the multifrontal method for sparse matrix solution theory and practice this paper presents an overview of the multifrontal method for the solution of large sparse symmetric positive definite linear systems the method is formulated in terms of frontal matrices update matrices and an assembly tree formal definitions of these notions are given based on the sparse matrix structure various advances to the basic method are surveyed they include the role of matrix reorderings the use of supernodes and other implementatjon techniques the use of the method in different computational environments is also described

3d finite difference modeling of elastic wave propagation in the laplace fourier domain abstractwith the recent interest in the laplace fourier domain full waveform inversion we have developed new heterogeneous 3d fourth and second order staggered grid finite difference schemes for modeling seismic wave propagation in the laplace fourier domain our approach is based on the integro interpolation technique for the velocity stress formulation in the cartesian coordinate system five averaging elastic coefficients and three averaging densities are necessary to describe the heterogeneous medium with harmonic averaging of the bulk and shear moduli and arithmetic averaging of density in the fourth order approximation we improved the accuracy of the scheme using a combination of integral identities for two elementary volumes small and large around spatial gridpoints where the wave variables are defined two solution approaches are provided both of which are solved with iterative krylov methods in the first approach the stress variables are eliminated and a linear system for the veloc

a finite volume solution to the geophysical electromagnetic forward problem using unstructured grids abstractthe application of unstructured grids can improve the solution of total field electromagnetic problems because these grids allow efficient local refinement of the mesh at the locations of high field gradients unstructured grids also provide the flexibility required for representing arbitrary topography and subsurface interfaces we investigated the generalization of the standard yee s staggered scheme to unstructured tetrahedral voronoi grids using a finite volume approach we discretized the helmholtz equation for the electric field in the frequency domain and solved the problem to find the projection of the total electric field along the edges of the tetrahedral elements to compute the electric and magnetic fields at the observation points an interpolation technique was employed which uses the edge vector interpolation functions of the tetrahedral elements to verify the presented scheme two examples were evaluated which revealed the computation of the total and secondary fields due to ele

a bayesian approach for predicting with polynomial regression of unknown degree this article compares three methods for computing the posterior probabilities of the possible orders in polynomial regression models these posterior probabilities are used for forecasting using bayesian model averaging it is shown that bayesian model averaging provides a closer relationship between the theoretical coverage of the high density predictive interval hdpi and the observed coverage than those corresponding to selecting the best model the performance of the different procedures is illustrated with simulations and some known engineering data

an optimal 9 point finite difference frequency space 2 d scalar wave extrapolator in this study a new finite difference technique is designed to reduce the number of grid points needed in frequency space domain modeling the new algorithm uses optimal nine point operators for the approximation of the laplacian and the mass acceleration terms the coefficients can be found by using the steepest descent method so that the best normalized phase curves can be obtained this method reduces the number of grid points per wavelength to 4 or less with consequent reductions of computer memory and cpu time that are factors of tens less than those involved in the conventional second order approximation formula when a band type solver is used on a scalar machine

the variable selection problem abstract the problem of variable selection is one of the most pervasive model selection problems in statistical applications often referred to as the problem of subset selection it arises when one wants to model the relationship between a variable of interest and a subset of potential explanatory variables or predictors but there is uncertainty about which subset to use this vignette reviews some of the key developments that have led to the wide variety of approaches for this problem

robust stochastic approximation approach to stochastic programming in this paper we consider optimization problems where the objective function is given in a form of the expectation a basic difficulty of solving such stochastic optimization problems is that the involved multidimensional integrals expectations cannot be computed with high accuracy the aim of this paper is to compare two computational approaches based on monte carlo sampling techniques namely the stochastic approximation sa and the sample average approximation saa methods both approaches the sa and saa methods have a long history current opinion is that the saa method can efficiently use a specific say linear structure of the considered problem while the sa approach is a crude subgradient method which often performs poorly in practice we intend to demonstrate that a properly modified sa approach can be competitive and even significantly outperform the saa method for a certain class of convex stochastic problems we extend the analysis to the case of convex concave stochastic saddle point problems and present in our opinion highly encouraging results of numerical experiments

p sv wave propagation in heterogeneous media velocity stress finite difference method i present a finite difference method for modeling p sv wave propagation in heterogeneous media this is an extension of the method i previously proposed for modeling sh wave propagation by using velocity and stress in a discrete grid the two components of the velocity cannot be defined at the same node for a complete staggered grid the stability condition and the p wave phase velocity dispersion curve do not depend on the poisson s ratio while the s wave phase velocity dispersion curve behavior is rather insensitive to the poisson s ratio therefore the same code used for elastic media can be used for liquid media where s wave velocity goes to zero and no special treatment is needed for a liquid solid interface typical physical phenomena arising with p sv modeling such as surface waves are in agreement with analytical results the weathered layer and corner edge models show in seismograms the same converted phases obtained by previous authors this method gives stable results for step discontinuities as shown for a liquid layer above an elastic half space the head wave preserves the correct amplitude finally the corner edge model illustrates a more complex geometry for the liquid solid interface as the poisson s ratio v increases from 0 25 to 0 5 the shear converted phases are removed from seismograms and from the time section of the wave field

