genetic algorithm approach on multi criteria minimum spanning tree problem minimum spanning tree mst problem is of high importance in network optimization the multi criteria mst mc mst is a more realistic representation of the practical problem in the real world but it is difficult for the traditional network optimization technique to deal with in this paper a genetic algorithm ga approach is developed to deal with this problem without neglecting its network topology the proposed method adopts the prufer number as the tree encoding and applies the multiple criteria decision making mcdm technique and nondominated sorting technique to make the ga search give out all pareto optimal solutions either focused on the region near the ideal point or distributed all along the pareto frontier compared with the enumeration method of pareto optimal solution the numerical analysis shows the efficiency and effectiveness of the ga approach on the mc mst problem

superconvergence of the local discontinuous galerkin method for the linearized korteweg de vries equation we study the superconvergence property of the local discontinuous galerkin ldg method for solving the linearized korteweg de vries kdv equation we prove that if the piecewise p k polynomials with k 1 are used the ldg solution converges to a particular projection of the exact solution with the order k 3 2 when the upwind flux is used for the convection term and the alternating flux is used for the dispersive term numerical examples are provided at the end to support the theoretical results

compensating q effects in viscoelastic media by adjoint based least squares reverse time migration low values of p and s wave quality factors qp and qs result in strong intrinsic seismic attenuation in both seismic modeling and imaging we use a linearized waveform inversion approach by generalizing least squares reverse time migration lsrtm for viscoelastic media q lsrtm to compensate the attenuation loss for both p and s images we use the first order particle velocity stress and memory variable equations with explicit q in the formulations based on the generalized standard linear solid gsls as the forward modeling operator the linearized two way viscoelastic modeling operator is obtained with modulus perturbations introduced for the relaxed p and s moduli the viscoelastic adjoint operator and the p and s imaging conditions for modulus perturbations are derived using the adjoint state method and an augmented lagrangian functional q lsrtm solves the viscoelastic linearized modeling operator for generating synthetic data and the adjoint operator is used for back propagating the data re

expander graphs and their applications a major consideration we had in writing this survey was to make it accessible to mathematicians as well as to computer scientists since expander graphs the protagonists of our story come up in numerous and often surprising contexts in both fields but perhaps we should start with a few words about graphs in general they are of course one of the prime objects of study in discrete mathematics however graphs are among the most ubiquitous models of both natural and human made structures in the natural and social sciences they model relations among species societies companies etc in computer science they represent networks of communication data organization computational devices as well as the flow of computation and more in mathematics cayley graphs are useful in group theory graphs carry a natural metric and are therefore useful in geometry and though they are just one dimensional complexes they are useful in certain parts of topology e g knot theory in statistical physics graphs can represent local connections between interacting parts of a system as well as the dynamics of a physical process on such systems the study of these models calls then for the comprehension of the significant structural properties of the relevant graphs but are there nontrivial structural properties which are universally important expansion of a graph requires that it is simultaneously sparse and highly connected expander graphs were first defined by bassalygo and pinsker and their existence first proved by pinsker in the early 70s the property of being an expander seems significant in many of these mathematical computational and physical contexts it is not surprising that expanders are useful in the design and analysis of communication networks what is less obvious is that expanders have surprising utility in other computational settings such as in the theory of error correcting codes and the theory of pseudorandomness in mathematics we will encounter e g their role in the study of metric embeddings and in particular in work around the baum connes conjecture expansion is closely related to the convergence rates of markov chains and so they play a key role in the study of monte carlo algorithms in statistical mechanics and in a host of practical computational applications the list of such interesting and fruitful connections goes on and on with so many applications we will not even

