multiscattering inversion for low model wavenumbers abstracta successful full waveform inversion implementation updates the low wavenumber model components first for a proper description of the wavefield propagation and slowly adds the high wavenumber potentially scattering parts of the model the low wavenumber components can be extracted from the transmission parts of the recorded wavefield emanating directly from the source or the transmission parts from the single or double scattered wavefield computed from a predicted scatter field acting as secondary sources we use a combined inversion of data modeled from the source and those corresponding to single and double scattering to update the velocity model and the component of the velocity perturbation responsible for the single and double scattering the combined inversion helps us access most of the potential model wavenumber information that may be embedded in the data a scattering angle filter is used to divide the gradient of the combined inversion so initially the high wavenumber low scatterin

waveform inversion of reflection seismic data for kinematic parameters by local optimization a reformulation of the usual least squares waveform inversion problem is proposed to retrieve from seismic data the kinematic parameters the two dimensional 2d background velocity and the source and cable depth by local optimization these last parameters are of paramount importance for a successful inversion of very high resolution vhr seismic data which we are interested in in our inversion the source and cable depth parameters are treated in the same way as the background velocity to avoid the problem of local minima a change of unknowns is performed the depth reflectivity is replaced by its dual variable called the time reflectivity in this way the current value of the reflectivity is stored in the time domain and is strongly decoupled from the current value of the velocity field and cable depth the increase in modeling complexity an additional prestack migration is required for each function evaluation is compensated by the enlargement of the attraction domain of the global minimum which allows the use of a local optimization technique a numerical implementation with born approximation and ray tracing is detailed in which the derivatives of the travel time are computed via an adjoint state technique for more efficiency numerical results illustrate the behavior of the new objective function and inversion of synthetic and real vhr data for kinematic parameters is performed

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

least squares reverse time migration with an angle dependent weighting factor least squares reverse time migration lsrtm produces higher quality images than conventional rtm however directly using the standard gradient formula the inverted images suffer from low wavenumber noise using a simple high pass filter on the gradient can alleviate the effect of the low wavenumber noise but owing to the illumination issue the amplitudes are not balanced and in the deep part they are often weak these two issues can be mitigated by the iterative approach but it needs more iterations we introduced an angle dependent weighting factor to weight the gradient of lsrtm to suppress the low wavenumber noise and also to emphasize the gradient in the deep part an optimal step length for the l2 norm objective function is also presented to scale the gradient to the right order two numerical examples performed with the data synthesized on the sigsbee2a and marmousi models indicate that when using this weighted gradient combined with the preconditioned l bfgs algorithm with the optimal step length only a few iterations can achieve satisfying results

velocity inversion using a stratified reference we present an inversion algorithm for backscattered stacked seismic data which will reconstruct the velocity profile in realistic earth conditions the basic approach follows that of the original constant reference speed cohen and bleistein paper 1979a in that hig hfrequency asymptotics and perturbation methods are used however we use a reference speed which may vary with depth and this greatly enhances the validity of the perturbation assumption and hence the inversion results the new algorithm enjoys the same economies and stability properties of the original algorithm making it very competitive with current migration schemes four major assumptions are made 1 the acoustic wave equation is an adequate model 2 stacked data have amplitude information worth preserving fairly accurately 3 the actual reflectivity coefficients can be adequately modeled as perturbations from a continuous reference velocity which depends only on the depth variable and 4 the subsurface can be adequately

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

instantaneous control of brownian motion a controller continuously monitors a storage system such as an inventory or bank account whose content z zt t 0 fluctuates as a 2 brownian motion in the absence of control holding costs are incurred continuously at rate h zt at any time the controller may instantaneously increase the content of the system incurring a proportional cost of r times the size of the increase or decrease the content at a cost of l times the size of the decrease we consider the case where h is convex on a finite interval and h outside this interval the objective is to minimize the expected discounted sum of holding costs and control costs over an infinite planning horizon it is shown that there exists an optimal control limit policy characterized by two parameters a and b a b roughly speaking this policy exerts the minimum amounts of control sufficient to keep zt a b for all t 0 put another way the optimal control limit policy imposes on z a lower reflecting barrier at a an

