asymptotic formulas for the eigenvalues of the timoshenko beam abstract asymptotic formulas are derived for the eigenvalues of a free ended timoshenko beam which has variable mass density and constant beam parameters otherwise these asymptotic formulas show how the eigenvalues and hence how the natural frequencies of such a beam depend on the material and geometric parameters which appear as coefficients in the timoshenko differential equations

heights of heegner cycles and derivatives of l series 0 introduction 99 1 intersections and heights 103 2 kuga sato varieties and cm cycles 117 3 heights of cm cycles 124 4 heights of heegner cycles 137 5 proof of the main identity and the consequences 147 references 151

flux normalized wavefield decomposition and migration of seismic data separation of wavefields into directional components can be accomplished by an eigenvalue decomposition of the accompanying system matrix in conventional pressurenormalized wavefield decomposition the resulting one way wave equations contain an interaction term which depends on the reflectivity function applying directional wavefield decomposition using flux normalized eigenvalue decomposition and disregarding interaction between up and downgoing wavefields these interaction terms were absent by also applying a correction term for transmission loss the result was an improved estimate of the up and downgoing wavefields in the wave equation angle transform a crosscorrelation function in local offset coordinates was fourier transformed to produce an estimate of reflectivity as a function of slowness or angle we normalized this wave equation angle transform with an estimate of the planewave reflection coefficient the flux normalized oneway wave propagation scheme was applied to imaging and to the normalized wave equation angle transform on synthetic and field data this proved the effectiveness of the new methods

application of unsplit convolutional perfectly matched layer for scalar arbitrarily wide angle wave equation abstracta classical split perfectly matched layer pml method has recently been applied to the scalar arbitrarily wide angle wave equation awwe in terms of displacement however the classical split pml obviously increases computational cost and cannot efficiently absorb waves propagating into the absorbing layer at grazing incidence our goal was to improve the computational efficiency of awwe and to enhance the suppression of edge reflections by applying a convolutional pml cpml we reformulated the original awwe as a first order formulation and incorporated the cpml with a general complex frequency shifted stretching operator into the renewed formulation a staggered grid finite difference fd method was adopted to discretize the first order equation system for wavefield depth continuation the first order awwe with the cpml saved memory compared with the original second order awwe with the conventional split pml with the help of numerical examples we verified the correctness of the staggered

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

reducing two way splitting error of ffd method in dual domains the fourier finite difference ffd method is very popular in seismic depth migration but its straightforward 3d extension creates two way splitting error due to ignoring the cross terms of spatial partial derivatives traditional correction schemes either in the spatial domain by the implicit finite difference method or in the wavenumber domain by phase compensation lead to substantially increased computational costs or numerical difficulties for strong velocity contrasts we propose compensating the two way splitting error in dual domains alternately in the spatial and wavenumber domains via fourier transform first we organize the expanded square root operator in terms of two way splitting ffd plus the usually ignored cross terms second we select a group of optimized coefficients to maximize the accuracy of propagation in both inline and crossline directions without yet considering the diagonal directions finally we further optimize the constant coefficient of the compensation part to further improve the overall accuracy of the operator in implementation the compensation terms are similar to the high order corrections of the generalized screen method but their functions are to compensate the two way splitting error rather than the expansion error numerical experiments show that optimized one term compensation can achieve nearly perfect circular impulse responses and the propagation angle with less than 1 error for all azimuths is improved up to 60 from 35 compared with traditional single domain methods our scheme can handle lateral velocity variations even for strong velocity contrasts much more easily with only one additional fourier transform based on the two way splitting ffd method which helps retain the computational efficiency

dynamics of timoshenko beam on linear and nonlinear foundation phase relations significance of the second spectrum stability abstract some controversial and unclear issues in the timoshenko beam tb theory are analyzed with emphasis on features of two branches of frequencies taking into account the elastic foundation creates additional opportunities for comparative analysis of both the branches relationship between the second branch modes and so called thickness shear mode is discussed phase relations between the deformation and force factors in beam for both the branches are established which generalize the previously established relations it is shown that the elastic foundation can change these relations for the first branch based on this analysis the significance of the second spectrum is assessed and peculiarities of dispersion relations for frequency phase and group velocities are explained free transverse waves in infinitely long beam on nonlinear cubic foundation are studied using the multiple scales method the obtained nonlinear schrodinger equation enables us to check stability of the harmonic waves in infinite beam and to determine diapasons of their modulational instability

another second order beam vibration theory explicit bending warping flexibility and restraint abstract a new symmetric beam bending vibration theory which explicitly includes bending warping is presented the governing differential equations are of sixth order and the boundary conditions allow enforcement of total fixity of the end of a beam the motivation for the study was the desire to determine if consideration of bending warping restraint in bending theory is as important as is torsion warping restraint in torsion theory natural frequency predictions are given and compared with other theories for constant cross section cantilevered beams of two types short deep rectangular beams and short thin shallow circular cylindrical shells the accuracy of the new theory is comparable with murty s levinson s bickford s and rehfield s theories for beams with rectangular cross sections it is also comparable with cowper s theory for general symmetric cross sections

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 separable kernel decomposition method for approximating the dsr continuation operator we develop a separable kernel decomposition method for approximating the double square root dsr continuation operator in one way migrations in this paper this new approach is a further development of separable approximations of the single square root ssr operator the separable approximation of the dsr operator generally involves solving a complicated nonlinear system of integral equations instead of solving this nonlinear system directly our new method consists of repeatedly applying the separable kernel technique developed for the two variable ssr operator to the multivariable dsr operator numerical experiments demonstrate the efficiency of the proposed method we illustrate the fast convergence of the obtained separable approximation we also demonstrate the capability of this novel approximation for imaging an area with geologic complexities through synthetic data

coupled bending torsional dynamic stiffness matrix of an axially loaded timoshenko beam element abstract analytical expressions for the coupled bending torsional dynamic stiffness matrix terms of an axially loaded uniform timoshenko beam element are derived in an exact sense by solving the governing differential equations of motion of the element the symbolic computing package reduce has been used to generate an analytical expression for each of the dynamic stiffness terms in a concise form for check purposes numerical values of the dynamic stiffness matrix terms were obtained using the derived explicit expressions as well as by an alternative nonanalytical method based on matrix inversions and matrix multiplications stiffnesses obtained from both methods agreed with each other to machine accuracy application of the developed theory is discussed with particular reference to an established algorithm the influence of axial force shear deformation and rotatory inertia on the natural frequencies of a bending torsion coupled beam with cantilever end conditions is demonstrated by numerical results such results are not generally available in the literature therefore results obtained by partially restricting the present theory are compared with the existing literature wherever possible the results indicate that the method is accurate and efficient

number theoretic generalization of the monster denominator formula the denominator formula for the monster lie algebra is the product expansion for the modular function in terms of the hecke system of modular functions this formula can be reformulated entirely number theoretically namely it is equivalent to the description of the generating function for the as a weight 2 modular form in with a pole at z although these results rely on the fact that has genus 0 here we obtain a generalization framed in terms of polar harmonic maass forms for all of the modular curves in this survey of recent work we discuss this generalization and we offer an introduction to the theory of polar harmonic maass forms we conclude with applications to formulas of ramanujan and green s functions

