analytic calculation of phase and group velocities of p waves in orthorhombic media abstractwe have developed an approximate method to calculate the p wave phase and group velocities for orthorhombic media two forms of analytic approximations for p wave velocities in orthorhombic media were built by analogy with the five parameter moveout approximation and the four parameter velocity approximation for transversely isotropic media respectively they are called the generalized moveout approximation gma type approximation and the fomel approximation respectively we have developed approximations for elastic and acoustic orthorhombic media we have characterized the elastic orthorhombic media in voigt notation and we can describe the acoustic orthorhombic media by introducing the modified alkhalifah s notation our numerical evaluations indicate that the gma type and fomel approximations are accurate for elastic and acoustic orthorhombic media with strong anisotropy and the gma type approximation is comparable with the approximation recently proposed by sripanich and fomel potential

estimating anisotropy parameters and traveltimes in the τ p domain the presence of anisotropy influences many aspects of seismic wave propagation and has therefore implications for conventional processing schemes to estimate the anisotropy we need both forward modelling and inversion tools exact forward modelling in anisotropic media is generally done by raytracing however we present a new and fast method using the p transform to calculate exact p and sv reflection moveout curves in stratified laterally homogeneous anisotropic media which requires no ray tracing results are exact even if the sv waves display cusps in addition we show how the same method can be used for parameter estimation since inversion for anisotropic parameters is very nonunique we develop expressions requiring only a reduced number of parameters nevertheless predictions using these expressions are more accurate than taylor series expansions and are also able to handle cusps in the sv traveltime curves in addition layer stripping is a linear process therefore both effective aver

wide angle phase slowness approximations in vti media an anisotropic medium with vertical symmetry axis vti often presents a good model for describing real rocks propagation of quasi p and quasi sv waves in such media requires an expression of the vertical phase slowness a complicated function of the horizontal phase slowness and the medium parameters for converted wave phase shift migration methods it is desired to have slowness expressions that are simple and accurate at wide angles of propagation taylor series representations of the squared vertical slowness for quasi p and quasi sv waves result in new wide angle phase slowness approximations based on truncated series and continued fraction representations slowness approximations that are exact for both vertical propagation and at a horizontal slowness corresponding to horizontally traveling qp waves are derived the approximation for quasi sv waves can be used in phase shift migration in media where the quasi sv wavefront contains triplications these approximations are tested on several models and compared to previously published approximations the numerical tests suggest that the new continued fraction approximations are more accurate they can be used in phase shift migration algorithms which are more efficient for large angles than the existing approximations

generalized nonelliptic moveout approximation in τ p domain we have derived a generalized nonelliptic moveout approximation in the p domain that is a companion to the generalized nonhyperbolic moveout approximation in the t x domain approximation requires five parameters three parameters that can be determined from the zero slowness ray and two parameters that can be determined from one additional ray we convert the generalized approximation from the p domain to the t x domain and vice versa the generalized nonelliptic moveout approximation in the p domain admits other moveout approximations as special cases several analytical and numerical tests showed a superior accuracy of the generalized nonelliptic moveout approximation

theory of interval traveltime parameter estimation in layered anisotropic media abstractmoveout approximations for reflection traveltimes are typically based on a truncated taylor expansion of traveltime squared around the zero offset the fourth order taylor expansion involves normal moveout velocities and quartic coefficients we have derived general expressions for layer stripping second and fourth order parameters in horizontally layered anisotropic strata and specified them for two important cases horizontally stacked aligned orthorhombic layers and azimuthally rotated orthorhombic layers in the first of these cases the formula involving the out of symmetry plane quartic coefficients has a simple functional form and possesses some similarity to the previously known formulas corresponding to the 2d in symmetry plane counterparts in vertically transversely isotropic vti media the error of approximating effective parameters by using approximate vti formulas can be significant in comparison with the exact formulas that we have derived we have proposed a framework for derivin