harder better faster stronger convergence rates for least squares regression we consider the optimization of a quadratic objective function whose gradients are only accessible through a stochastic oracle that returns the gradient at any given point plus a zero mean finite variance random error we present the first algorithm that achieves jointly the optimal prediction error rates for least squares regression both in terms of forgetting of initial conditions in o 1 n 2 and in terms of dependence on the noise and dimension d of the problem as o d n our new algorithm is based on averaged accelerated regularized gradient descent and may also be analyzed through finer assumptions on initial conditions and the hessian matrix leading to dimension free quantities that may still be small while the optimal terms above are large in order to characterize the tightness of these new bounds we consider an application to non parametric regression and use the known lower bounds on the statistical performance without computational limits which happen to match our bounds obtained from a single pass on the data and thus show optimality of our algorithm in a wide variety of particular trade offs between bias and variance

a new iterative solver for the time harmonic wave equation the time harmonic wave equation also known as the helmholtz equation is obtained if the constant density acoustic wave equation is transformed from the time domain to the frequency domain its discretization results in a large sparse linear system of equations in two dimensions this system can be solved efficiently by a direct method in three dimensions direct methods cannot be used for problems of practical sizes because the computational time and the amount of memory required become too large iterative methods are an alternative these methods are often based on a conjugate gradient iterative scheme with a preconditioner that accelerates its convergence the iterative solution of the time harmonic wave equation has long been a notoriously difficult problem in numerical analysis recently a new preconditioner based on a strongly damped wave equation has heralded a breakthrough the solution of the linear system associated with the preconditioner is approximated by another iterative method the multigrid method the multigrid method fails for the original wave equation but performs well on the damped version the performance of the new iterative solver is investigated on a number of 2d test problems the results suggest that the number of required iterations increases linearly with frequency even for a strongly heterogeneous model where earlier iterative schemes fail to converge complexity analysis shows that the new iterative solver is still slower than a time domain solver to generate a full time series we compare the time domain numeric results obtained using the new iterative solver with those using the direct solver and conclude that they agree very well quantitatively the new iterative solver can be applied straightforwardly to 3d problems

spectral element method for elastic and acoustic waves in frequency domain numerical techniques in time domain are widespread in seismic and acoustic modeling in some applications however frequency domain techniques can be advantageous over the time domain approach when narrow band results are desired especially if multiple sources can be handled more conveniently in the frequency domain moreover the medium attenuation effects can be more accurately and conveniently modeled in the frequency domain in this paper we present a spectral element method sem in frequency domain to simulate elastic and acoustic waves in anisotropic heterogeneous and lossy media the sem is based upon the finite element framework and has exponential convergence because of the use of gll basis functions the anisotropic perfectly matched layer is employed to truncate the boundary for unbounded problems compared with the conventional finite element method the number of unknowns in the sem is significantly reduced and higher order accuracy is obtained due to its spectral accuracy to account for the acoustic solid interaction the domain decomposition method ddm based upon the discontinuous galerkin spectral element method is proposed numerical experiments show the proposed method can be an efficient alternative for accurate calculation of elastic and acoustic waves in frequency domain

bayes model averaging with selection of regressors when a number of distinct models contend for use in prediction the choice of a single model can offer rather unstable predictions in regression stochastic search variable selection with bayesian model averaging offers a cure for this robustness issue but at the expense of requiring very many predictors here we look at bayes model averaging incorporating variable selection for prediction this offers similar mean square errors of prediction but with a vastly reduced predictor space this can greatly aid the interpretation of the model it also reduces the cost if measured variables have costs the development here uses decision theory in the context of the multivariate general linear model in passing this reduced predictor space bayes model averaging is contrasted with single model approximations a fast algorithm for updating regressions in the markov chain monte carlo searches for posterior inference is developed allowing many more variables than observations to be contemplated we discuss the merits of absolute rather than proportionate shrinkage in regression especially when there are more variables than observations the methodology is illustrated on a set of spectroscopic data used for measuring the amounts of different sugars in an aqueous solution