wide angle fd and ffd migration using complex pade approximations seismic migration by downward continuation using the one way wave equation approximations has two shortcomings imaging steep dip reflectors and handling evanescent waves complex pade approximations allow a better treatment of evanescent modes stabilizing finite difference migration without requiring special treatment for the migration domain boundaries imaging of steep dip reflectors can be improved using several terms in the pade expansion we discuss the implementation and evaluation of wide angle complex pade approximations for finite difference and fourier finite difference migration methods the dispersion relation and the impulsive response of the migration operator provide criteria to select the number of terms and coefficients in the pade expansion this ensures stability for a prescribed maximum propagation direction the implementations are validated on the marmousi model data set and seg eage salt model data

experimental study of the timoshenko beam theory predictions abstract the theory of flexural vibrations proposed by timoshenko almost 90 years ago has been the subject of several recent papers in the timoshenko beam theory a critical frequency f c is expected and for frequencies f larger than f c some authors argue that a second spectrum exists this is still the matter of much debate in order to settle this question we have performed experimental measurements of flexural normal mode frequencies and amplitudes which are much more precise than what has been obtained up to now for some purposes cylindrical rods are more convenient whereas in other cases rectangular beams are easier to deal with so we analyze rods and beams of different shapes and sizes we show that below f c all observed resonant peaks can be associated with one of the compressional torsional or timoshenko resonances and that no resonant peaks are left undetected the validity of timoshenko beam theory below f c and that only one spectrum appears in this regime is therefore proved the comparison between theory and experiment for rectangular beams also shows that our experimental setup is appropriate so we can use it confidently for f f c the measurements we report here in this regime show the existence of a second spectrum at least for free free boundary conditions

a closed form solution of a bernoulli euler beam on a viscoelastic foundation under harmonic line loads fourier transform is used in this paper to solve the problem of steady state response of a beam on a viscoelastic foundation subjected to a harmonic line load the solution is constructed in the form of the convolution of the green function of the beam the theorem of residue is employed to evaluate the generalized integral such that a closed form solution can be achieved all the different combinations of damping and vibration frequency are discussed and analytical solutions are presented as a special case the solution of the beam on a winkler foundation is also discussed the validation of the solution is verified by considering the static solution of the beam and comparing the degraded solution to a well known result the closed form expression of the result can be used to construct algorithms for the inverse problems of non destructive testing of pavement structures using vibration devices

migration of primaries and multiples using an imaging condition for amplitude normalized separated wavefields abstractmigration of primary and multiple reflections leads to enhanced subsurface illumination and to increased image resolution a joint migration approach using the complete wavefield requires properly imaged primaries and multiples of all orders in recent works primaries and multiples have therefore been imaged separately using the upgoing and downgoing pressure wavefields obtained by decomposing dual sensor streamer data the matches between the corresponding depth images are still not found to be sufficiently accurate so new and more appropriate imaging conditions need to be found we reviewed the classical imaging approach used in one way wave equation migration with the aim of extending it for simultaneous migration of primaries and all orders of multiples based on rayleigh s reciprocity theorem and well known theoretical developments we derived a new imaging condition described in terms of the upgoing pressure wavefield and a filtered version of the downgoing vertical velocity wavefield t

on the shear coefficient in timoshenko s beam theory abstract some existing formulations for the shear coefficient in timoshenko s beam theory are discussed especially through evaluation of the accuracy to which natural frequencies of simply supported prismatic thin walled beams can be obtained the main conclusion drawn is that if a consistent expression for the shear coefficient such as those given by cowper 1 or stephen 2 is used in timoshenko s beam theory then very high accuracies can be expected for the natural frequencies even for wavelengths of the same magnitude as the transverse dimension of the beam it is noted that no reduction of the moment of inertia due to shear lag effects should be made as these effects are included in the consistent formulas for the shear coefficient finally some apparently paradoxical results indicating that a reduction in shear stiffness occurs in rare cases if more material is added to a section are discussed and explained as resulting from the use of integrated rather than pointwise deflection measures in the derivation of consistent shear coefficient expressions the results are discussed in the light of the importance of the shear stiffness of the hull girder in ship hull vibration analysis

frequency analysis of finite beams on nonlinear kelvin voight foundation under moving loads abstract the vibration of an euler bernoulli beam resting on a nonlinear kelvin voight viscoelastic foundation traversed by a moving load is studied in the frequency domain the objective is to obtain the frequency responses of the beam and the effects of different parameters on the system response the parameters include the magnitude and speed of the moving load and the foundation nonlinearity and its damping coefficient the solution is obtained by using the galerkin method in conjunction with the multiple scales method msm the governing nonlinear partial differential equations of motion are discretized into sets of nonlinear ordinary differential equations subsequently the solution is calculated for different harmonics by using the msm as one of the powerful perturbation techniques the steady state responses of the main harmonic as well as its two super harmonics are then obtained as a case study a conventional railway track is dynamically simulated and the jump phenomenon in the response is observed for three harmonics moreover a thorough stability analysis of the system is carried out

extended local born fourier migration method a migration approach based on a local application of the born approximation within each extrapolation interval contains a singularity that can make direct application unstable previous authors have suggested adding an imaginary part to the vertical wavenumber to eliminate the singularity however their method requires that the reference slowness must be the maximum slowness of a given layer consequently the slowness perturbations are larger than those when the average slowness is selected as a reference slowness therefore its applicability is limited we develop an extended local born fourier migration method that circumvents the singularity problem of the local born solution and makes it possible to choose the average slowness as a reference slowness it is computationally efficient because of the use of a fast fourier transform algorithm it can handle wider angles or steeper interfaces and scattering effects of heterogeneities more accurately than the split step fourier ssf method which acco

globally optimized finite difference extrapolator for strongly vti media implicit finite difference fd migration is unconditionally stable and is popular in handling strong velocity variations but its extension to strongly transversely anisotropic media with vertical symmetric axis media is difficult traditional local optimizations generate the optimized coefficients for each pair of thomsen anisotropy parameters independently which can degrade results substantially for large anisotropy variations and lead to a huge table we developed an implicit fd method using the analytic taylor series expansion and used a global optimization scheme to improve its accuracy for wide phase angles we first extended the number of the constant coefficients then we relaxed the coefficient of the time delay extrapolation term by tuning a small factor such that error is less than 0 1 finally we optimized the constant coefficients using a simulated annealing algorithm by constraining that all the error functions on a fine grid of the whole anisotropic region did not exceed 0 5 simultaneously the extended number of the constant coefficients and the relaxed coefficient greatly enhanced the flexibility of matching the dispersion relation and significantly improved the ability of handling strong anisotropy over a much wider range compared with traditional local optimization our scheme does not need any table and table lookup for each order of the fd method only one group of optimized coefficients is enough to handle strong variations in velocity and anisotropy more importantly our global optimization scheme guarantees the accuracy for various possible ranges of anisotropy parameters no matter how strong the anisotropy is for the globally optimized second order fd method the accurate phase angle is up to 58 and the increase is about 18 22 for the globally optimized fourth order fd method the accurate phase angle is up to 77 and the increase is about 22 27