weak anisotropy moveout approximations for p waves in homogeneous layers of monoclinic or higher anisotropy symmetries we have used so called weak anisotropy wa parameterization as an alternative to the parameterization of generally anisotropic media by a stiffness tensor wa parameters consist of linear combinations of normalized stiffness tensor elements controlling various seismic signatures hence they are theoretically extractable from seismic data they are dimensionless and can be designed to have the same order of magnitude wa parameters similarly to thomsen type parameters have a clear physical interpretation they are however applicable to anisotropy of any symmetry strength and orientation they are defined in coordinate systems independent of the symmetry elements of the studied media expressions using wa parameters naturally simplify as the anisotropy becomes weaker or as the anisotropy symmetry increases we expect that due to these useful properties wa parameterization can potentially provide a framework for seismic data processing in generally anisotropic media using the wa parameterization we have derived and tested approximate p wave moveout formulas for a homogeneous layer of up to monoclinic symmetry underlain by a horizontal reflector coinciding with a symmetry plane the derived traveltime formulas represent an expansion of the traveltime with respect to small wa parameters for the comparison with standard moveout formulas we expressed ours in the common form of nonhyperbolic moveout containing normal moveout velocity and a quartic coefficient as functions of the wa parameters the accuracy of our formulas depends strongly on the deviation of ray and phase velocity directions controlled by the deviation of the ray and phase velocities the errors do not generally increase with increasing offset nor do they increase with decreasing anisotropy symmetry the accuracy of our formulas is comparable with or better than the accuracy of commonly used formulas

moveout approximations for p and sv waves in vti media abstractwe tested alternative expressions for the p and sv wave moveout formulas in vti media based on the weak anisotropy wa approximation our moveout formulas represent expansions with respect to small parameters which are related to deviations of anisotropy from isotropy first order p wave formulas depend on four parameters two way zero offset traveltime t0 related to the vertical velocity 0 the depth h of the single horizontal reflector and two wa parameters w and w the first order sv wave formulas depend on three parameters again on t0 now related to the sv wave vertical velocity 0 depth h and the wa version of parameter the second order formulas are slightly more complicated the p and sv wave formulas depend on an additional parameter r the ratio of the sv and p wave vertical velocities the sv wave formula depends in addition on the wa parameter w because the dependence of the moveout formulas on r is very weak r can be specified as a typical sv to p wave velocity rat

velocity independent marchenko focusing in time and depth imaging domains for media with mild lateral heterogeneity abstractthe marchenko method retrieves green s functions between the acquisition surface and any arbitrary point in the medium the process generally involves solving an inversion starting with an initial focusing function e g a direct wave green s function from the desired subsurface position typically obtained using an approximate velocity model we have formulated the marchenko method in the time imaging domain in that domain we recognize that the traveltime of the direct wave green s function is related to the cheop s traveltime pyramid commonly used in time domain processing which in turn can be readily obtained from the local slopes of the common midpoint gathers this observation allows us to substitute the velocity model based initial focusing operator with that from a data driven slope estimation process moreover we found that working in the time imaging domain allows for the specification of the desired subsurface position in terms of vertical time which is connected to the cartesian depth position via the time to depth conversion our results suggest that the prior velocity model is only required when specifying the position in depth but this requirement can be circumvented by making use of the time imaging domain within its usual assumptions e g mild lateral heterogeneity provided that those assumptions are satisfied the estimated green s functions from the proposed method have comparable quality to those obtained with the knowledge of a prior velocity model

quadratic wavefront and traveltime approximations in inhomogeneous layered media with curved interfaces a quadratic approximation for the square of the traveltime from a source region to a receiver region is given for a three dimensional 3 d medium consisting of inhomogeneous layers with curved interfaces the square of the traveltime being a function of source and receiver coordinates is developed in a taylor series around a reference source and receiver point the relationships of the traveltime parameters to the ray parameters and the wavefront curvature matrices are shown using midpoint half offset coordinates gives a simplified traveltime function compared to using source receiver coordinates only in the case that the reference source point and the reference receiver point coincide zero offset approximation for a medium consisting of homogeneous layers with plane dipping interfaces the traveltime approximation is further simplified the derived traveltime approximation is shown to be exact for a reflection from a plane dipping interface in a homogeneous medium explicit expressions for the tr

theory of differential offset continuation i introduce a partial differential equation to describe the process of prestack reflection data transformation in the offset midpoint and time coordinates the equation is proved theoretically to provide correct kinematics and amplitudes on the transformed constant offset sections solving an initial value problem with the proposed equation leads to integral and frequency domain offset continuation operators which reduce to the known forms of dip moveout operators in the case of continuation to zero offset

