated data very well we use local ssa to remove noise and to better preserve the steeply dipping components the local ssa operator corresponds to a local low rank constraint applied in the inversion process the migration operator used in the study is the reverse time migration rtm operator tests using the marmousi model showed the superior per

accelerating extended least squares migration with weighted conjugate gradient iteration abstractleast squares migration lsm iteratively achieves a mean square best fit to seismic reflection data provided that a kinematically accurate velocity model is available the subsurface offset extension adds extra degrees of freedom to the model thereby allowing lsm to fit the data even in the event of significant velocity error this type of extension also implies additional computational expense per iteration from crosscorrelating source and receiver wavefields over the subsurface offset and therefore places a premium on rapid convergence we have accelerated the convergence of extended least squares migration by combining the conjugate gradient algorithm with weighted norms in range data and domain model spaces that render the extended born modeling operator approximately unitary we have developed numerical examples that demonstrate that the proposed algorithm dramatically reduces the number of iterations required to achieve a given level of fit or gradient reduction compared with conjuga

modeling of seismic data in the downward continuation approach seismic data are commonly modeled by a high frequency single scattering approximation this amounts to a linearization in the medium coefficient about a smooth background the discontinuities are contained in the medium perturbation the high frequency part of the wavefield in the background medium is described by a geometrical optics representation it can also be described by a one way wave equation based on this we derive a downward continuation operator for seismic data this operator solves a pseudodifferential evolution equation in depth the so called double square root equation we consider the modeling operator based on this equation if the rays in the background that are associated with the reflections due to the perturbation are nowhere horizontal the singular part of the data is described by the solution to an inhomogeneous double square root equation

topology of hitchin systems and hodge theory of character varieties the case a1 for g gl2 pgl2 sl2 we prove that the perverse ltration associated with the hitchin map on the rational cohomology of the moduli space of twisted g higgs bundles on a compact riemann surface c agrees with the weight ltration on the rational cohomology of the twisted g character variety of c when the cohomologies are identied via non abelian hodge theory the proof is accomplished by means of a study of the topology of the hitchin map over the locus of integral spectral curves