fourier finite difference migration many existing migration schemes cannot simultaneously handle the two most important problems of migration imaging of steep dips and imaging in media with arbitrary velocity variations in all directions for example phase shift k migration is accurate for nearly all dips but it is limited to very simple velocity functions on the other hand finite difference schemes based on one way wave equations consider arbitrary velocity functions but they attenuate steeply dipping events we propose a new hybrid migration method named fourier finite difference migration wherein the downward continuation operator is split into two downward continuation operators one operator is a phase shift operator for a chosen constant background velocity and the other operator is an optimized finite difference operator for the varying component of the velocity function if there is no variation of velocity then only a phase shift operator will be applied automatically on the other hand if there is a strong variatio

eigenvalue analysis of timoshenko beams and axisymmetric mindlin plates by the pseudospectral method a study of the free vibration of timoshenko beams and axisymmetric mindlin plates is presented the analysis is based on the chebyshev pseudospectral method which has been widely used in the solution of fluid mechanics problems clamped simply supported free and sliding boundary conditions of timoshenko beams are treated and numerical results are presented for different thickness to length ratios eigenvalues of the axisymmetric vibration of mindlin plates with clamped simply supported and free boundary conditions are presented for various thickness to radius ratios

riemannian wavefield extrapolation riemannian spaces are described by nonorthogonal curvilinear coordinates we generalize one way wavefield extrapolation to semiorthogonal riemannian coordinate systems that include but are not limited to ray coordinate systems we obtain a one way wavefield extrapolation method that can be used for waves propagating in arbitrary directions in contrast to downward continuation which is used for waves propagating mainly in the vertical direction ray coordinate systems can be initiated in many different ways for example from point sources or from plane waves incident at various angles since wavefield propagation happens mostly along the extrapolation direction we can use inexpensive finite difference or mixed domain extrapolators to achieve high angle accuracy the main applications of our method include imaging of steeply dipping or overturning reflections

representation of seismic sources in the one way wave equations one way extrapolation of downgoing and upgoing acoustic waves plays an essential role in the current practice of seismic migration berkhout 1985 stolt and benson 1986 claerbout 1985 gardner 1985 generally one way wave equations are derived for the source free situation sources are then represented as boundary conditions for the one way extrapolation problem this approach is valid provided the source representation is done with utmost care for instance it is not correct to represent a monopole source by a spatial delta function and to use this as input data for a standard one way extrapolation scheme this yields an erroneous directivity pattern as illustrated below

variational derivation of governing differential equations for truncated version of bresse timoshenko beams abstract in this paper we provide a variational derivation of modified bresse timoshenko equations this process leads to the additional term to so called truncated set of bresse timoshenko equations this new term is associated with the modified slope inertia the two sets of truncated versions of the bresse timoshenko equations are contrasted with each other as well as with the original bresse timoshenko equations on the example of the a beam that is simply supported at its both ends and b the cantilever beam with end mass it is concluded that application of either truncated sets of equations is advantageous over the original bresse timoshenko equations as far as two truncated sets of equations are concerned the variationally derived set appears to be preferable

compensation for transmission losses based on one way propagators in the mixed domain most true amplitude migration algorithms based on oneway wave equations involve corrections of geometric spreading and seismic q attenuation however few papers discuss the compensation of transmission losses ctl based on one way wave equations here we present a method to compensate for transmission losses using one way wave propagators for a 2d case the scheme is derived from the lippmann schwinger integral equation the ctl scheme is composed of a transmission term and a phase shift term the transmission term compensates amplitudes while the wave propagates through subsurfaces the transmission term is a function of the vertical wavenumbers of two adjacent heterogeneous screens the phase shift term is a fourier finite difference ffd propagator implemented in a mixed domain via fourier transform the transmission term can be flexibly incorporated into the conventional phaseshift migration algorithm i e ffd at every depth step we analyze the effects of frequency lateral velocity contrast and vertical velocity ratio on the accuracy of the presented formulae numerical examples from a flat model and a fault model with lateral velocity variations are presented to demonstrate the ability of the proposed scheme for compensation of transmission losses

the stabilization of high order multistep schemes for the laguerre one way wave equation solver abstract this paper considers spectral finite difference methods of a high order of accuracy for solving the one way wave equation using the laguerre integral transform with respect to time as the base in order to provide a high spatial accuracy and stability the richardson method can be employed however such an approach requires high computer costs therefore we consider alternative algorithms based on the adams multistep schemes to reach the stability for the one way equation the stabilizing procedures using the spline interpolation were developed this made it possible to efficiently implement a predictor corrector type method thus decreasing computer costs the stability and accuracy of the procedures proposed have been studied based on the implementation of the migration algorithm within a problem of seismic prospecting

mindlin plate theory best shear coefficient and higher spectra validity mindlin plate theory predicts three frequency spectra or equivalently three branches to a phase velocity dispersion diagram the lowest of which thew1mode provides rotatory inertia and shear deformation corrections to classical thin plate theory employing consistent truncation procedures to both the mindlin and the exact rayleigh lamb frequency equations valid for long wavelength and low phase velocity one finds thatw1mode agreement is achieved when the shear coefficient takes the valuek 5 6 v the mindlin prediction is then less than 0 5 in error when the wavelength is equal to the plate thickness and less than 1 in error as wavelength approaches zero the previously dismissed mindlinhmode is seen to be in exact frequency or phase velocity agreement with the second slowestshwave in the infinite plate as long as the shear coefficient for this mode takes the valuek 2 12 however thew2mode as with the second frequency spectrum of timoshenko beams should be regarded as the inevitable but meaningless consequence of an otherwise remarkable approximate engineering theory

wave equation based seismic illumination analysis we present a wave equation based method for seismic illumination analysis a one way wave equation based generalized screen propagator is used to extrapolate the wavefields from sources and receivers to the subsurface target a local plane wave analysis is used at the target to calculate localized directionalenergyfluxesforbothsourceandreceiver wavefields we construct an illumination matrix using these energy fluxes to quantify the target illumination conditions the target geometry information is used to manipulate the illumination matrix and generate different types of illumination measures the wave equation based approach can properly handle forward multiple scattering phenomena including focusing defocusing diffraction and interference effects itcanbedirectlyappliedtocomplexvelocitymodels velocity model smoothing and fresnel zone smoothing are not required different illumination measurements derived from this method can be applied to target oriented or volumetric illumination analyses this new method is flexible and practical for illumination analysis in complex 2d and 3d velocity models with nontrivial acquisition and target

imaging the total wavefields by reflectivity estimation using amplitude normalized wavefield decomposition field data example abstractto combine imaging information of up and downgoing wavefields using primaries and multiples we have developed an imaging framework based on the subsurface impulse response the latter is expressed at every depth level by a fredholm integral equation defined in terms of the amplitude normalized upgoing pressure and downgoing vertical velocity wavefields the subsurface impulse response can be obtained by inverting the matrix form of the integral equation by performing a multidimensional deconvolution mdd of the upgoing components by the downgoing components this procedure gives the reflectivity matrix by assuming locally reacting media or lateral shift invariance we can describe how the mdd can reduce to simpler deconvolution imaging conditions based on an example acquired with dual sensor towed streamers and simultaneous sources we have developed a real data application of imaging the total up and downgoing wavefields by reflectivity inversion the total upgoing wavefield used in the im