applying finite element analysis to the memory variable formulation of wave propagation in anelastic media finite element methods are applied to solution of seismic wave motion in linear viscoelastic media using the memory variable formalism the displacements are represented as a superposition of a set of basis functions it is shown that if memory variables are represented using the spatial derivatives of those basis functions rather than the basis functions themselves the equations to be solved are simpler and require less computer memory using this formulation results for sh waves in one and two dimensions are calculated using a simple explicit finite element in space finite difference in time scheme these results agree with those found with a method of lines solution results in a homogeneous medium also agree with the frequency domain solutions of kjartansson s constant q method

quick and accurate q parameterization in viscoelastic wave modeling we introduce a procedure for including the attenuation factor q in a consistent manner in seismic modeling and show 3d examples the q fitting over a chosen frequency band involves two algorithms the first creates starting values of relaxation times and the second does nonlinear inversion using the results of the first as initial values the resulting q function gives a good approximation to a constant q over the chosen frequency band the algorithm is combined with a finite difference f d code that includes topographies in 3d seismic media the velocity stress formulation for viscoelastic wave modeling is used with an arbitrary number of relaxation mechanisms to model a desired q behavior these equations are discretized by high order f ds in the interior of the medium and we gradually reduce the f d order to two at the stress free surface where we implement our free surface boundary conditions the seismic f d algorithm is applied to a marine seismic experiment with and without viscoelasticity to emphasize the importance of including physical attenuation and dispersion in seismic modeling their inclusion even for marine surveys is clearly important for lossy ocean bottoms our procedure for more accurate modeling of physical dispersion and attenuation may increase future motivation to include viscoelasticity in seismic inversion

finite difference modeling of viscoelastic and anisotropic wave propagation using the rotated staggered grid we describe the application of the rotated staggered grid rsg finite difference technique to the wave equations for anisotropic and viscoelastic media the rsg uses rotated finite difference operators leading to a distribution of modeling parameters in an elementary cell where all components of one physical property are located only at one single position this can be advantageous for modeling wave propagation in anisotropic media or complex media including high contrast discontinuities because no averaging of elastic moduli is needed the rsg can be applied both to displacement stress and to velocity stress finite difference fd schemes whereby the latter are commonly used to model viscoelastic wave propagation with a von neumann style anlysis we estimate the dispersion error of the rsg scheme in general anisotropic media in three different simulation examples all based on previously published problems we demonstrate the application and the accuracy of the proposed numerical approach

three dimensional gravity inversion using graph theory to delineate the skeleton of homogeneous sources abstractwe developed a new strategy based on graph theory concepts to invert gravity data using an ensemble of simple point masses our method consisted of a genetic algorithm with elitism to generate a set of possible solutions each estimate was associated to a graph to solve the minimum spanning tree mst problem to produce unique and stable estimates we restricted the position of the point masses by minimizing the statistical variance of the distances of an mst jointly with the data misfit function during the iterations of the genetic algorithm hence the 3d spatial distribution of the point masses identified the skeleton of homogeneous gravity sources in addition our method also gave an estimation of the anomalous mass of the source so together with the anomalous mass the skeleton could aid other 3d methods with promising geometric a priori parameters several tests with different values of regularizing parameter were made to bespeak this new regularizing strategy the inversion results ap

finite difference modeling of viscoelastic materials with quality factors of arbitrary magnitude to minimize acoustic noise designers of sonic logging tools often consider coatings of viscoelastic materials with very high attenuation properties efficient finite difference modeling of viscoelastic materials is a topic of current research to model viscoelastic materials in the time domain through finite differences efficiently one needs to replace the time convolution which enters in the stress strain relations by a set of first order differential equations this procedure is equivalent to computing a rational approximation of a certain form to the frequency dependent complex modulus of viscoelasticity known schemes for computing such approximations are designed to treat materials with low attenuation such as underground formations but fail to produce accurate or even physically meaningful results for highly attenuative materials we propose a novel scheme that allows one to construct for a given frequency range a uniformly optimal rational approximation for the most widely used model of mater