improving multifrontal methods by means of block low rank representations matrices coming from elliptic partial differential equations pdes have been shown to have a low rank property well defined off diagonal blocks of their schur complements can be approximated by low rank products given a suitable ordering of the matrix which gives to the blocks a geometrical meaning such approximations can be computed using an svd or a rank revealing qr factorization the resulting representation offers a substantial reduction of the memory requirement and gives efficient ways to perform many of the basic dense algebra operations several strategies have been proposed to exploit this property we propose a low rank format called block low rank blr and explain how it can be used to reduce the memory footprint and the complexity of direct solvers for sparse matrices based on the multifrontal method we present experimental results that show how the blr format delivers gains that are comparable to those obtained with hierarchical formats such as hierarchical matrices h matrices and hierarchically semi separable hss matrices but provides much greater flexibility and ease of use which are essential in the context of a general purpose algebraic solver

sh wave propagation in heterogeneous media velocity stress finite difference method a new finite difference fd method is presented for modeling sh wave propagation in a generally heterogeneous medium this method uses both velocity and stress in a discrete grid density and shear modulus are similarly discretized avoiding any spatial smoothing therefore boundaries will be correctly modeled under an implicit formulation standard problems quarter plane propagation sedimentary basin propagation are studied to compare this method with other methods finally a more complex example a salt dome inside a two layered medium shows the effect of lateral propagation on seismograms recorded at the surface a corner wave always in phase with the incident wave and a head wave will appear which will pose severe problems of interpretation with the usual vertical migration methods

the intrinsic bayes factor for model selection and prediction abstract in the bayesian approach to model selection or hypothesis testing with models or hypotheses of differing dimensions it is typically not possible to utilize standard noninformative or default prior distributions this has led bayesians to use conventional proper prior distributions or crude approximations to bayes factors in this article we introduce a new criterion called the intrinsic bayes factor which is fully automatic in the sense of requiring only standard noninformative priors for its computation and yet seems to correspond to very reasonable actual bayes factors the criterion can be used for nested or nonnested models and for multiple model comparison and prediction from another perspective the development suggests a general definition of a reference prior for model comparison

a predictive approach to the analysis of designed experiments abstract viewing the analysis of designed experiments as a model selection problem we introduce the use of a predictive bayesian criterion in this context based on the predictive density of a replicate experiment pdre a calibration of the criterion is provided to assist in the model choice the relationships of the proposed criterion to other prevalent criteria such as aic bic and mallows s c p are given an information theoretic criterion based on the pdre s of two competing models is also introduced and compared with the usual f statistic for two nested models examples are given to illustrate the proposed methodology

bayesian variable selection in linear regression abstract this article is concerned with the selection of subsets of predictor variables in a linear regression model for the prediction of a dependent variable it is based on a bayesian approach intended to be as objective as possible a probability distribution is first assigned to the dependent variable through the specification of a family of prior distributions for the unknown parameters in the regression model the method is not fully bayesian however because the ultimate choice of prior distribution from this family is affected by the data it is assumed that the predictors represent distinct observables the corresponding regression coefficients are assigned independent prior distributions for each regression coefficient subject to deletion from the model the prior distribution is a mixture of a point mass at 0 and a diffuse uniform distribution elsewhere that is a spike and slab distribution the random error component is assigned a normal distribution with mean 0 and standard deviation

source receiver compression scheme for full waveform seismic inversion we have developed a source receiver compression approach for reducing the computational time and memory usage of the acoustic and elastic full waveform inversions by detecting and quantifying the extent of redundancy in the data we assembled a reduced set of simultaneous sources and receivers that are weighted sums of the physical sources and receivers used in the survey because the numbers of these simultaneous sources and receivers could be significantly less than those of the physical sources and receivers the computational time and memory usage of any gradient type inversion method such as steepest descent nonlinear conjugate gradient contrast source inversion and quasi newton methods could be reduced the scheme is based on decomposing the data into their principal components using a singular value decomposition approach and the data reduction is done through the elimination of the small eigenvalues consequently this would suppress the effect of noise in the data moreover taking advantage

on the equivalence between kernel quadrature rules and random feature expansions we show that kernel based quadrature rules for computing integrals can be seen as a special case of random feature expansions for positive definite kernels for a particular decomposition that always exists for such kernels we provide a theoretical analysis of the number of required samples for a given approximation error leading to both upper and lower bounds that are based solely on the eigenvalues of the associated integral operator and match up to logarithmic terms in particular we show that the upper bound may be obtained from independent and identically distributed samples from a specific non uniform distribution while the lower bound if valid for any set of points applying our results to kernel based quadrature while our results are fairly general we recover known upper and lower bounds for the special cases of sobolev spaces moreover our results extend to the more general problem of full function approximations beyond simply computing an integral with results in l2 and l norm that match known results for special cases applying our results to random features we show an improvement of the number of random features needed to preserve the generalization guarantees for learning with lipshitz continuous losses

precision complexity and bayesian model determination summary in this paper a new criterion for model determination is proposed an explicit decision theoretic approach to model selection is employed to derive a bayes decision rule the operational characteristics of the criterion are discussed and consistency is shown