considerations on second order beam theories abstract by comparing the phase velocity predictions of various higher order beam theories with the second order approximations to the exact frequency equations for circular and thin rectangular section beams it is concluded that equivalence requires the inclusion of rotatory inertia shear deflection lateral contraction inertia and transverse direct stress corrections to the euler bernoulli beam theory in addition it is shown that cowper s expression for the shear coefficient in the constitutive equation for shear is in fact identical to the original timoshenko definition the apparent difference arising through employing averaged rather than centre line displacement quantities

on two geometric theta lifts the theta correspondence has been an important tool in studying cycles in locally symmetric spaces of orthogonal type in this paper we establish for the orthogonal group o p 2 an adjointness result between borcherds s singular theta lift and the kudla millson lift we extend this result to arbitrary signature by introducing a new singular theta lift for o p q on the geometric side this lift can be interpreted as a differential character in the sense of cheeger and simons for the cycles under consideration

a comparison of infinite timoshenko and euler bernoulli beam models on winkler foundation in the frequency and time domain abstract this paper deals with the dynamic analysis of infinite beam models the translational and the rotational dynamic stiffness of both timoshenko and euler bernoulli beams on winkler foundation are derived and compared in the frequency domain the situation of vanishing elastic foundation is included as a special case here special emphasis is placed on the asymptotic behaviour of the derived stiffness expressions for high frequencies since this is of importance in case of transient excitations it is shown that the dynamic stiffness of the infinite timoshenko beam follows a linear function of i whereas rational powers of i are involved in case of euler bernoulli s model the stiffness formulations can be transformed into the time domain using the mixed variables technique this is based on a rational approximation of the low frequency force displacement relationship and a subsequent algebraic splitting process at the same time the high frequency asymptotic dynamic stiffness is transformed into the time domain in closed form it is shown that the timoshenko beam is equivalent to a simple dashpot in the high frequency limit whereas euler bernoulli s beam model leads to fractional derivatives of the unknown state variables in an equivalent time domain description this finding confirms the superiority of timoshenko s model especially for high frequencies and transient excitations numerical examples illustrate the differences with respect to the two beam models and demonstrate the applicability of the proposed method for the time domain transformation of force displacement relationships

theory of true amplitude one way wave equations and true amplitude common shot migration one way wave operators are powerful tools for forward modeling and migration here we describe a recently developed true amplitude implementation of modified one way operators and present some numerical examples by true amplitude one way forward modeling we mean that the solutions are dynamically correct as well as kinematically correct that is ray theory applied to these equations yields the upward and downward traveling eikonal equations of the full wave equation and the amplitude satisfies the transport equation of the full wave equation the solutions of these equations are used in the standard wave equation migration imaging condition the boundary data for the downgoing wave is also modified from the one used in the classic theory because the latter data is not consistent with a point source for the full wave equation when the full wave form solutions are replaced by their ray theoretic approximations the imaging formula reduces to the common shot kirchhoff inversion formula in this sense

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

migration with arbitrarily wide angle wave equations we develop a new scalar migration technique that is highly accurate for imaging steep dips in heterogeneous media this method is based on arbitrarily wide angle wave equations awwes that are highly accurate space domain one way wave equations and have a form similar to the 15 equation the accuracy of the proposed method is increased by introducing auxiliary variables as well as adjusting the parameters of the approximation poststack migration is carried out by downward continuation using the awwe for which we have developed a stable explicit double marching scheme up to 80 accuracy is achieved by second order awwe migration with only 2 3 times the computational effort of the 15 equation and requiring almost the same storage we illustrate the performance of awwe migration using impulse response graphs a single dipping reflector and a slice of the seg eage salt model

arbitrarily wide angle wave equations for complex media by combining various ideas related to one way wave equations owwes half space stiffness relation special finite element discretization and complex coordinate stretching a systematic procedure is developed for deriving a series of highly accurate space domain versions of owwes the resulting procedure is applicable to complex media where the governing equation full wave equation is a second order differential system making the procedure applicable for general heterogeneous anisotropic porous viscoelastic media owing to their high accuracy in representing waves propagating in an arbitrarily wide range of angles the resulting equations are named arbitrarily wide angle wave equations awwes in order to illustrate the proposed procedure awwes are derived for one way propagation in acoustic as well as elastic media while acoustic awwes can be considered as modified versions of well known space domain owwes based on rational approximations of the square root operator the elastic awwes are significantly different from the existing elastic owwes unlike the existing elastic owwes elastic awwes are displacement based and are applicable to general anisotropic media furthermore awwes are simple in their form and appear amenable to easy numerical implementation

the second spectrum of timoshenko beam theory further assessment a review of contributions and views on the second spectrum of timoshenko beam theory tbt over the past two decades together with some new results are presented it is shown that the timoshenko frequency equation factorises not solely for hinged hinged end conditions as is often claimed but also for guided guided and guided hinged these new cases may be regarded as portions of a multi span hinged hinged beam a higher derivative lagrangian that leads directly to the well known fourth order timoshenko beam equation is reviewed a simple relationship between the so called ostrogradski energy and the mechanical energy is derived for hinged hinged end conditions it is shown that the ostrogradski energy is positive for the first spectrum but negative for the second within some branches of physics this would be sufficient to conclude that the second spectrum is unphysical a numerical example presented by levinson and cooke is re examined using both tbt and exact plane stress elastodynamic theory agreement is excellent for the first spectrum however the second spectrum predictions are not in consistent agreement with any single mode of vibration for long wavelength it is very close to the second asymmetric mode but as wavelength shortens it becomes closer to the second symmetric then the third asymmetric modes the conclusion remains unchanged the second spectrum predictions of tbt should be disregarded

a stable one way wave propagator for vti media for the purpose of one way wave equation imaging a pseudoscreen propagator is developed for transversely isotropic media with vertical axes of symmetry the phase shift for propagation through a depth slice is decomposed into three terms a gazdag phase shift for propagation in a laterally homogeneous reference medium a correction for lateral variability of vertical propagation and a remaining wide angle term for oblique directions of propagation based on rational function approximation for this remaining wide angle term a fourier finite difference ffd approach with four way splitting is applied fourth order pade approximation is unsatisfactory in anelliptic media for large propagation angles with respect to the vertical direction therefore a method of coefficient optimization is developed in conjunction with a method of choosing an adequate homogeneous reference medium in a depth slice by symmetrizing the finite difference operators and because of the choice of the optimized coefficients the propagator is stable in the sense that the least squares norm of the wavefield measured for a frequency depth slice does not grow with increasing depth of propagation a small amount of artificial damping is applied to suppress artifacts that appear at the critical angle defined by the velocities in the reference medium and the actual medium synthetic examples confirm that good kinematic accuracy is achieved for a wide range of propagation angles typically up to 60

anisotropic complex pade hybrid finite difference depth migration standard real valued finite difference fd and fourier finite difference ffd migrations cannot handle evanescent waves correctly which can lead to numerical instabilities in the presence of strong velocity variations a possible solution to these problems is the complex pade approximation which avoids problems with evanescent waves by rotating the branch cut of the complex square root we have applied this approximation to the acoustic wave equation for vertical transversely isotropic media to derive more stable fd and hybrid fd ffd migrations for such media our analysis of the dispersion relation of the new method indicates that it should provide more stable migration results with fewer artifacts and higher accuracy at steep dips our studies lead to the conclusion that the rotation angle of the branch cut that should yield the most stable image is 60 for fd migration as confirmed by numerical impulse responses and work with synthetic data