reverse time wave field extrapolation imaging and inversion the scattered wave field propagated backward in time into an arbitrary background medium is related via a volume integral to perturbations in velocity about the background which are expressed as a scattering potential in general there is no closed form expression for the kernel of this integral representation although it can be expressed asymptotically as a superposition of plane waves backpropagated from the receiver array when the receiver array completely surrounds the scatterer the kernel reduces to the imaginary part of the green s function for the background medium this integral representation is used to relate the images obtained by imaging algorithms to the actual scattering potential two such relations are given 1 for the migrated image obtained by deconvolving the extrapolated field with the incident field and 2 for the reconstructed image obtained by applying a one way wave operator to the extrapolated field and then deconvolving by the incident field the migrated image highlig

a brief comparison of some kirchhoff integral formulas for migration and inversion kirchhoff migration has traditionally been viewed as an imaging procedure usually few claims are made regarding the amplitudes in the imaged section in recent years a number of inversion formulas similar in form to those of kirchhoff migration have been proposed a kirchhoff type inversion produces not only an image but also an estimate of velocity variations or perhaps reflection coefficients the estimate is obtained from the peak amplitudes in the image in this paper prestack kirchhoff migration and inversion formulas for the one parameter acoustic wave equation are compared following a heuristic approach based on the imaging principle a migration formula is derived which turns out to be identical to one proposed by bleistein for inversion prestack kirchhoff migration and inversion are thus seen to be the same both in terms of the image produced and the peak amplitudes of the output

accommodating lateral velocity changes in kirchhoff migration by means of fermat s principle when velocity varies laterally as well as with depth an exact kirchhoff depth migration requires that rays be traced from each depth point in the section to each source receiver location because such a procedure is prohibitively expensive kirchhoff migration is usually carried out by using a velocity function that depends only on depth this paper introduces a new method based on fermat s principle which is a compromise between these two extremes the slowness reciprocal velocity function is written as the sum of two functions the first of which is large and depends only on depth while the other is small and varies both with depth and position along the line raypaths are traced for the first slowness function and are used to calculate migration curves for each depth point these same raypaths are used to calculate traveltime perturbations due to the laterally varying part of the slowness the traveltime perturbations are added to the migration curve to obtain an approximation to the exact migrat

2d frequency domain elastic full waveform inversion using the block diagonal pseudo hessian approximation abstractelastic full waveform inversion efwi of multicomponent seismic data is a powerful tool for estimating the subsurface elastic parameters with high accuracy however the trade offs between multiple parameters increase the nonlinearity of efwi although the conventional diagonal approximate hessian matrix describes the illumination and limited bandwidth effects it ignores the trade off effects and decreases the convergence rate of efwi we have developed a block diagonal pseudo hessian operator for 2d frequency domain efwi to take into account the approximate trade offs among the p wave compressional wave velocity s wave shear wave velocity and density without extra computational costs on forward simulations the hessian matrix tends toward a block diagonal matrix as the frequency grows to infinity thus the proposed block diagonal pseudo hessian matrix is more accurate at higher frequencies the inverse of the block diagonal pseudo hessian matrix is used as a preconditioner for the nonlin

inversion of normal incidence seismograms the well known instability of kunetz s 1963 inversion algorithm can be explained by the progressive manner in which the calculations are done descending from the surface and by the fact that completely different impedances can yield indistinguishable synthetic seismograms those difficulties can be overcome by using an iterative algorithm for the inversion of the one dimensional 1 d wave equation together with a stabilizing constraint on the sums of the jumps of the desired impedance for computational efficiency the synthetic seismogram is computed by the method of characteristics and the gradient of the error criterion is computed by optimal control techniques adjoint state equation the numerical results on simulated data confirm the expected stability of the algorithm in the presence of measurement noise tests include noise levels of 50 percent the inversion of two field sections demonstrates the practical feasibility of the method and the importance of taking into account all internal a

