structure and properties of hughston s stochastic extension of the schrodinger equation hughston has recently proposed a stochastic extension of the schrodinger equation expressed as a stochastic differential equation on projective hilbert space we derive new projective hilbert space identities which we use to give a general proof that hughston s equation leads to state vector collapse to energy eigenstates with collapse probabilities given by the quantum mechanical probabilities computed from the initial state we discuss the relation of hughston s equation to earlier work on norm preserving stochastic equations and show that hughston s equation can be written as a manifestly unitary stochastic evolution equation for the pure state density matrix we discuss the behavior of systems constructed as direct products of independent subsystems and briefly address the question of whether an energy based approach such as hughston s suffices to give an objective interpretation of the measurement process in quantum mechanics

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

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

stochastic sequential reduction of commutative hamiltonians this paper introduces a class of stochastic processes constructed by conditioning cadlag processes to take a predetermined set of marginal laws at fixed points in time we collectively refer to the elements of this class of processes as random n bridges rnbs where n refers to the cardinality of the set of conditioning laws which can be chosen arbitrarily we prove that if the underlying cadlag process is markov then its rnb is markov for n 1 and has a particular markov like property for n 1 as a canonical subclass we construct brownian rnbs brnbs and provide their anticipative representation as well as their non anticipative semimartingale representation this motivates the second part of this paper wherein we apply brnbs to describe an energy based state reduction in a composite quantum system involving a set of commuting observables measured successively at different times the highly tractable so called energy based quantum state reduction models are significant in their role as the most important alternative to the popular cls model we ask the random variables composing the anticipative form of a brnb to take the eigenvalues of a set of compatible hamiltonians and provide a new stochastic schrodinger evolution on a complex hilbert space that models sequential reduction dynamics of the eigenstates through a single wave function by doing so we also extend the brody hughston finite time collapse model to a many body setup with sequential measurements

to a unified approach to 3 d seismic reflection imaging part ii theory m tygel j schleicher and p hubral geophysics 61 759 775 it came to our attention that in the paper of tygel et al 1996 formula a 5 is incorrect first it is different from the cited equation 17 of tygel et al 1995 second the derivation of formula 17 in appendix a of tygel et al 1995 is invalid for the purposes of tygel et al 1996 because in the 1995 paper a rotation is treated whereas in tygel et al 1996 a projection matrix is needed further investigation of the subject was needed to resolve these inconsistencies

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

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

a unified approach to 3 d seismic reflection imaging part ii theory diffraction stack and isochrone stack integrals are quantitatively described and employed they constitute an asymptotic transform pair both integrals are the key tools of a unified approach to seismic reflection imaging that can be used to solve a multitude of amplitude preserving target oriented seismic imaging or image transformation problems these include for instance the generalizations of the kinematic map transformation examples discussed in part i all image transformation problems can be addressed by applying both stacking integrals in sequence whereby the macro velocity model the measurement configuration or the ray code of the considered elementary reflections may change from step to step this leads to weighted kirchhoff or generalized radon type summations along certain stacking surfaces or inplanats for which true amplitude ta weights are provided to demonstrate the value of the proposed imaging theory which is based on analytically chaining the two stacking integrals and using certain inherent dualities we examine in detail the amplitude preserving configuration transform and remigration for the case of a 3 d laterally inhomogeneous velocity medium

martingale models for quantum state reduction stochastic models for quantum state reduction give rise to statistical laws that are in most respects in agreement with those of quantum measurement theory here we examine the correspondence of the two theories in detail making a systematic use of the methods of martingale theory an analysis is carried out to determine the magnitude of the fluctuations experienced by the expectation of the observable during the course of the reduction process and an upper bound is established for the ensemble average of the greatest fluctuations incurred we consider the general projection postulate of l uders applicable in the case of a possibly degenerate eigenvalue spectrum and derive this result rigorously from the underlying stochastic dynamics for state reduction in the case of both a pure and a mixed initial state we also analyse the associated lindblad equation for the evolution of the density matrix and obtain an exact time dependent solution for the state reduction that explicitly exhibits the transition from a general initial density matrix to the l uders density matrix finally we apply girsanov s theorem to derive a set of simple formulae for the dynamics of the state in terms of a family of geometric brownian motions thereby constructing an explicit unravelling of the lindblad equation