optimized chebyshev fourier migration a wide angle dual domain method for media with strong velocity contrasts a wide angle propagator is essential when imaging complex media with strong lateral velocity contrasts in one way waveequation migration we have developed a dual domain one way propagator using truncated chebyshev polynomials and a globally optimized scheme our method increases the accuracy of the expanded square root operator by adding two high order terms to the traditional split step fourier propagator first we approximate the square root operator using taylor expansion around the reference background velocity then we apply the first kindchebyshevpolynomialstoeconomizetheresultsofthe taylor expansion finally we optimize the constant coefficients using the globally optimized scheme which are fixed and feasible for arbitrary velocity models theoretical analysis and numericalexperimentshavedemonstratedthatthemethodhasvery highaccuracyandexceedstheunoptimizedfourierfinite difference propagator for the entire range of practical velocity contrasts the accurate propagation angle of the method is always about 60 under the relative error of 1 for complex media with weak moderate and even strong lateral velocity contrasts the method allows us to handle wide angle propagations and strong lateral velocity contrast simultaneously by using fourier transform alone only four 2d fourier transforms are required for each step of 3d wavefield extrapolation and the computing cost is similar to that of the fourier finite difference method compared with the finite difference method our method has no twowaysplittingerrori e azimuthal anisotropyerrorfor3dcases and almost no numerical dispersion for coarse grids in addition it has strong potential to be accelerated when an enhanced fast fouriertransformalgorithmemerges

a new rectangular beam theory abstract a new theory for beams of rectangular cross section which includes warping of the cross sections is presented in the present work by satisfying the shear free conditions on the lateral surfaces of the beam a pair of coupled equations of motion are obtained such that no arbitrary shear coefficient is required it is shown that the uncoupled equation for the transverse displacement is the same as the corresponding equation in timoshenko beam theory provided that for the timoshenko equation the shear coefficient is taken to be 5 6 this value lies within the range of values 0 822 0 870 appearing in the literature for the beam of rectangular cross section results for two typical static examples are given for both the new theory and timoshenko beam theory these results are compared with the solutions of the comparable problems in the linear theory of elasticity for the end loaded cantilever beam the new theory predicts the same result for the neutral surface deflection as does the linear theory of elasticity while timoshenko beam theory underestimates the shear correction term by 20 for the uniformly loaded and simply supported case both beam theories provide the same overestimate of the central deflection when compared with the theory of elasticity solution

a root finding technique to compute eigenfrequencies for elastic beams in this manuscript a method to compute eigenfrequencies for elastic beams is presented for beam modeling three fundamental effects are considered bending rotary inertia and shear deformation the method consists on enclosing each eigenfrequency in an interval where the characteristic function is monotonic then a root finding technique is used to compute the eigenfrequency to any desired accuracy the method is applied successfully to equations involving bending and either rotary inertia or shear deformation

the second frequency spectrum of timoshenko beams abstract a second spectrum of frequencies was reported in early analytical work on the vibrations of timoshenko beams however in subsequent finite element modelling this phenomenon was either ignored or not definitively classified and recorded in fact from a recent finite element analysis with a high precision element it was even concluded that there is no separate second spectrum of frequencies except for the special case of hinged hinged beams and it was asserted that previous investigators had misinterpreted some frequencies thus introducing the notion of second frequencies in this paper a simple linear beam element with independent displacement fields and reduced integration to eliminate shear locking is used and enables one to detect the second spectrum accurately guidelines are provided which help to identify and classify the frequencies into two separate spectra

the hartley transform in seismic imaging phase shift migration techniques that attempt to account for lateral velocity variations make substantial use of the fast fourier transform fft generally the hermitian symmetry of the complex valued fourier transform causes computational redundancies in terms of the number of operations and memory requirements in practice a combination of the fft with the well known real to complex fourier transform is often used to avoid such complications as an alternative means to the fourier transform we introduce the inherently real valued non symmetric hartley transform into phase shift migration techniques by this we automatically avoid the hermitian symmetry resulting in an optimized algorithm that is comparable in efficiency to algorithms based on the real to complex fft we derive the phase shift operator in the hartley domain for migration in two and three dimensions and formulate phase shift plus interpolation split step migration and split step double square root prestack migration in terms of the

optimum split step fourier 3d depth migration developments and practical aspects we present an efficient scheme for depth extrapolation of wide angle 3d wavefields in laterally heterogeneous media the scheme improves the so called optimum split step fourier method by introducing a frequency independent cascaded operator with spatially varying coefficients the developments improve the approximation of the optimum split step fourier cascaded operator to the exact phase shift operator of a varying velocity in the presence of strong lateral velocity variations and they naturally lead to frequency dependent varying step depth extrapolations that reduce computational cost significantly the resulting scheme can be implemented alternatively in spatial and wavenumber domains using fast fourier transforms ffts the accuracy of the first order approximate algorithm is similar to that of the second order optimum split step fourier method in modeling wide angle propagation through strong laterally varying media similar to the optimum split step fourier method the scheme is superior to methods such as the generalized screen and fourier finite difference we demonstrate the scheme s accuracy by comparing it with 3d two way finite difference modeling comparisons with the 3d prestack kirchhoff depth migration of a real 3d data set demonstrate the practical application of the proposed method

critical contrasting of three versions of vibrating bresse timoshenko beam with a crack abstract this paper deals with natural frequencies of short or thick beams taking into account effects of rotary inetria and shear deformation three versions of the bresse timoshenko beam theory are considered and contrasted in the context of the problem of beam vibration in presence of the single crack namely the a original bresse timoshenko theory b elishakoff s truncated version and c the proposed slope inertia based version the original bresse timoshenko equations contain fourth order derivative with respect to time that is not present in the classical bernoulli euler theory whose correction was the declared goal of s p timoshenko who himself did not overestimate the importance of this term the truncated version of the bresse timoshenko was suggested by elishakoff with no fourth order time derivative incorporated in the theory in this paper variational derivation of the truncated version of the bresse timoshenko beam theory is conducted with attendant additional term containing the fourth derivative with respect to axial coordinate and second derivative with respect to time it appears imperative to conduct thorough comparison of the above three theories in the context of beams containing a crack which is often present in engineering structures the applicability of these three versions with attendant extensive numerical calculations is specifically studied for the titled problem it is demonstrated that the original bresse timoshenko beam theory as an overcomplicated theory is not needed for crack analysis simpler theories namely the truncated version of this theory or the one based on slope inertia are sufficient for crack analysis in all treated cases