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

zero sum stochastic differential games and backward equations abstract in this paper we are concerned with the applications to stochastic zero sum differential games of the results on backward stochastic differential equations using these techniques the existence of a saddle point in the bounded case is obtained if the isaacs condition holds furthermore this technique is also a very simple approach for the existence of an optimal strategy in the case of controlled diffusions

asymptotic pseudounitary stacking operators stacking operators are widely used in seismic imaging and seismic data processing examples include kirchhoff datuming migration offset continuation dip moveout and velocity transform two primary approaches exist for inverting such operators the first approach is iterative least squares optimization which involves the construction of the adjoint operator the second approach is asymptotic inversion where an approximate inverse operator is constructed in the high frequency asymptotics adjoint and asymptotic inverse operators share the same kinematic properties but their amplitudes weighting functions are defined differently this paper describes a theory for reconciling the two approaches i introduce a pair of asymptotic pseudounitary operators which possess both the property of being adjoint and the property of being asymptotically inverse the weighting function of the asymptotic pseudounitary stacking operators is shown to be completely defined by the derivatives of the operator kinematics i exemplify the general theory by considering several particular examples of stacking operators simple numerical experiments demonstrate a noticeable gain in efficiency when the asymptotic pseudounitary operators are applied for preconditioning iterative least squares optimization

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

additive control of stochastic linear systems with finite horizon we consider a dynamic system whose state is governed by a linear stochastic differential equation with time dependent coefficients the control acts additively on the state of the system our objective is to minimize an integral cost which depends upon the evolution of the state and the total variation of the control process it is proved that the optimal cost is the unique solution of an appropriate free boundary problem in a space time domain by using some decomposition arguments the problems of a two sided control i e optimal corrections and the case with constraints on the resources i e finite fuel can be reduced to a simpler case of only one sided control i e a monotone follower these results are applied to solving some examples by the so called method of similarity solutions

adapted solution of a backward stochastic differential equation abstract let wt t 0 1 be a standard k dimensional weiner process defined on a probability space f p and let ft denote its natural filtration given a f1 measurable d dimensional random vector x we look for an adapted pair of processes x t y t t 0 1 with values in rd and rd k respectively which solves an equation of the form x t t 1 f s x s y s ds t 1 g s x s y s dw s x a linearized version of that equation appears in stochastic control theory as the equation satisfied by the adjoint process we also generalize our results to the following equation x t t 1 f s x s y s ds t 1 g s x s y s dw s x under rather restrictive assumptions on g

singular optimal controls of stochastic recursive systems and hamilton jacobi bellman inequality abstract in this paper we study the optimal singular controls for stochastic recursive systems in which the control has two components the regular control and the singular control under certain assumptions we establish the dynamic programming principle for this kind of optimal singular controls problem and prove that the value function is a unique viscosity solution of the corresponding hamilton jacobi bellman inequality in a given class of bounded and continuous functions at last an example is given for illustration

ray kirchhoff migration in inhomogeneous media a kirchhoff method that avoids possible singularities on the surface of integration and is more accurate than previous kirchhoff methods has been developed for seismic migration in laterally inhomogeneous media it is based on a newly derived integral solution to the acoustic wave equation this solution indicates that wave fields in an inhomogeneous medium can be expressed as a summation of ray solutions determined by the transport and extended eikonal equations the extended eikonal equation is in turn solved by an asymptotic series for implementation a perturbation scheme is developed for solving the ray and transport equations in addition to computing higher order terms of the asymptotic series this scheme can be used to avoid most of the ray tracing required for wave extrapolation in a medium where vertical variations may be large but lateral variations are small compared to velocity itself where an analytic ray solution for the reference velocity used in the perturbation scheme exists ray kirchhoff migration in such a medium can be carried out without numerical ray tracing

on the principle of smooth fit for a class of singular stochastic control problems for diffusions this paper considers the principle of smooth fit for a class of one dimensional singular stochastic control problems allowing the system to be of nonlinear diffusion type the existence and the uniqueness of a convex c 2 solution to the corresponding variational inequality are obtained it is proved that this solution gives the value function of the control problem and the optimal control process is constructed as an example of the degenerate case it is proved that the conclusion is also true for linear systems and the explicit formula for the smooth fit points is derived