a unified approach to 3 d seismic reflection imaging part i basic concepts given a 3 d seismic record for an arbitrary measurement configuration and assuming a laterally and vertically inhomogeneous isotropic macro velocity model a unified approach to amplitude preserving seismic reflection imaging is provided this approach is composed of 1 a weighted kirchhoff type diffraction stack integral to transform migrate seismic reflection data from the measurement time domain into the model depth domain and of 2 a weighted kirchhoff type isochrone stack integral to transform demigrate the migrated seismic image from the depth domain back into the time domain both the diffraction stack and isochrone stack integrals can be applied in sequence i e they can be chained for different measurement configurations or different velocity models to permit two principally different amplitude preserving image transformations these are 1 the amplitude preserving transformation directly in the time domain of one 3 d seismic record section into another one pertaining to a different

geometry of stochastic state vector reduction the state space of a quantum mechanical system is a complex projective space the space of rays in the associated hilbert space the state space comes equipped with a natural riemannian metric the fubini study metric and a compatible symplectic structure the operations of ordinary quantum mechanics can thus be reinterpreted in the language of differential geometry it is interesting in this spirit to scrutinize the probabilistic assumptions that are brought in at various stages in the analy sis of quantum dynamics particularly in connection with state vector reduction a promising approach to understanding reduction studied recently by a number of authors involves the use of nonlinear stochastic dynamics to modify the ordinary linear schrodinger evolution here we use stochastic differential geometry to give a systematic geometric formulation for such stochastic models of state vector collapse in this picture the conventional schrodinger evolution which corresponds to the unitary flow associated with a killing vector of the fubini study metric is replaced by a more general stochastic flow on the state manifold in the simplest example of such a flow the volatility term in the stochastic differential equation for the state trajectory is proportional to the gradient of the expectation of the hamiltonian the conservation of energy is represented by the requirement that the actual process followed by the expectation of the hamiltonian as the state evolves should be a martingale this requirement implies the existence of a nonlinear term in the drift vector of the state process which is always oriented opposite to the direction of in creasing energy uncertainty as a consequence the state vector necessarily collapses to an energy eigenstate and a martingale argument can be used to show that the probability of collapse to a given eigenstate from any particular initial state is in fact given by precisely the usual quantum mechanical probability

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

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

finite time stochastic reduction models a new energy based stochastic extension of the schrodinger equation for which the wave function collapses after the passage of a finite amount of time is proposed an exact closed form solution to the dynamical equation valid for all finite dimensional quantum systems is presented and used to verify explicitly that reduction of the state vector to an energy eigenstate occurs a time change technique is introduced to construct a clock variable that relates the asymptotic and the finite time collapse models by means of a nonlinear transformation

a pendulum theorem we consider n linked pendulums which are inverted and balanced on top of one another and establish a general theorem which shows how they may be stabilized by small vertical oscillations of the support

parameter inversion and angle migration in anisotropic elastic media seismic parameter inversion using the inverse generalized radon transform grt was originally formulated as an integral over source and receiver surface coordinates involving a jacobian from angle to surface coordinates by formulating the inversion directly in terms of angle coordinates at the subsurface image point i obtain simplified inversion formulas for point scatterers and subsurface reflectors amplitude preserving angle migration based on the inverse grt gives an estimate of the scattering coefficient in the born integral as a function of scattering angle and azimuth the scattering coefficient is directly related to the linearized plane wave reflection coefficient between two anisotropic media which limits the result to small angles and small contrasts in the elastic parameters i use the kirchhoff modeling integral for anisotropic media to formulate a kirchhoff angle migration integral that is valid for large angles and large parameter contrasts it gives an estimate of the plane wave refle

elastic least squares reverse time migration abstractleast squares migration can produce images with improved resolution and reduced migration artifacts compared with conventional imaging we have developed a method for elastic least squares reverse time migration lsrtm based on a new perturbation imaging condition that yields scalar images of squared p and s velocity perturbations these perturbation images do not suffer from polarity reversals that are common for more conventional elastic imaging methods we use 2d synthetic and field data examples to demonstrate the proposed lsrtm algorithm using the perturbation imaging condition our results show that elastic lsrtm improves the energy focusing and illumination of the elastic images and it attenuates artifacts resulting for instance from sparseness in the wavefield sampling and crosstalk of the p and s modes compared with rtm images the lsrtm images provide more accurate relative amplitude information that is useful for reservoir characterization

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