an efficient solution methodology to study the response of a beam on viscoelastic and nonlinear unilateral foundation static response abstract many engineering materials and foundations such as soils demonstrate nonlinear and viscoelastic behaviour yet it is challenging to develop static and dynamic models of systems that include these materials and are able to predict the behaviour over a wide range of loading conditions this research is focused on a specific example a pinned pinned beam interacting with polyurethane foam foundation two cases when the foundation can react in tension and compression as well as only in compression are considered the model developed here is capable of predicting the response to static as well as dynamic forces whether concentrated or distributed galerkin s method is used to derive modal amplitude equations in the tensionless foundation case the contact region changes with beam motion and the estimation of the co ordinates of the lift off points is embedded into the solution procedure an efficient solution technique is proposed that is capable of handling cases where there are multiple contact and non contact regions depending on the loading profiles a high number of modes may need to be included in the solution and to speed up computation time a convolution method is used to evaluate the integral terms in the model the adaptability of the solution scheme to complicated loading patterns is demonstrated via examples the solution approach proposed is applicable to dynamic loadings as well and in these cases the automated treatment of complicated response patterns makes the convolution approach particularly attractive the influence of various parameters on the static response is discussed

directional illumination analysis using the local exponential frame we have developed an efficient method of directional illuminationanalysisinthelocalangledomainusinglocalexponential frame beamlets the space domain wavefields with different shot receiver geometries are decomposed into the local angle domain by using the local exponential beamlets which form a tight frame with the redundancy ratio two and are implemented by a linear combination of local cosine and localsinetransforms becauseofthefastalgorithmsofthelocal cosine sine transforms this method is much more efficient than the previously used decomposition methods in directional illumination analysis such as the local slant stackingmethodandthegabor daubechiesframemethod theresults of directional illumination di maps and the acquisitiondipresponsesadrforthe2dseg eagesaltmodel and the 45 shot 3d seg eage model demonstrated the validity and feasibility of our method compared with the illumination results using local slant stacking decomposition thenewmethodproducesilluminationmapsofsimilarquality but it does so a few times faster furthermore because of its high computational efficiency and saving in memory usage the new method makes the 3d directional illumination analysisreadilyapplicableintheindustry

stable wide angle fourier finite difference downward extrapolation of 3 d wavefields i present an unconditionally stable implicit finite difference operator that corrects the constant velocity phase shift operator for lateral velocity variations the method is based on the fourier finite difference ffd method contrary to previous results my correction operator is stable even when the medium velocity has sharp discontinuities and the reference velocity is higher than the medium velocity the stability of the new correction enables the definition of a new downward continuation method based on the interpolation of two wavefields the first wavefield is obtained by applying the ffd correction starting from a reference velocity lower than the medium velocity the second wavefield is obtained by applying the ffd correction starting from a reference velocity higher than the medium velocity the proposed fourier finite difference plus interpolation ffdpi method combines the advantages of the ffd technique with the advantages of interpolation a simple and economical procedure for defining

fourier finite difference wave propagation we introduce a novel technique for seismic wave extrapolation in time the technique involves cascading a fourier transform operator and a finite difference operator to form a chain operator fourier finite differences ffd we derive the ffd operator from a pseudoanalytical solution of the acoustic wave equation two dimensional synthetic examples demonstrate that the ffd operator can have high accuracy and stability in complex velocity media applying the ffd method to the anisotropic case overcomes some disadvantages of other methods such as the coupling of qp waves and qsv waves the ffd method can be applied to enhance accuracy and stability of seismic imaging by reverse time migration

dynamics of transversely vibrating beams using four engineering theories abstract in this paper the full development and analysis of four models for the transversely vibrating uniform beam are presented the four theories are the euler bernoulli rayleigh shear and timoshenko first a brief history of the development of each beam model is presented second the equation of motion for each model and the expressions for boundary conditions are obtained using hamilton s variational principle third the frequency equations are obtained for four sets of end conditions free free clamped clamped hinged hinged and clamped free the roots of the frequency equations are presented in terms of normalized wave numbers the normalized wave numbers for the other six sets of end conditions are obtained using the analysis of symmetric and antisymmetric modes fourth the orthogonality conditions of the eigenfunctions or mode shape and the procedure to obtain the forced response using the method of eigenfunction expansion is presented finally a numerical example is shown for a non slender beam to signify the differences among the four beam models

prestack migration by split step dsr the double square root dsr prestack migration equation though defined for depth variable velocity can be used to image media with strong velocity variations using a phase shift plus interpolation pspi or split step correction the split step method is based on applying a phase shift correction to the extrapolated wavefield a correction that attempts to compensate for the lateral velocity variations i show how to extend dsr prestack migration to lateral velocity media and exemplify the method by applying the new algorithm to the marmousi data set the split step dsr migration is very fast and offers excellent imaging results

toward a new engineering theory of bending fundamentals the classical engineering theory of bending due to bernoulli and euler serves as a cornerstone for structural analysis and design limitations of this theory however become apparent in flexural wave propagation studies it predicts infinite phase velocity as the wavelength becomes shorter this theoretical deficiency is corrected by timoshenko theory which accounts for transverse shear deformation a thorough study of exact elasticity solutions reveals that there are two additional effects that are the same order as transverse shear in bending behavior a new theory accounting for them is presented along with several applications the new equations are no more complicated than those of timoshenko ty pe theory yet they yield solutions which are exact or indistinguishable from exact in the examples studied

improving low order eigenfrequency estimates derived from the wave propagation method for an euler bernoulli beam we state the wpm when combined with the perturbation procedure as developed in this work enables us to capture the entire low medium and high range of the spectrum of vibration of an euler bernoulli beam both asymptotically and numerically

wide angle one way wave equations a one way wave equation also known as a paraxial or parabolic wave equation is a differential equation that permits wave propagation in certain directions only such equations are used regularly in underwater acoustics in geophysics and as energy absorbing numerical boundary conditions the design of a one way wave equation is connected with the approximation of 1 s2 1 2 on 1 1 by a rational function which has usually been carried out by pade approximation this article presents coefficients for l2 l and other alternative classes of approximants that have better wide angle behavior for theoretical results establishing the well posedness of these wide angle equations see the work of trefethen and halpern well posedness of one way wave equations and absorbing boundary conditions math comput 47 421 435 1986

extended local rytov fourier migration method we develop a novel depth migration method termed the extended local rytov fourier elrf migration method it is based on the scalar wave equation and a local application of the rytov approximation within each extrapolation interval wavefields are fourier transformed back and forth between the frequency space and frequency wavenumber domains during wavefield extrapolation the lateral slowness variations are taken into account in the frequency space domain the method is efficient due to the use of a fast fourier transform algorithm under the small angle approximation the elrf method leads to the split step fourier ssf method that is unconditionally stable the elrf method and the extended local born fourier elbf method that we previously developed can handle wider propagation angles than the ssf method and account for the phase and amplitude changes due to the lateral variations of slowness whereas the ssf method only accounts for the phase changes the stability of the elrf method is controlled

recursive integral time extrapolation methods for waves a comparative review abstractwe compared several families of algorithms for recursive integral time extrapolation rite algorithms for waves in isotropic and anisotropic media these methods allow simulating accurate wave extrapolation with little numerical dispersion even when using larger time steps than are usually possible for conventional finite difference methods these various rite algorithms all share the use of mixed space wavenumber domain operators derived from fourier integral solutions of single mode wave equations we evaluated a taxonomy for rite methods based on how they approximated the influence of medium heterogeneity one family of methods uses mixed domain series expansions to provide accurate approximations to heterogeneous extrapolators even for large time steps we compared several methods for deriving coefficients for such series approximations another family of methods uses interpolation between different homogeneous extrapolations to approximate heterogeneous time extrapolation such methods can b