near surface attenuation estimation using wave propagation modeling the effect of the near surface on seismic land data can be so severe that static corrections are insufficient full waveform inversion followed by redatuming may be an alternative but inversion will work only if the starting model is sufficiently close to the true model as a first step toward determining a viscoelastic near surface model we assume that existing methods can provide a horizontally layered velocity and density model because near surface attenuation is strongest we propose a method to estimate the p wave attenuation based on viscoacoustic finite difference modeling we compare energy decay along traveltime curves of reflection and refraction events in the modeled and observed seismic data for a range of attenuation parameters the best match provides an estimate of the attenuation first we estimate only the attenuation of the top layer and study the sensitivity to depth and velocity perturbations then we consider multiple layers we apply the method to synthetic and real data and investigate the effect of source wavelet and topography the method is robust against depth and velocity perturbations smaller than 10 the results are sensitive to the source wavelet incorporating the surface topography in the computed traveltimes reduces the uncertainty of the attenuation estimates especially for deeper layers

viscoacoustic waveform inversion of velocity structures in the time domain abstractbecause of the conversion of elastic energy into heat seismic waves are attenuated and dispersed as they propagate the attenuation effects can reduce the resolution of velocity models obtained from waveform inversion or even cause the inversion to produce incorrect results using a viscoacoustic model consisting of a single standard linear solid we discovered a theoretical framework of viscoacoustic waveform inversion in the time domain for velocity estimation we derived and found the viscoacoustic wave equations for forward modeling and their adjoint to compensate for the attenuation effects in viscoacoustic waveform inversion the wave equations were numerically solved by high order finite difference methods on centered grids to extrapolate seismic wavefields the finite difference methods were implemented satisfying stability conditions which are also presented numerical examples proved that the forward viscoacoustic wave equation can simulate attenuative behaviors very well in amplitude

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

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

the solution of nonlinear inverse problems and the levenberg marquardt method although the levenberg marquardt damped least squares method is an extremely powerful tool for the iterative solution of nonlinear problems its theoretical basis has not been described adequately in the literature this is unfortunate because levenberg and marquardt approached the solution of nonlinear problems in different ways and presented results that go far beyond the simple equation that characterizes the method the idea of damping the solution was introduced by levenberg who also showed that it is possible to do that while at the same time reducing the value of a function that must be minimized iteratively this result is not obvious although it is taken for granted moreover levenberg derived a solution more general than the one currently used marquardt started with the current equation and showed that it interpolates between the ordinary least squares method and the steepest descent method in this tutorial the two papers are combined into a unified presentation which will help the reader gain a better understanding of what happens when solving nonlinear problems because the damped least squares and steepest descent methods are intimately related the latter is also discussed in particular in its relation to the gradient when the inversion parameters have the same dimensions and units the direction of steepest descent is equal to the direction of minus the gradient in other cases it is necessary to introduce a metric i e a definition of distance in the parameter space to establish a relation between the two directions although neither levenberg nor marquardt discussed these matters their results imply the introduction of a metric some of the concepts presented here are illustrated with the inversion of synthetic gravity data corresponding to a buried sphere of unknown radius and depth finally the work done by early researchers that rediscovered the damped least squares method is put into a historical context

dithering of absorbers for efficient finite difference modeling of viscoelastic wave propagation viscoelastic modeling traditionally uses multiple relaxation mechanisms absorbers per wave type in each cell of the numerical grid we show that one absorber per cell suffices when the absorbers of different relaxation frequencies and absorption strengths are dithered in a supercell i e a very small area of several nearby cells the relaxation frequencies used in a supercell can be preselected to cover the bandwidth required in the modeling and distributed locally in a random manner to minimize numerical scattering the absorption strengths however must be found iteratively to yield an average absorption that matches the desired medium over the bandwidth we have tested the dithered absorber method for modeling a variety of 2 d media using a staggered grid finite difference scheme for a uniform medium our result agrees well with the exact analytic solution for inhomogeneous models the dithered absorber solution agrees well with the finite difference solution computed with five absorbers per cel