normal moveout revisited inhomogeneous media and curved interfaces the equation of normal moveout t2 t02 x2 va2 is valid for a reflection from the base of a single homogeneous and isotropic bed but is only an approximation in the real world of multilayered inhomogeneous media and curved interfaces using the theory of geometrical optics we can find another second order equation which represents hyperbolas that are also symmetrical about the time axis however the centers of these hyperbolas do not coincide with the center of coordinates but are shifted along the time axis the equation describing this second type of hyperbola is t tp t0 2 tp2 x2 v12 where tp is the time of focusing depth and v1 the velocity of the input medium this equation is not only more accurate than the usual normal moveout but its use is more economical on a vector computer because the traditional dynamic correction is a static correction in the tp analysis this procedure makes it possible to compute velocities for all the samples of all the stacked traces and produces a velocity secti

generalized moveout approximation for qp and qsv waves in a homogeneous transversely isotropic medium the moveout approximations can be used in kinematic modeling velocity analysis and time migration the generalized moveout approximation involves five approximation parameters and has several known approximations as special cases a method is demonstrated for determining parameters of the generalized nonhyperbolic moveout approximation for qp and qsv waves in a homogeneous transversely isotropic medium with vertical symmetry axis vti medium the additional parameters for the generalized approximation are computed from the hyperbolic asymptote at infinite offset comparison with a few well known moveout approximations for higher order terms in the taylor series and asymptotic behavior shows that the generalized moveout approximation is superior to other nonhyperbolic approximations a few numerical examples for qp and qsv waves in a vti medium also indicate that the generalized approximation performs the best

velocity independent time domain seismic imaging using local event slopes by estimating local event slopes in prestack seismic reflection data it is possible to accomplish all common time domain imaging tasks from normal moveout to prestack time migration without the need to estimate seismic velocities or any other attributes local slopes contain complete information about the reflection geometry once they are estimated seismic velocities and all other moveout parameters turn into data attributes and are directly mappable from the prestack data domain into the time migrated image domain i develop an analytical theory which defines the transformation from data and local slopes to the image space for different time domain imaging operators computational experiments with synthetic and field data examples confirm theoretical expectations and demonstrate practical effectiveness of the proposed method

generalized nonhyperbolic moveout approximation reflection moveout approximations are commonly used for velocity analysis stacking and time migration a novel functional form for approximating the moveout of reflection traveltimes at large offsets is introduced in comparison with the classic hyperbolic approximation which uses only two parameters zero offset time and moveout velocity this form involves five parameters that can be determined in a known medium from zero offset computations and from tracing one nonzero offset ray it is called a generalized approximation because it reduces to some known three parameter forms with a particular choice of coefficients by testing the accuracy of the proposed approximation with analytical and numerical examples the new approximation is shown to bring an improvement in accuracy of several orders of magnitude compared to known analytical approximations which makes it as good as exact for many practical purposes

signal and traveltime parameter estimation using singular value decomposition abstractsignal detection and traveltime parameter estimation can be performed by computing a coherence function in a data window centered around a traveltime function defined by its parameters we used singular value decomposition of the data matrix not eigendecomposition of a covariance matrix to review the most commonly used coherence measures this resulted in a new reduced semblance coefficient defined from the first eigenimage assuming that the signal amplitude was the same on all data channels as in classical semblance in a second signal model the time signal was constant on each channel but the amplitude changed then the semblance coefficient is the square of the first singular value divided by the energy of the data two normalized crosscorrelation coefficients derived from the first eigenimage can also be used as a coherence measure the normalized crosscorrelation of the spatial singular vector with a vector with all elements equal to one and the normalized crosscorrelation of the tem

automatic data extrapolation to zero offset along local slope abstractvelocity independent seismic data processing requires information about the local slope in the data from estimates of local time and space derivatives of the data a total least squares algorithm gives an estimate of the local slope at each data point total least squares minimizes the orthogonal distance from the data points the local time and space derivatives to the fitted straight line defining the local slope this gives a more consistent estimate of the local slope than standard least squares because it takes into account uncertainty in the temporal and spatial derivatives the total least squares slope estimate is the same as the one obtained from using the structure tensor with a rectangular window function the estimate of the local slope field is used to extrapolate all traces in a seismic gather to the smallest recorded offset without using velocity information extrapolation to zero offset is done using a hyperbolic traveltime function in which slope information replaces the knowled