an absorbing boundary condition for acoustic wave propagation using a mesh free method abstractwe have developed an absorbing boundary condition for acoustic wave propagation using a mesh free method without sacrificing the flexibility of the mesh free framework when we simulate acoustic wave propagation using a numerical method artificial reflections from model edges induced by a truncated computational domain should be avoided although many absorbing boundary conditions have been developed most of them have been based on a regular latticed alignment of grids or nodes and the efficiency of such absorbing boundary conditions for irregular arrangement of grids or nodes has not been examined yet we have studied the artificial reflections generated at the boundaries of a model for a mesh free method and we have proposed a novel approach for suppressing the artifacts the method uses a hybrid approach with a transition zone in which the wavefield is estimated by a weighted average of solutions from the one and two way wave equations numerical experiments indicate that the proposed met

diagnostic measures for model criticism abstract we discuss the problem of model criticism with emphasis on developing summary diagnostic measures we approach model criticism by identifying possible troublesome features of the currently entertained model embedding the model in an elaborated model and measuring the value of elaborating this requires three elements a model elaboration a prior distribution and a utility function each triplet generates a different diagnostic measure we focus primarily on the measure given by a kullback leibler divergence between the marginal prior and posterior distributions on the elaboration parameter we also develop a linearized version of this diagnostic and use it to show that our procedure is related to other tools commonly used for model diagnostics such as bayes factors and the score function one attraction of this approach is that it allows model criticism to be performed jointly with parameter inference and prediction also this diagnostic approach aims at maintaining an exploratory nature t

massively parallel structured direct solver for equations describing time harmonic qp polarized waves in tti media abstractwe considered the discretization and approximate solutions of equations describing time harmonic qp polarized waves in 3d inhomogeneous anisotropic media the anisotropy comprises general tilted transversely isotropic symmetries we are concerned with solving these equations for a large number of different sources we considered higher order partial differential equations and variable order finite difference schemes to accommodate anisotropy on the one hand and allow higher order accuracy to control sampling rates for relatively high frequencies on the other hand we made use of a nested dissection based domain decomposition in a massively parallel multifrontal solver combined with hierarchically semiseparable matrix compression techniques the higher order partial differential operators and the variable order finite difference schemes require the introduction of separators with variable thickness in the nested dissection the development of these and their integration with the multifrontal

universality characteristic kernels and rkhs embedding of measures over the last few years two different notions of positive definite pd kernels universal and characteristic have been developing in parallel in machine learning universal kernels are proposed in the context of achieving the bayes risk by kernel based classification regression algorithms while characteristic kernels are introduced in the context of distinguishing probability measures by embedding them into a reproducing kernel hilbert space rkhs however the relation between these two notions is not well understood the main contribution of this paper is to clarify the relation between universal and characteristic kernels by presenting a unifying study relating them to rkhs embedding of measures in addition to clarifying their relation to other common notions of strictly pd conditionally strictly pd and integrally strictly pd kernels for radial kernels on d all these notions are shown to be equivalent

immersive boundary conditions theory implementation and examples abstractmany applications in computational geophysics involve the modeling of seismic wave propagation on a set of closely related subsurface models in such scenarios it is of interest to recompute the seismic wavefields locally only in the regions of change instead of in the full subsurface model we have developed a method for local acoustic wavefield recomputation that makes it possible to fully immerse a local modeling domain within a larger domain of arbitrary extent and complexity such that the wave propagation in the full domain is completely accounted for the method enables wavefield modeling on much smaller local domains while relying on the up front generation of a large number of green s functions and a wavefield extrapolation step at each time step of the simulation a kirchhoff helmholtz extrapolation integral is used to predict the interaction of the wavefield leaving the local domain with the exterior domain the outward propagating wavefield and the wavefield reentering the local d

3d scalar wave absorbing boundary conditions with optimal coefficients in the frequency space domain abstractto improve the computational efficiency for the solution of the 3d helmholtz equation in the frequency space domain high order compact forms of finite differences are preferred we applied a pointwise pade approximation to develop a 3d 27 point fourth order compact finite difference fd stencil in the grid interior with a space differentiated source term for the scalar wave equation this has similar high accuracy 4 5 grid points per the shortest wavelength to another 27 point fourth order fd stencil using a parsimonious mixed grid and staggered grid combination but is much simpler for absorbing boundary conditions abcs a damping zone is expensive and a perfectly matched layer can not be straightforwardly introduced into the compact fd form for the second order wave equation thus we developed 3d one way wave equation owwe abcs with adjustable coefficients they have different angle approximations and fd forms for the six faces twelve edges and eight corners in 3d models to fit wi