dynamic analyses of plane frames by integral equations for bars and timoshenko beams abstract since especially for higher frequencies timoshenko s theory gives more reliable results than euler bernoulli s theory systems of beams like frames under arbitrary dynamic excitations should be analyzed on the basis of this refined theory in this paper after deriving the basic fundamental solutions of a lateral unit point force and of a unit single moment the deflection and the rotation of the beam cross section are given as integral equation forms of the governing second order differential equation system in the laplace and the frequency domain since in frameworks axial displacements also occur the well known integral equations for bars under tension or compression are added to complete the modelling of plane frame structures from these the unknown boundary values may be exactly determined via point collocation at the beam ends and solution of the resulting algebraic system with these boundary values the exact deflection and rotation can also be found at arbitrary interior points finally the convolution quadrature method is applied to the laplace domain integral equation formulation in order to analyze the time dependent wave propagation process two examples are presented to demonstrate the method s exactness compared with conventional finite element results a single beam where the shift of the resonance frequencies when using the timoshenko theory instead of euler bernoulli s theory is shown and a two storey framework where in addition to the frequence response the causality of the propagating wave is studied

comparison of weights in prestack amplitude preserving kirchhoff depth migration three different theoretical approaches to amplitude preserving kirchhoff depth migration are compared each of them suggests applying weights in the diffraction stack migration to correct for amplitude loss resulting from geometric spreading the weight functions are given in different notations but as is shown all of these expressions are similar a notation that is well suited for implementation is suggested entirely in terms of green s function quantities amplitudes or point source propagators for the most common prestack configurations common shot and common offset and 3 d 2 5 d and 2 d migrations expressions of the weights are given in this notation the quantities needed for calculation of the weights can be computed easily e g by dynamic ray tracing

preconditioned acoustic least squares two way wave equation migration with exact adjoint operator abstractwe have investigated the problem of designing the forward operator and its exact adjoint for two way wave equation least squares migration we study the problem in the time domain and pay particular attention to the individual operators that are required by the algorithm we derive our algorithm using the language of linear algebra and establish a simple path to design forward and adjoint operators that pass the dot product test we also found that the exact adjoint operator is not equal to the classic reverse time migration algorithm for instance one must pay particular attention to boundary conditions to compute the exact adjoint that accurately passes the dot product test forward and adjoint operators are adopted to solve the so called least squares reverse time migration problem via the method of conjugate gradients we also examine a preconditioning strategy to invert extended images

optimized implicit finite difference and fourier finite difference migration for vti media propagation velocity of seismic waves in heterogeneous vti media depends not only on spatial location but also on their propagation direction which leads to a much more complex dispersion relation than in isotropic media as a result designing implicit finite difference fd schemes for wavefield extrapolation in anisotropic media through analytic taylor series expansion is more difficult implicit fd and fourier finite difference ffd schemes are developed for vertical transversely isotropic vti media based on function fitting the dispersion relation of vti media is approximated with a rational function and its coefficients are estimated by weighted least squares optimization because these coefficients are functions of thomsen anisotropy parameters e and and vary laterally in heterogeneous vti media they are calculated before wavefield extrapolation and stored in a table implicit fd and ffd schemes for vti media are almost the same as for isotropic media except that coefficients are looke

optimization of the parameters in complex pade fourier finite difference migration abstractcomplex pade fourier finite difference migration is a stable one way wave equation technique that allows for better treatment of evanescent modes than its real counterpart in this way producing fewer artifacts as for real fourier finite difference ffd migration its parameters can be optimized to improve the imaging of steeply dipping reflectors the dip limitation of the ffd operator depends on the variation of the velocity field we have developed a wide angle approximation for the one way continuation operator by means of optimization of the pade coefficients and the most important velocity dependent parameter we have evaluated the achieved quality of the approximate dispersion relation in dependence on the chosen function of the ratio between the model and reference velocities under consideration of the number of terms in the pade approximation and the branch cut rotation angle the optimized parameters are chosen based on the migration results and the computational cost we found that by

stable depth extrapolation of seismic wavefields by a neumann series several migration methods fail to work when applied to complex geological structures with strong lateral heterogeneity the generalized migration in the frequency wavenumber f k domain based on a convolution with a slowness inverse of velocity dependent operator is capable of downward continuation of wavefield in media with strong vertical and lateral variations of velocity unfortunately this method as presented in the literature is potentially unstable we propose a new stable extrapolator based on the solution of the integral fredholm equation which describes a one way wave equation in the form of a neumann series the resulting algorithm of depth migration is implemented in both the frequency wavenumber f k and frequency space f x domains and takes into account arbitrary lateral gradients of velocity using a low frequency filter in x f domain that is the sum of the power series the computation time of depth migration by a neumann series is slightly longer than for split step fourier

subsurface reflectivity estimation from imaging of primaries and multiples using amplitude normalized separated wavefields abstractseveral studies have shown the benefits of including multiple reflections together with primaries in the structural imaging of subsurface reflectors however to characterize the reflector properties there is a need to compensate for propagation effects due to multiple scattering and to properly combine the information from primaries and all orders of multiples from this perspective and based on the wave equation and rayleigh s reciprocity theorem recent works have suggested computing the subsurface image from the green s function reflection response or reflectivity by inverting a fredholm integral equation in the frequency space domain by following claerbout s imaging principle and assuming locally reacting media the integral equation may be reduced to a trace by trace deconvolution imaging condition for a complex overburden and considering that the structure of the subsurface is angle dependent this trace by trace deconvolution does not properly solve the fredholm integral equation we

one way wave equation migration in log polar coordinates we obtained acoustic wave and wavefield extrapolation equations in log polar coordinates lpcs and tried to enhance the imaging to achieve this goal it was necessary to decrease the angle between the wavefield extrapolation axis and wave propagation direction in the one way wave equation migration wem if we were unable to carry it out more reflection wave energy would be lost in the migration process it was concluded that the wavefield extrapolation operator in lpcs at low frequencies has a large wavelike region and at high frequencies it can mute the evanescent energy in these coordinate systems an extrapolation operator can readily lend itself to high order finitedifference schemes therefore even with the use of inexpensive operators wem in lpcs can clearly image varied horizontal and vertical events in complex geologic structures using wide angle and turning waves in these coordinates we did not encounter any problems with reflections from opposing dips dispersion played important roles not only as a filter operator but also as a gain function prestack and poststack migration results were obtained with extrapolation methods in different coordinate systems and it was concluded that migration in lpcs can image steeply dipping events in a much better way when compared with other methods