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

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

waveform inversion using a logarithmic wavefield although waveform inversion has been studied extensively since its beginning 20 years ago applications to seismic field data have been limited and most of those applications have been for global seismology or engineering seismology scale problems not for exploration scale data as an alternative to classical waveform inversion we propose the use of a new objective function constructed by taking the logarithm of wavefields allowing consideration of three types of objective function namely amplitude only phase only or both in our waveform inversion we estimate the source signature as well as the velocity structure by including functions of amplitudes and phases of the source signature in the objective function we compute the steepest descent directions by using a matrix formalism derived from a frequency domain finite element finite difference modeling technique our numerical algorithms are similar to those of reverse time migration and waveform inversion based on the adjoint state of the wave equation in order to demonstrate the practical applicability of our algorithm we use a synthetic data set from the marmousi model and seismic data collected from the korean continental shelf for noisefree synthetic data the velocity structure produced by our inversion algorithm is closer to the true velocity structure than that obtained with conventional waveform inversion when random noise is added the inverted velocity model is also close to the true marmousi model but when frequencies below 5 hz are removed from the data the velocity structure is not as good as those for the noise free and noisy data for field data we compare the time domain synthetic seismograms generated for the velocity model inverted by our algorithm with real seismograms and find that the results show that our inversion algorithm reveals short period features of the subsurface although we use wrapped phases in our examples we still obtain reasonable results we expect that if we were to use correctly unwrapped phases in the inversion algorithm we would obtain better results

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

solving forward backward stochastic differential equations explicitly a four step scheme in this paper we investigate the nature of the adapted solutions to a class of forward backward stochastic differential equations sdes for short in which the forward equation is non degenerate we prove that in this case the adapted solution can always be sought in an ordinary sense over an arbitrarily prescribed time duration via a direct four step scheme using this scheme we further prove that the backward components of the adapted solution are determined explicitly by the forward components via the solution of a certain quasilinear parabolic pde system moreover the uniqueness of the adapted solutions over an arbitrary time duration as well as the continuous dependence of the solutions on the parameters can all be proved within this unified framework some special cases are studied separately in particular we derive a new form of the integral representation of the clark haussmann ocone type for functionals or functions of diffusions in which the conditional expectation is no longer needed

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

the link of kirchhoff migration and demigration to kirchhoff and born modeling the kirchhoff approximation provides a representation of seismic data as a summation of imaged data along isochron surfaces demigration the asymptotic inversion of this representation provides a migration as a summation of seismic data along diffraction surfaces we replace born inversion techniques with kirchhoff inversion techniques and further show the link between the kirchhoff and born representations after the born linearized reflection coefficient is replaced by the kirchhoff reflection coefficient

kirchhoff integral extrapolation and migration of nonplanar data the form of the kirchhoff integral commonly used for migration of seismic data assumes that the observation surface is a flat horizontal plane this restriction is not necessary in theory or practice a useful integral expression can be obtained for an arbitrary observation surface by use of the kirchhoff approximation for relating the field on the observation surface to its normal derivative kirchhoff migration is usually carried out by an integral that represents summation over receivers for a given shot summation over shots is not explicitly handled by invoking the principle of reciprocity and applying the kirchhoff integral twice one arrives at an integral expression that explicitly represents shot and receiver summation contains the factors needed to compensate for nonplanar survey geometry and can be used for wave extrapolation as well as migration adaptation of this integral to handling 2 d or line survey data is straightforward and with the usual approximations concerning spherical wave s