predictive model selection we consider the problem of selecting one model from a large class of plausible models a predictive bayesian viewpoint is advocated to avoid the specification of prior probabilities for the candidate models and the detailed interpretation of the parameters in each model using criteria derived from a certain predictive density and a prior specification that emphasizes the observables we implement the proposed methodology for three common problems arising in normal linear models variable subset selection selection of a transformation of predictor variables and estimation of a parametric variance function interpretation of the relative magnitudes of the criterion values for various models is facilitated by a calibration of the criteria relationships between the proposed criteria and other well known criteria are examined

a helmholtz iterative solver for 3d seismic imaging problems a preconditioned iterative solver for the 3d frequency domain wave equation applied to seismic problems is evaluated the preconditioner corresponds to an approximate inverse of a heavily damped wave equation deduced from the undamped wave equation the approximate inverse is computed with one multigrid cycle numerical results show that the method is robust and that the number of iterations increases roughly linearly with frequency when the grid spacing is adapted to keep a constant number of discretization points per wavelength to evaluate the relevance of this iterative solver the number of floating point operations required for two imaging problems are roughly evaluated this rough estimate indicates that the time domain migration approach is more than one order of magnitude faster the full wave form tomography based on a least squares formulation and a scale separation approach has the same complexity in both domains

a dispersion minimizing finite difference scheme and preconditioned solver for the 3d helmholtz equation in this paper a new 27 point finite difference method is presented for solving the 3d helmholtz equation with perfectly matched layer pml which is a second order scheme and pointwise consistent with the equation an error analysis is made between the numerical wavenumber and the exact wavenumber and a refined choice strategy based on minimizing the numerical dispersion is proposed for choosing weight parameters a full coarsening multigrid based preconditioned bi cgstab method is developed for solving the linear system stemming from the helmholtz equation with pml by the finite difference scheme the shifted laplacian is extended to precondition the 3d helmholtz equation and a spectral analysis is given the discrete preconditioned system is solved by the bi cgstab method with a multigrid method used to invert the preconditioner approximately full coarsening multigrid is employed and a new matrix based prolongation operator is constructed accordingly numerical results are presented to demonstrate the efficiency of both the new 27 point finite difference scheme with refined parameters and the preconditioned bi cgstab method with the 3d full coarsening multigrid

nonparametric stochastic approximation with large step sizes we consider the random design least squares regression problem within the reproducing kernel hilbert space rkhs framework given a stream of independent and identically distributed input output data we aim to learn a regression function within an rkhs mathcal h even if the optimal predictor i e the conditional expectation is not in mathcal h in a stochastic approximation framework where the estimator is updated after each observation we show that the averaged unregularized least mean square algorithm a form of stochastic gradient given a sufficient large step size attains optimal rates of convergence for a variety of regimes for the smoothnesses of the optimal prediction function and the functions in mathcal h

waveform inversion including well constraints anisotropy and attenuation the goal of full waveform inversion wfi is to derive high fidelity earth models for seismic imaging from the full waveforms of the acquired seismic data the attractiveness of wfi lies mainly in its limited number of approximations at least in a theoretical sense in contrast to other model determination techniques such as semblance or ray based tomography despite this various methodologies must be utilized to make the technique viable with today s computing technology and restrictions of seismic acquisition these are collectively referred to as waveform inversion strategies and in this article we discuss mainly regularization and preconditioning strategies as the wavefields need to be accurately modeled to represent the kinematics of all the waves during wfi iterations the inclusion of anisotropy often helps to improve the wfi results in the first section of this article we introduce forward modeling and its adjoint based on acoustic wave equations in vertical transversely isotropic vti med

3d frequency domain seismic inversion with controlled sloppiness seismic waveform inversion aims at obtaining detailed estimates of subsurface medium parameters such as the spatial distribution of soundspeed from multiexperiment seismic data a formulation of this inverse problem in the frequency domain leads to an optimization problem constrained by a helmholtz equation with many right hand sides application of this technique to industry scale problems faces several challenges first we need to solve the helmholtz equation for high wave numbers over large computational domains second the data consist of many independent experiments leading to a large number of pde solves this results in high computational complexity both in terms of memory and cpu time as well as input output costs finally the inverse problem is highly nonlinear and a lot of art goes into preprocessing and regularization ideally an inversion needs to be run several times with different initial guesses and or tuning parameters in this paper we discuss the requirements of the various components pde solver optimization method dots when applied to large scale three dimensional seismic waveform inversion and combine several existing approaches into a flexible inversion scheme for seismic waveform inversion the scheme is based on the idea that in the early stages of the inversion we do not need all the data or very accurate pde solves we base our method on an existing preconditioned krylov solver carp cg and use ideas from stochastic optimization to formulate a gradient based quasi newton optimization algorithm that works with small subsets of the right hand sides and uses inexact pde solves for the gradient calculations we propose novel heuristics to adaptively control both the accuracy and the number of right hand sides we illustrate the algorithms on synthetic benchmark models for which significant computational gains can be made without being sensitive to noise and without losing the accuracy of the inverted model