modeling of a constant q methodology and algorithm for an efficient and optimally inexpensive viscoelastic technique linear anelastic phenomena in wave propagation problems can be well modeled through a viscoelastic mechanical model consisting of standard linear solids in this paper we present a method for modeling of constant q as a function of frequency based on an explicit closed formula for calculation of the parameter fields several standard linear solids connected in parallel can be tuned through a single parameter to yield an excellent constant q approximation the proposed method enables substantial savings in computations and memory requirements experiments show that the new method also yields higher accuracy in the modeling of q than e g the pade approximant method

least squares migration data domain versus image domain using point spread functions abstract conventional amplitude inversion assumes that the migrated image preserves relative amplitude information however illumination effects caused by complex geologic settings undersampled acquisition geometry and limited recording aperture pose a challenge to even the most advanced imaging algorithms in addition standard depth migration images can suffer from lack of resolution caused by wavelet stretch attenuation and suboptimal deghosting least squares migration lsm can mitigate many of these problems and produce better resolved migration images suitable for avo inversion however whether formulated in the data domain or the image domain lsm is an inversion algorithm and is sensitive to inaccuracies in the source wavelet velocity model data preprocessing and the propagator used practical considerations to mitigate these problems under nonideal conditions and cost reduction strategies differ between the data and image domain formulations the relative merits of each approach are ev

two efficient modeling schemes for fractional laplacian viscoacoustic wave equation abstractrecently a decoupled fractional laplacian viscoacoustic wave equation has been developed based on the constant q model to describe wave propagation in heterogeneous media we have developed two efficient modeling schemes to solve the decoupled fractional laplacian viscoacoustic wave equation both schemes can cope with spatial variable order fractional laplacians conveniently and thus are applicable for modeling viscoacoustic wave propagation in heterogeneous media both schemes are based on fast fourier transform and have a spectral accuracy in space the first scheme solves a modified wave equation with constant order fractional laplacians instead of spatial variable order fractional laplacians due to separate discretization of space and time the first scheme has only first order accuracy in time differently the second scheme is based on an analytical wave propagator and has a higher accuracy in time to increase computational efficiency of the second modeling scheme we have adopted the

numerical modeling of transient two dimensional viscoelastic waves this paper deals with the numerical modeling of transient mechanical waves in linear viscoelastic solids dissipation mechanisms are described using the generalized zener model no time convolutions are required thanks to the introduction of memory variables that satisfy local in time differential equations by appropriately choosing the relaxation parameters it is possible to accurately describe a large range of materials such as solids with constant quality factors the evolution equations satisfied by the velocity the stress and the memory variables are written in the form of a first order system of pdes with a source term this system is solved by splitting it into two parts the propagative part is discretized explicitly using a fourth order ader scheme on a cartesian grid and the diffusive part is then solved exactly jump conditions along the interfaces are discretized by applying an immersed interface method numerical experiments of wave propagation in viscoelastic and fluid media show the efficiency of this numerical modeling for dealing with challenging problems such as multiple scattering configurations

an efficient seismic modeling in viscoelastic isotropic media abstractenergy is absorbed and attenuated when seismic waves propagate in real earth media hence the viscoelastic medium needs to be considered there are many ways to construct the viscoelastic body in which the generalized standard linear viscoelastic body is the most representative one for viscoelastic wave propagation and imaging it is very important to obtain a compact and efficient viscoelastic equation because of this we derived a set of simplified viscoelastic equations in isotropic media on the basis of the standard linear solid body and the constitutive relation for a linear viscoelastic isotropic solid the simplified equations were composed of the linear equations of momentum conservation the stress strain relations and the memory variable equations during the derivation of the equations the lame differentiation matrix which has a similar form to the stiffness matrix and indicates the relations between viscoelastic and elastic stiffness matrices was introduced to simplify the memo