efficient traveltime calculations for kirchhoff migration acoustic kirchhoff migration is performed for each point x z in an output depth section by an integrating or weigthed summing procedure each trace of the input time section contained in an aperture about x contributes to the sum and the contribution from a trace at x x consists of the element whose time is equal to the total time required for a seismic wave to travel from the source to the point x z to x x 0 that is kirchhoff migration sees every subsurface location as a possible point scatterer and explicitly reverses the operation of scattering the diffracted seismic energy from such an obstacle onto an array of receivers as a function of time continuous reflectors are considered ensembles of point scatterers and they are imaged by summing over traces for each point on the reflector if the array of nx nz output depth points for a two dimensional 2 d poststack migration contains nxnz points and on the average o nx points are included in each summation aperture kirchhoff summation

singular control of stochastic linear systems with recursive utility we formulate a class of singular stochastic control problem with recursive utility where the cost function is determined by a backward stochastic differential equation some characteristics of the value function of the control problem are obtained by the method of approximation via penalization and the optimal control process is constructed

migration based traveltime waveform inversion of 2 d simple structures a synthetic example migration based traveltime mbtt formulation provides algorithms for automatically determining background velocities from full waveform surface seismic reflection data using local optimization methods in particular it addresses the difficulty of the nonconvexity of the least squares data misfit function the method consists of parameterizing the reflectivity in the time domain through a migration step and providing a multiscale representation for the smooth background velocity we present an implementation of the mbtt approach for a 2 d finite difference fd full wave acoustic model numerical analysis on a 2 d synthetic example shows the ability of the method to find much more reliable estimates of both long and short wavelengths of the velocity than the classical least squares approach even when starting from very poor initial guesses this enlargement of the domain of attraction for the global minima of the least squares misfit has a price each evaluation of the new objective function requires besides the usual fd full wave forward modeling an additional full wave prestack migration hence the fd implementation of the mbtt approach presented in this paper is expected to provide a useful tool for the inversion of data sets of moderate size

prestack kirchhoff inversion of common offset data in trying to resolve complex geologic structures the pitfalls in employing the cdp method become evident additionally stacking multioffset traces corrupts the amplitudes necessary for stratigraphic analysis in order to preserve whatever structural and amplitude information is in the data prestack processing should be performed given common offset data and the velocity above a reflector prestack acoustic kirchhoff inversion resolves the location of the interface when amplitude information has been preserved in the data the method additionally calculates the reflection coefficient at each interface point for band limited seismic data the inversion operator produces a sinc like picture of the reflector with the peak amplitude of this band limited singular function equal to the angularly dependent reflection coefficient the inversion development is based upon high frequency kirchhoff data which are inserted into a general 3 d inversion operator asymptotically evaluating the four resulting integrals by the method of four dimensional stationary phase permits an inversion amplitude function to be chosen so that the inversion operator produces a singular function of support on the reflector weighted by the reflection coefficient specializing the three dimensional inversion operator to two and one half dimensions allows for processing of single lines of common offset data synthetic examples illustrate the accuracy of the method for constant velocity kirchhoff data as well as the problems in applying constant velocity data to multivelocity models

waveform inversion using a back propagation algorithm and a huber function norm waveforminversionfacesdifficultieswhenappliedtoreal seismic data including the existence of many kinds of noise the 1 norm is more robust to noise with outliers than the least squares method nevertheless the least squares method is preferred as an objective function in many algorithms because the gradient of the 1 norm has a singularity when the residual becomes zero we propose a complex valued huber function for frequency domain waveform inversion that combines the 2 norm for small residuals with the 1 norm for large residuals we also derive a discretized formula for the gradient of the huber function through numericaltestsonsimplesyntheticmodelsandmarmousidata we find the huber function is more robust to outliers and coherent noise we apply our waveform inversion algorithm to field data taken from the continental shelf under the east sea in korea in this setting we obtain a velocity model whose syntheticshotprofilesaresimilartotherealseismicdata