accuracy of finite difference and finite element modeling of the scalar and elastic wave equations numerical solutions of the scalar and elastic wave equations have greatly aided geophysicists in both forward modeling and migration of seismic wave fields in complicated geologic media and they promise to be invaluable in solving the full inverse problem this paper quantitatively compares finite difference and finite element solutions of the scalar and elastic hyperbolic wave equations for the most popular implicit and explicit time domain and frequency domain techniques in addition to versatility and ease of implementation it is imperative that one choose the most cost effective solution technique for a fixed degree of accuracy to be of value a solution technique must be able to minimize 1 numerical attenuation or amplification 2 polarization errors 3 numerical anisotropy 4 errors in phase and group velocities 5 extraneous numerical parasitic modes 6 numerical diffraction and scattering and 7 errors and transmission coefficients this paper shows that in homogeneous media the

a new slant on seismic imaging migration and integral geometry a new approach to seismic migration formalizes the classical diffraction or common tangent stack by relating it to linearized seismic inversion and the generalized radon transform this approach recasts migration as the problem of reconstructing the earth s acoustic scattering potential from its integrals over isochron surfaces the theory rests on a solution of the wave equation with the geometrical optics green function and an approximate inversion formula for the generalized radon transform the method can handle both complex velocity models and nearly arbitrary configurations of sources and receivers in this general case the method can be implemented as a weighted diffraction stack with the weights determined by tracing rays from image points to the experiment s sources and receivers when tested on a finite difference simulation of a deviated well vertical seismic profile a hybrid experiment which is difficult to treat with conventional wave equation methods the algorithm accurately reconstructed faulted earth models analytical reconstruction formulas are derived from the general formula for zero offset and fixed offset surface experiments in which the background velocity is constant the zero offset inversion formula resembles standard kirchhoff migration our analysis provides a direct connection between the experimental setup source and receiver positions source wavelet background velocity and the spatial resolution of the reconstruction synthetic examples illustrate that the lateral resolution in seismic images is described well by the theory and is improved greatly by combining surface data and borehole data the best resolution is obtained from a zero offset experiment that surrounds the region to be imaged

comparison of multifrequency selection strategies for simultaneous source full waveform inversion abstractthe computational cost of full waveform inversion fwi is a major obstacle to estimating the velocity model for large scale problems one way to reduce the overall cost of waveform inversion is by adopting a simultaneous source strategy in other words multiple sources are simultaneously fired to simulate supershot gathers and thereby reduce the number of seismic modeling simulations that are performed during the inversion however the use of simultaneous sources introduces crosstalk artifacts that arise from the interference among the sources that constitute a supershot we analyzed the influence of different simultaneous multifrequency selection strategies on crosstalk artifacts our analysis focused on a frequency domain fwi algorithm that is implemented with simultaneous sources that are randomly encoded with random time shifts in the multiscale conventional fwi strategies a finite set of discrete frequencies was selected and the inversion was carried out sequentially from low to high fr

velocity model building by 3d frequency domain full waveform inversion of wide aperture seismic data we assessed 3d frequency domain fd acoustic fullwaveforminversionfwidataasatooltodevelophigh resolution velocity models from low frequency global offset data the inverse problem was posed as a classic leastsquares optimization problem solved with a steepest descent method inversion was applied to a few discrete frequencies allowing management of a limited subset of the 3d data volume the forward problem was solved with a finite difference frequency domain method based on a massively paralleldirectsolver allowingefficientmultiple shotsimulations the inversion code was fully parallelized for distributedmemory platforms taking advantage of a domain decomposition of the modeled wavefields performed by the direct solver after validation on simple synthetic tests fwi was applied to two targets channel and thrust system of the 3d seg eageoverthrustmodel correspondingto3ddomains of 78 752 25 km and 13 513 54 65 km respectively the maximum inverted frequencies are 15 and 7 hz for the two applications amaximum of 30 dual core biprocessor nodes with 8 gb of shared memory per node were used for the second target the main structures were imaged successfully at a resolution scale consistent with the inverted frequencies our study confirms the feasibility of 3d frequency domain fwi of global offset data on large distributed memory platforms to develop high resolution velocity models these high velocity models may provide accurate macromodelsforwave equationprestackdepthmigration