3d wavefield extrapolation with optimum split step fourier method a one way propagator is proposed for more accurately modeling wide angle wavefields in the presence of severe lateral variations of the velocity the method adds a higher order correction to improve the split step fourier method by directly designing a cascaded operator that matches the exact phase shift operator of a varying velocity using an optimization scheme the coefficients in the cascaded operator are determined according to the local velocity distribution and the prescribed angular range of wavefield propagation the proposed algorithm is implemented alternately in spatial and wavenumber domains using fast fourier transforms as in the split step fourier and generalized screen methods this algorithm can achieve higher accuracy than the generalized screen method for wide angle wavefields although the same numerical scheme is used with comparable computational cost no extra error arises for the proposed algorithm when used for 3d wave propagation in contrast to methods that introduce an implic

true amplitude one way wave equation migration in the mixed domain one way wave equation migration is a powerful imaging tool for locating accurately reflectors in complex geologic structures however the classical formulation of one way wave equations does not provide accurate amplitudes for the reflectors when dynamic information is required after migration such as studies for amplitude variation with angle or when the correct amplitudes of the reflectors in the zero offset images are needed some modifications to the one way wave equations are required the new equations which are called true amplitude one way wave equations provide amplitudes that are equivalent to those provided by the leading order of the ray theoretical approximation through the modification of the transverse laplacian operator with dependence of lateral velocity variations the introduction of a new term associated with the amplitudes and the modification of the source representation in a smoothly varying vertical medium the extrapolation of the wavefields with the true amplitude one way

finite difference modeling with adaptive variable length spatial operators most finite difference simulation algorithms use fixed length spatial operators to compute spatial derivatives the choice of length is dictated by computing cost stability and dispersion criteria that are satisfied globally we propose finite difference schemes with adaptive variable length spatial operators to decrease computing costs significantly without reducing accuracy these schemes adopt long operators in regions of low velocity and short operators in regions of high velocity two methods automatically determine variable operator lengths dispersion analysis along with 1d and 2d modeling demonstrates the validity and efficiency of our schemes in addition a hybrid absorbing boundary condition helps reduce unwanted reflections from model boundaries our scheme is more efficient than those based on variable grid methods for modeling migration and inversion of models with complex velocity structures because the latter require local grid refinement which usually increases memory requirements

a perturbation approach for the eigenfrequency analysis of separable kirchhoff thin plate problems on a rectangle we introduce a method for the estimation of the eigenfrequencies for separable kirchhoff thin plate problems on a rectangle particularly those involving the free boundary condition we believe that this paper is the first to give such a treatment for the low end of the spectrum the method is an adaptation of an asymptotic perturbation method used for the treatment of various beam problems and in the setting of the plate is shown to be a generalization and refinement of the asymptotic methods of bolotin and keller and rubinow the wave propagation method we compare our results with those of our own legendre tau approximation and where available with numerical results extant in the literature excellent agreement is found in each case

exact dynamic stiffness elements based on one dimensional higher order theories for free vibration analysis of solid and thin walled structures in this paper an exact dynamic stiffness formulation using one dimensional 1d higher order theories is presented and subsequently used to investigate the free vibration characteristics of solid and thin walled structures higher order kinematic fields are developed using the carrera unified formulation which allows for straightforward implementation of any order theory without the need for ad hoc formulations classical beam theories euler bernoulli and timoshenko are also captured from the formulation as degenerate cases the principle of virtual displacements is used to derive the governing differential equations and the associated natural boundary conditions an exact dynamic stiffness matrix is then developed by relating the amplitudes of harmonically varying loads to those of the responses the explicit terms of the dynamic stiffness matrices are also presented the resulting dynamic stiffness matrix is used with particular reference to the wittrick williams algorithm to carry out the free vibration analysis of solid and thin walled structures the accuracy of the theory is confirmed both by published literature and by extensive finite element solutions using the commercial code msc nastran

migration of seismic data by phase shift plus interpolation under the horizontally layered velocity assumption migration is defined by a set of independent ordinary differential equations in the wavenumber frequency domain the wave components are extrapolated downward by rotating their phases this paper shows that one can generalize the concepts of the phase shift method to media having lateral velocity variations the wave extrapolation procedure consists of two steps in the first step the wave field is extrapolated by the phase shift method using l laterally uniform velocity fields the intermediate result is l reference wave fields in the second step the actual wave field is computed by interpolation from the reference wave fields the phase shift plus interpolation pspi method is unconditionally stable and lends itself conveniently to migration of three dimensional data the performance of the methods is demonstrated on synthetic examples the pspi migration results are then compared with those obtained from a finite difference method

on the two frequency spectra of timoshenko beams abstract this paper is concerned with the question of whether there are indeed two distinct spectra of frequencies for the transverse vibrations of timoshenko beams as has been claimed by a number of prior authors for the case of the simply supported beam and more recently for beams supported in an arbitrary manner elementary analysis leads to the conclusion that there is only a single frequency spectrum in the particular case of the simply supported beam the two frequency spectra viewpoint may be expedient as a device to compute frequencies but does not serve otherwise to explain the complex dynamical behavior of timoshenko beams

a boundary condition in pade series for frequency domain solution of wave propagation in unbounded domains a boundary condition satisfying the radiation condition at infinity is frequently required in the numerical simulation of wave propagation in an unbounded domain in a frequency domain analysis using finite elements this boundary condition can be represented by the dynamic stiffness matrix of the unbounded domain defined on its boundary a method for determining a pade series of the dynamic stiffness matrix is proposed in this paper this method starts from the scaled boundary finite element equation which is a system of ordinary differential equations obtained by discretizing the boundary only the coefficients of the pade series are obtained directly from the ordinary differential equations which are not actually solved for the dynamic stiffness matrix the high rate of convergence of the pade series with increasing order is demonstrated numerically this technique is applicable to scalar waves and elastic vector waves propagating in anisotropic unbounded domains of irregular geometry it can be combined seamlessly with standard finite elements copyright 2006 john wiley sons ltd

generalized moonshine ii borcherds products the goal of this paper is to construct infinite dimensional lie algebras by using infinite product identities and to use these lie algebras to reduce the generalized moonshine conjecture to a pair of hypotheses about group actions on vertex algebras and lie algebras we expect the lie algebras that we construct to manifest as algebras of physical states in an orbifold conformal field theory yet to be fully constructed with symmetries given by the monster simple group we introduce vector valued modular functions attached to families of modular functions of different levels and we prove infinite product identities for a distinguished class of automorphic functions on a product of two half planes we recast this result using the borcherds harvey moore singular theta lift and show that the vector valued functions attached to completely replicable modular functions with integer coefficients lift to automorphic functions with infinite product expansions at all cusps for each element of the monster simple group we construct an infinite dimensional lie algebra such that its denominator formula is an infinite product expansion of the automorphic function arising from that element s mckay thompson series these lie algebras have the unusual property that their simple roots and all root multiplicities are known we show that under certain hypotheses characters of groups acting on these lie algebras form functions on the upper half plane that are either constant or invariant under a genus zero congruence group

finite element modelling of infinite euler beams on kelvin foundations exposed to moving loads in convected co ordinates the paper deals with the finite element method fem solution of the problem with loads moving uniformly along an infinite euler beam supported by a linear elastic kelvin foundation with linear viscous damping initially the problem is formulated in a moving co ordinate system following the load using a galilean co ordinate transformation and subsequently the analytical solution to the homogeneous beam problem is shown to be used in more complicated cases where no analytical solutions can be found a numerical approach of the same problem is then suggested based on the fem absorbing boundary conditions to be applied at the ends of the modelled part of the infinite beam are derived the quality of the numerical results for single frequency harmonic excitation is tested by comparison with the indicated analytical solution finally the robustness of the boundary condition is tested for a ricker pulse excitation in the time domain