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

optimal control of inflation a central bank problem this paper models the action of the central bank on the dynamics of the nominal interest rate with the aim of controlling inflation the problem is set up as a two dimensional bounded variation control problem it is shown that its variational formulation leads to a stochastic differential game with stopping times between the conservative and the expansionist tendencies of the bank

inverse extrapolation of primary seismic waves forward wave field extrapolation operators simulate propagation effects from one depth level to another inverse wave field extrapolation operators eliminate those propagation effects since forward wave field extrapolation can be described in terms of spatial convolution inverse wave field extrapolation can be described in terms of spatial deconvolution a simple approximation to a stable spatially band limited deconvolution operator is obtained by taking the complex conjugate of the convolution operator a one way version of the kirchhoff integral that contains the conjugate complex green s function is derived unlike the situation with respect to the forward problem the modification of the closed surface integral into an open boundary integral involves an approximation that is identical to the approximation in the conjugate complex deconvolution approach this approximation neglects the evanescent field and causes a second order amplitude error for a plane acquisition surface the one way kirchhoff

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

efficient 2 5 d true amplitude migration kirchhoff depth migration is a widely used algorithm for imaging seismic data in both two and three dimensions to perform the summation at the heart of the algorithm standard kirchhoff migration requires a traveltime map for each source and receiver true amplitude kirchhoff migration in 2 5 d x z media additionally requires maps of amplitudes out of plane spreading factors and takeoff angles these quantities are necessary for calculating the true amplitude weight term in the summation the increased input output i o and computational expense of including the true amplitude weight term is often not justified by significant improvement in the final muted and stacked image for this reason some authors advocate neglecting the weight term in the kirchhoff summation entirely for most everyday imaging purposes we demonstrate that for nearly the same expense as ignoring the weight term a much better solution is possible we first approximate the true amplitude weight term by the weight term for co

block row recursive least squares migration abstractrecursive estimates of large systems of equations in the context of least squares fitting is a common practice in different fields of study for example recursive adaptive filtering is extensively used in signal processing and control applications the necessity of solving least squares problems recursively stems from the need for fast real time signal processing strategies the computational cost of least squares solvers can also limit the applicability of this technique in geophysical problems we have considered a recursive least squares solution for least squares wave equation migration with sliding windows involving several fixed rank downdating and updating computations this technique can be applied for dynamic and stationary processes if we use enough data in each windowed setup the spectrum of the preconditioned system is clustered around one and the method will converge superlinearly with probability one numerical experiments were performed to test the effectiveness of the technique

a unified approach to acoustical reflection imaging ii the inverse problem using the forward matrix model as derived in part i a j berkhout j acoust soc am 93 xxx 1993 it is shown that the first and main part of numerical acoustic imaging consists of a wave field extrapolation process by double matrix inversion physically the wave field extrapolation process means that the downward propagation effects and the upward propagation effects are eliminated from the measurements next the reflection information is extracted from the wave field extrapolation result optionally the reflection information is translated to discipline oriented material parameters by some data fitting process double focusing i e focusing in emission and focusing in detection is closely related to the above numerical imaging process finally it is shown that imaging of zero offset or puls echo data can be formulated by single matrix inversion involving phase shifts only

regularity of the value function for a two dimensional singular stochastic control problem it is desired to control a two dimensional brownian motion by adding a possibly singularly continuous process to it so as to minimize an expected infinite horizon discounted running cost the hamilton jacobi bellman characterization of the value function v is a variational inequality which has a unique twice continuously differentiable solution the optimal control process is constructed by solving the skorokhod problem of reflecting the two dimensional brownian motion along a free boundary in the nabla v direction

least squares gaussian beam migration abstractleast squares migration lsm can be effective to mitigate the limitation of finite seismic acquisition balance the subsurface illumination and improve the spatial resolution of the image but it requires iterations of migration and demigration to obtain the desired subsurface reflectivity model the computational efficiency and accuracy of migration and demigration operators are crucial for applying the algorithm we have developed a test of the feasibility of using the gaussian beam as the wavefield extrapolating operator for the lsm denoted as least squares gaussian beam migration our method combines the advantages of the lsm and the efficiency of the gaussian beam propagator our numerical evaluations including two synthetic data sets and one marine field data set illustrate that the proposed approach could be used to obtain amplitude balanced images and to broaden the bandwidth of the migrated images in particular for the low wavenumber components

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