3d laplace domain waveform inversion using a low frequency time domain modeling algorithm we have developed a laplace domain full waveform inversion technique based on a time domain finite difference modeling algorithm for efficient 3d inversions theoretically the laplace domain green s function multiplied by a constant can be obtained regardless of the frequency content in the time domain source wavelet therefore we can use low frequency sources and large grids for efficient modeling in the time domain we laplace transform time domain seismograms to the laplace domain and calculate the residuals in the laplace domain then we back propagate the laplace domain residuals in the time domain using a predefined time domain source wavelet with the amplitude of the residuals the back propagated wavefields are transformed to the laplace domain again to update the velocity model the inversion results are long wavelength velocity models on large grids similar to those obtained by the original approach based on laplace domain modeling inversion examples with 2d gulf of mexico field data revealed that the method yielded long wavelength velocity models comparable with the results of the original laplace domain inversion methods a 3d seg eage salt model example revealed that the 3d laplace domain inversion based on time domain modeling method can be more efficient than the inversion based on laplace domain modeling using an iterative linear system solver

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

3d finite difference frequency domain modeling of controlled source electromagnetic data direct solution and optimization for high accuracy three dimensional modeling of marine controlled source electromagnetic csem data is vital to improve the understanding of electromagnetic em responses collected in increasingly complex geologic settings a modeling tool for simulating 3d marine csem surveys based on a finite difference discretization of the helmholtz equation for the electric fields has been developed optimizations for csem simulations include the use of a frequency domain technique a staggering scheme that reduces inaccuracies especially for horizontal electric dipole sources located near the seafloor and a new interpolation technique that provides highly accurate em field values for receivers located in the immediate vicinity of the seafloor source singularities are eliminated through a secondary field approach in which the primary fields are computed analytically for a homogeneous or a 1d layered background the secondary fields are computed using the finite difference technique exploiting recent advances in computer technology and algorithmic developments the system of finite difference equations is solved using the mumps direct matrix solver in combination with the other optimizations this allows accurate em field computations for moderately sized models on small computer clusters the explicit availability of matrix factorizations is advantageous for multisource modeling and makes the algorithm well suited for future use within an inversion scheme comparisons of simulated data for 1 1d models to data generated using a 1d reflectivity technique and 2 3d models to published 3d data demonstrate the accuracy and benefits of the approach

an analysis of the difference of code lengths between two step codes based on mdl principle and bayes codes in this paper we discuss the difference in code lengths between the code based on the minimum description length mdl principle the mdl code and the bayes code under the condition that the same prior distribution is assumed for both codes it is proved that the code length of the bayes code is smaller than that of the mdl code by o 1 or o 1 for the discrete model class and by o 1 for the parametric model class because we can assume the same prior for the bayes code as for the code based on the mdl principle it is possible to construct the bayes code with equal or smaller code length than the code based on the mdl principle from the viewpoint of mean code length per symbol unit compression rate the bayes code is asymptotically indistinguishable from the mdl two stage codes

accurate viscoelastic modeling by frequency domain finite differences using rotated operators the viscoelastic wave equation is an integro differential equation that requires special methods when using time domain numerical finite difference methods in the frequency domain the integral terms are easily represented by complex valued elastic media properties there are further significant advantages to using the frequency domain if the forward or the inverse problem requires modeling or inverting a large number of prestack source gathers numerical modeling is expensive for seismic data because of the large number of wavelenghths typically separating sources from receivers which results in a need for a large number of grid points a major obstacle to using frequency domain methods is the consequent storage requirements to reduce these we maximize the accuracy and simultaneously minimize the spatial extent of the numerical operators we achieve this by extending earlier published methods introduced for the viscoacoustic case to the viscoelastic case this requires the formulation of two new nume

3d frequency domain finite difference viscoelastic wave modeling using weighted average 27 point operators with optimal coefficients abstractexperimental data suggest that the viscoelastic behavior of rocks is more easily and accurately described in the frequency domain than in the time domain supporting the idea of simulating seismic wave propagation in the frequency domain we evaluated weighted averaged 27 point finite difference operators for 3d viscoelastic wave modeling in the frequency domain within the proposed framework we developed general equations for normalized phase velocities that can be used with arbitrary finite difference operators three sets of weighting coefficients for second order central finite difference operators that minimize the numerical dispersion for up to five grid points per wavelength were found using a damped least squares ls criterion as well as a global optimization scheme based on l1 and l2 norm criteria the three sets produced very similar dispersion curves and improvement provided by global optimization appeared marginal in this respect we also evaluated a discrete form for the heterogen