traveltime and wavefront curvature calculations in three dimensional inhomogeneous layered media with curved interfaces the seismic rays and wavefront curvatures are determined by solving a system of nonlinear ordinary differential equations for media with constant velocity and for media with constant velocity gradient simplified solutions exist in a general inhomogeneous medium these equations must be solved by numerical approximations the integration of the ray tracing and wavefront curvature equations is then performed by a modified divided difference form of the adams pece predict evaluate correct evaluate formulas and local extrapolation the interfaces between the layers are represented by bicubic splines the changes in ray direction and wavefront curvature at the interfaces are computed using standard formulas for three dimensional media two quadratic traveltime approximations have been proposed both are based on a taylor series expansion with reference to a ray from a reference source point to a reference receiver point the first approximation corresponds to expanding the square of the traveltime in a ta

three parameter radon transform based on shifted hyperbolas abstractthe focusing power of the conventional hyperbolic radon transform decreases for long offset seismic data due to the nonhyperbolic behavior of moveout curves at far offsets furthermore conventional radon transforms are ineffective for processing data sets containing events of different shapes the shifted hyperbola is a flexible three parameter zero offset traveltime slowness and focusing depth function which is capable of generating linear and hyperbolic shapes and improves the accuracy of the seismic traveltime approximation at far offsets radon transform based on shifted hyperbolas thus improves the focus of seismic events in the transform domain we have developed a new method for effective decomposition of seismic data by using such three parameter radon transform a very fast algorithm is constructed for high resolution calculations of the new radon transform using the recently proposed generalized fourier slice theorem gfst the gfst establishes an analytic expression between the f

bending of layers weakened by cuts abstract in the present paper a method proposed by one of the authors is extended to a class of skew symmetric elastic problems for the stress analysis of a layer supported by sliding fixed supports and weakened by several stress raisers the corresponding boundary value problem is reduced to an infinite system of one dimensional singular integral equations of the second kind the expressions for the stress components in an elastic layer weakened by stress raisers are presented based on the developed analytical algorithm extensive numerical investigations have been conducted the results of these investigations are illustrated graphically exposing some novel qualitative and quantitative knowledge about stress concentration in the layer depending on some geometric parameters of stress raisers and poisson s ratio of a layer material

equilibrium of layers weakened by loaded cuts abstract in this paper a symmetric boundary value problem of the stress analysis for an equilibrium of layer with end supports covered by diaphragms and weakened by several loaded stress raisers is investigated the given boundary value problem is reduced to an infinite system of singular integral equations of the second kind the expressions for stress components in an elastic layer weakened by stress raisers are presented based on the developed analytical procedure extensive numerical investigations have been conducted the results of these investigations are illustrated graphically exposing some novel qualitative and quantitative knowledge about stress concentration in the layer depending on some geometric parameters of stress raisers and poisson s ratio of a layer material

traveltime approximations for a layered transversely isotropic medium we consider multiple transmitted reflected and converted qp qsv waves or multiple transmitted and reflected sh waves in a horizontally layered medium that is transversely isotropic with a vertical symmetry axis vti traveltime and offset horizontal distance between a source and receiver not necessarily in the same layer are expressed as functions of horizontal slowness these functions are given in terms of a taylor series in slowness in exactly the same form as for a layered isotropic medium the coefficients depend on the parameters of the anisotropic layers through which the wave has passed and there is no weak anisotropy assumption using classical formulas the traveltime or traveltime squared can then be expressed as a taylor series in even powers of offset these taylor series give rise to a shifted hyperbola traveltime approximation and a new continued fraction approximation described by four parameters that match the taylor series up to the sixth power in offset further approximations give several simplified continued fraction approximations all of which depend on three parameters zero offset traveltime nmo velocity and a heterogeneity coefficient the approximations break down when there is a cusp in the group velocity for the qsv wave numerical studies indicate that approximations of traveltime squared are generally better than those for traveltime a new continued fraction approximation that depends on three parameters is more accurate than the commonly used continued fraction approximation and the shifted hyperbola

an anelliptic approximation for geometrical spreading in transversely isotropic and orthorhombic media abstractthe relative geometric spreading along the raypath contributes to the amplitude decay of the seismic wave propagation that needs to be considered for amplitude variation with offset or other seismic data processing methods that require the true amplitude processing expressing the p wave geometric spreading factor in terms of the offset traveltime based parameters is a more practical and convenient way because these parameters can be estimated from the nonhyperbolic velocity analysis we have developed an anelliptic approximation for the relative geometric spreading of p wave in a homogeneous transversely isotropic medium with vertical symmetry axis vti and an orthorhombic ort medium under the acoustic anisotropy assumption the coefficients in our approximation are only defined within the symmetry planes and computed from fitting with the exact parametric expression for an ort model due to the symmetric behavior in different symmetry planes the other coefficients in the approximation can b