finite difference modeling of electromagnetic wave propagation in dispersive and attenuating media realistic modeling of electromagnetic wave propagation in the radar frequency band requires a full solution of maxwell s equations as well as an adequate description of the material properties we present a finite difference time domain fdtd solution of maxwell s equations that allows accounting for the frequency dependence of the dielectric permittivity and electrical conductivity typical of many near surface materials this algorithm is second order accurate in time and fourth order accurate in space conditionally stable and computationally only marginally more expensive than its standard equivalent without frequency dependent material properties empirical rules on spatial wavefield sampling are derived through systematic investigations of the influence of various parameter combinations on the numerical dispersion curves since this algorithm intrinsically models energy absorption efficient absorbing boundaries are implemented by surrounding the computational domain by a thin 2 dominant waveleng

q least squares reverse time migration with viscoacoustic deblurring filters abstractviscoacoustic least squares reverse time migration also denoted as q lsrtm linearly inverts for the subsurface reflectivity model from lossy data compared with conventional migration methods it can compensate for the amplitude loss in the migrated images due to strong subsurface attenuation and can produce reflectors that are accurately positioned in depth however the adjoint q propagators used for backward propagating the residual data are also attenuative thus the inverted images from q lsrtm with a small number of iterations are often observed to have lower resolution when compared with the benchmark acoustic lsrtm images from acoustic data to increase the resolution and accelerate the convergence of q lsrtm we used viscoacoustic deblurring filters as a preconditioner for q lsrtm these filters can be estimated by matching a simulated migration image to its reference reflectivity model numerical tests on synthetic and field data demonstrate that q lsrtm combined with viscoacoustic de

viscoelastic finite difference modeling real earth media disperse and attenuate propagating mechanical waves this anelastic behavior can be described well by a viscoelastic model we have developed a finite difference simulator to model wave propagation in viscoelastic media the finite difference method was chosen in favor of other methods for several reasons finite difference codes are more portable than for example pseudospectral codes moreover finite difference schemes provide a convenient environment in which to define complicated boundaries a staggered scheme of second order accuracy in time and fourth order accuracy in space appears to be optimally efficient because of intrinsic dispersion no fixed grid points per wavelength rule can be given instead we present tables which enable a choice of grid parameters for a given level of accuracy since the scheme models energy absorption natural and efficient absorbing boundaries may be implemented merely by changing the parameters near the grid boundary the viscoelastic scheme is

waveform inversion for attenuation estimation in anisotropic media abstractrobust estimation of attenuation coefficients remains a challenging problem especially for heterogeneous anisotropic media here we apply waveform inversion wi to perform attenuation analysis in heterogeneous vti transversely isotropic with a vertical symmetry axis media a time domain finite difference algorithm based on the standard linear solid model simulates nearly constant quality factor values in a specified frequency band we employ the adjoint state method to derive the gradients of the objective function based on the born approximation four parameters describing the attenuation coefficients of p and sv waves are updated simultaneously with a quasi newton optimization algorithm to remove the time shifts between the modeled and observed data caused by velocity errors we apply a local similarity technique the inversion still requires a sufficiently accurate velocity model to minimize the trade off between the contributions of velocity and attenuation to amplitudes the inversion

viscoacoustic modeling and imaging using low rank approximation abstracta constant q wave equation involving fractional laplacians was recently introduced for viscoacoustic modeling and imaging this fractional wave equation has a convenient mixed domain space wavenumber formulation which involves the fractional laplacian operators with a spatially varying power we have applied the low rank approximation to the mixed domain symbol which enables a space variable attenuation specified by the variable fractional power of the laplacians using the proposed approximation scheme we formulated the framework of the q compensated reverse time migration q rtm for attenuation compensation numerical examples using synthetic data demonstrated the improved accuracy of using low rank wave extrapolation with a constant q fractional laplacian wave equation for seismic modeling and migration in attenuating media low rank q rtm applied to viscoacoustic data is capable of producing images comparable in quality with those produced by conventional rtm from acoustic data