boosting with early stopping convergence and consistency boosting is one of the most significant advances in machine learning for classification and regression in its original and computationally flexible version boosting seeks to minimize empirically a loss function in a greedy fashion the resulting estimator takes an additive function form and is built iteratively by applying a base estimator or learner to updated samples depending on the previous iterations an unusual regularization technique early stopping is employed based on cv or a test set this paper studies numerical convergence consistency and statistical rates of convergence of boosting with early stopping when it is carried out over the linear span of a family of basis functions for general loss functions we prove the convergence of boosting s greedy optimization to the infinimum of the loss function over the linear span using the numerical convergence result we find early stopping strategies under which boosting is shown to be consistent based on i i d samples and we obtain bounds on the rates of convergence for boosting estimators simulation studies are also presented to illustrate the relevance of our theoretical results for providing insights to practical aspects of boosting as a side product these results also reveal the importance of restricting the greedy search step sizes as known in practice through the work of friedman and others moreover our results lead to a rigorous proof that for a linearly separable problem adaboost with e 0 step size becomes an l 1 margin maximizer when left to run to convergence

online learning with kernels kernel based algorithms such as support vector machines have achieved considerable success in various problems in batch setting where all of the training data is available in advance support vector machines combine the so called kernel trick with the large margin idea there has been little use of these methods in an online setting suitable for real time applications in this paper we consider online learning in a reproducing kernel hilbert space by considering classical stochastic gradient descent within a feature space and the use of some straightforward tricks we develop simple and computationally efficient algorithms for a wide range of problems such as classification regression and novelty detection in addition to allowing the exploitation of the kernel trick in an online setting we examine the value of large margins for classification in the online setting with a drifting target we derive worst case loss bounds and moreover we show the convergence of the hypothesis to the minimizer of the regularized risk functional we present some experimental results that support the theory as well as illustrating the power of the new algorithms for online novelty detection

an efficient multiscale method for time domain waveform tomography this efficient multiscale method for time domain waveform tomography incorporates filters that are more efficient than hamming window filters a strategy for choosing optimal frequency bands is proposed to achieve computational efficiency in the time domain a staggered grid explicit finite difference method with fourth order accuracy in space and second order accuracy in time is used for forward modeling and the adjoint calculation the adjoint method is utilized in inverting for an efficient computation of the gradient directions in the multiscale approach multifrequency data and multiple grid sizes are used to overcome somewhat the severe local minima problem of waveform tomography the method is applied successfully to 1d and 2d heterogeneous models it can accurately recover low and high wavenumber components of the velocity models the inversion result for the 2d model demonstrates that the multiscale method is computationally efficient and converges faster than a conventional single scale method

3d finite difference frequency domain modeling of visco acoustic wave propagation using a massively parallel direct solver a feasibility study we present a finite difference frequency domain method for 3d visco acoustic wave propagation modeling in the frequency domain the underlying numerical problem is the resolution of a large sparse system of linear equations whose right hand side term is the source this system is solved with a massively parallel direct solver we first present an optimal 3d finite difference stencil for frequency domain modeling the method is based on a parsimonious staggered grid method differential operators are discretized with second order accurate staggered grid stencils on different rotated coordinate systems to mitigate numerical anisotropy an antilumped mass strategy is implemented to minimize numerical dispersion the stencil incorporates 27 grid points and spans two grid intervals dispersion analysis shows that four grid points per wavelength provide accurate simulations in the 3d domain to assess the feasibility of the method for frequency domain full waveform inversion we computed simulations in the 3d seg eage overthrust model for frequencies 5 7 and 10 hz results confirm the huge memory requirement of the factorization several hundred figabytes but also the cpu efficiency of the resolution phase few seconds per shot heuristic scalability analysis suggests that the memory complexity of the factorization is o 35n 4 for a n 3 grid our method may provide a suitable tool to perform frequency domain full waveform inversion using a large distributed memory platform further investigation is still necessary to assess more quantitatively the respective merits and drawbacks of time and frequency domain modeling of wave propagation to perform 3d full waveform inversion

methods and criteria for model selection model selection is an important part of any statistical analysis and indeed is central to the pursuit of science in general many authors have examined the question of model selection from both frequentist and bayesian perspectives and many tools for selecting the best model have been suggested in the literature this paper considers the various proposals from a bayesian decision theoretic perspective

full wave directional illumination analysis in the frequency domain directional illumination analysis based on one way wave equations has been studied extensively however its inherent limitations e g one way propagation wide angle error and amplitude inaccuracy can severely hinder its applications for accurate survey design and true reflection imaging corrections in complex media we have analyzed the illumination in the frequency domain using full two way wave propagators considering the extensive computation and huge storage required for time domain methods and the fact that the illumination is frequency dependent this full wave analysis can provide frequency dependent full angle true amplitude illumination not only for the downgoing waves but also for the upgoing waves including turning waves and reflected waves two methods were considered to decompose the full wavefield into the local angle domain a direct full dimensional decomposition and more efficient split step decomposition composed of three lower dimensional decompositions the results of illumination analysis demonstrated the advantages of this method the two decomposition methods produced similar results