bayesian linearized rock physics inversion abstractthe estimation of rock and fluid properties from seismic attributes is an inverse problem rock physics modeling provides physical relations to link elastic and petrophysical variables most of these models are nonlinear therefore the inversion generally requires complex iterative optimization algorithms to estimate the reservoir model of petrophysical properties we have developed a new approach based on the linearization of the rock physics forward model using first order taylor series approximations the mathematical method adopted for the inversion is the bayesian approach previously applied successfully to amplitude variation with offset linearized inversion we developed the analytical formulation of the linearized rock physics relations for three different models empirical granular media and inclusion models and we derived the formulation of the bayesian rock physics inversion under gaussian assumptions for the prior distribution of the model the application of the inversion to real

convergence of traveltime power series for a layered vti medium for a multiply reflected sh wave or a multiply reflected and converted qp qsv wave in a layered vti medium traveltime and offset can be expressed by power series in horizontal slowness as for a stack of isotropic layers these can be used to express traveltime as a series of even powers of offset the existence and behavior of this power series depend on the derivative of offset with respect to horizontal slowness formally expressed as traveltime multiplied by nmo velocity squared when the nmo velocity squared is different from zero the power series always exists for sufficiently small offset the nmo velocity squared is always positive for sh waves and for qp waves in media with normal polarization for which there are no triplications in traveltime as function of offset for qsv wave propagation insome layers the nmo velocity squared may be positive zero or negative when it is positive the power series exists but there may be an off axis triplication in traveltime when the nmo squared veloci

periodic seeded arrays and automorphisms of the shift the automorphism group autfsy of the full 2 shift is conjectured to be generated by the shift and involutions we approach this problem by studying a certain family of automorphisms whose order was unknown but which we show to be finite and for which we find factorizations as products of involutions the result of this investigation is the explicit construction of a subgroup z of aut z2 z is generated by certain involutions g and turns out to have a number of curious properties for example gn and gk commute unless n and k are consecutive integers the order of gn k o o gk is independent of k and z contains elements of all orders the investigation is aided by the development of results about certain new types of arrays of o s and l s called periodic seeded arrays as well as the use of boyle and krieger s work on return numbers and periodic points

solution of the elastic boundary value problems for a layer with tunnel stress raisers abstract a novel procedure for solving three dimensional problems for elastic layer weakened by through thickness tunnel cracks has been developed and is presented in this paper this procedure reduces the given boundary value problem to an infinite system of one dimensional singular integral equations and is based on a system of homogeneous solutions for a layer integral representations of single and double layer potentials are used for metaharmonic and harmonic functions entering in the singular integral equations these representations provide a continuous extendibility of the stress vector while allowing a jump in the displacement vector in the transition through the cut expanding the potential and biharmonic solutions in the fourier series over the thickness coordinate yields the integral representations of the displacement vector and stress tensor the problem of reducing a denumerable set of the integral equations of the given boundary value problem to one to one correspondence with the set of unknown densities appearing in the fourier s coefficient representations has been settled efficiently numerical investigations show a rapid convergence of the proposed reduction procedure as applied to the solution of the infinite system of one dimensional integral equations numerical examples illustrate the proposed method and demonstrate its advantages

power gradient velocity model a proposed power gradient velocity model incorporates several well known velocity models as special cases the model covers a wide range of possible velocity distributions and has four parameters giving more flexibility in velocity model manipulation for this nonlinear model the kinematic characteristics offset traveltime parameteric equations traveltime parameters relative geometric spreading and propagator phase are computed the characteristics are investigated with respect to a parameter responsible for nonlinearity of velocity distribution the inversion of travel time parameters was studied in three and four parameter frameworks

3d generalized nonhyperboloidal moveout approximation abstractmoveout approximations are commonly used in velocity analysis and time domain seismic imaging we revisit the previously proposed generalized nonhyperbolic moveout approximation and develop its extension to the 3d multiazimuth case the advantages of the generalized moveout approximation are its high accuracy and its ability to reduce to several other known approximations with particular choices of parameters the proposed 3d functional form involves 17 independent parameters instead of five as in the 2d case these parameters can be defined by zero offset traveltime attributes and four additional far offset rays in our tests the proposed approximation achieves significantly higher accuracy than previously proposed 3d approximations

conic velocity model the effect of gradually increasing velocity with depth in compacted sedimentary layers is described by asymptotically bounded velocity models these models are defined by three intuitive parameters the velocity its vertical gradient at an initial depth level and a bounded velocity value recently we have introduced the exponential asymptotically bounded eab model which belongs to this family of velocity models we introduce another monotonically increasing and asymptotically bounded model which approaches the asymptotic value in a slower and smoother fashion its main attractiveness is the simplicity of ray tracing the ray trajectories in this model are elliptic hyperbolic or parabolic curves depending on the ray parameter horizontal slowness the linear velocity model with a circular trajectory is a particular case of the proposed model because the shapes of the ray trajectories correspond to the three types of conic sections we call the model conic we derive the relationships for lateral propagation distance traveltime and arc length of the raypath for arbitrary nonvertical rays and we develop procedures for initial value and boundary value ray tracing both ray tracing problems have an analytical closed form solution

long offset moveout for vti using pade approximation abstractthe approximation of normal moveout is essential for estimating the anisotropy parameters of the transversally isotropic media with vertical symmetry axis vti we have approximated the long offset moveout using the pade approximation based on the higher order taylor series coefficients for vti media for a given anellipticity parameter we have the best accuracy when the numerator is one order higher than the denominator i e l l 1 thus we suggest using 4 3 and 7 6 orders for practical applications a 7 6 pade approximation can handle a much larger offset and stronger anellipticity parameter we have further compared the relative traveltime errors between the pade approximation and several approximations our method shows great superiority to most existing methods over a wide range of offset normalized offset up to 2 or offset to depth ratio up to 4 and anellipticity parameter 0 0 5 the pade approximation provides us with an attractive high accuracy scheme with an error that

periodic points and automorphisms of the shift on etudie le groupe d automorphisme d un decalage de markov topologique a l aide des points periodiques et des ensembles instables

nonhyperbolic reflection moveout in anisotropic media the standard hyperbolic approximation for reflection moveouts in layered media is accurate only for relatively short spreads even if the layers are isotropic velocity anisotropy may significantly enhance deviations from hyperbolic moveout nonhyperbolic analysis in anisotropic media is also important because conventional hyperbolic moveout processing on short spreads is insufficient to recover the true vertical velocity hence the depth we present analytic and numerical analysis of the combined influence of vertical transverse isotropy and layering on long spread reflection moveouts qualitative description of nonhyperbolic moveout on intermediate spreads offset to depth ratio x z 1 7 2 is given in terms of the exact fourth order taylor series expansion for p sv and p sv traveltime curves valid for multilayered transversely isotropic media with arbitrary strength of anisotropy we use this expansion to provide an analytic explanation for deviations from hyperbolic moveout such as the strong

a new parameterization for acoustic orthorhombic media abstractwe have defined a group of new parameterizations for p wave in acoustic orthorhombic ort media with three cross term normal moveout velocities and three cross term anellipticity parameters the corresponding perturbation based approximations for traveltime in ort model are developed using the new parameterizations the perturbation coefficients are computed by solving the eikonal equation in corresponding parameterization eight types of parameterization are defined based on the different elliptical background model and selection of anellipticity parameters as the traveltime can be converted from the group velocity inverse the sensitivity of the group velocity inverse to anellipticity parameters is analyzed for different parameterizations and different range of offsets to stabilize the perturbation series and improve the accuracy the shanks transform is applied from the comparison of traveltime after the shanks transform using different parameterizations we have concluded that the paramete

3d traveltime computation for quasi p wave in orthorhombic media using dynamic programming abstracta fractured area such as a fault area usually induces orthorhombic anisotropy ignoring orthorhombic anisotropy may degrade the subsurface image by creating a well mistie and blurring the image traveltime computation is essential for many processing techniques such as depth imaging and tomography solving the ray tracing system and eikonal equation are two popular methods for traveltime computation in isotropic media however because the ray tracing system becomes complex and the eikonal equation becomes highly nonlinear their applications in orthorhombic media become complex we have developed an alternative 3d traveltime computation method in orthorhombic media based on dynamic programming to avoid solving the complex ray tracing system and nonlinear eikonal equation it adopts an explicitly expressed group velocity from the moveout approximation to describe the propagation of the wavepath and computes the traveltime by fermat s principle similar to depth extrapolation it computes the t

on one skew symmetric problem of electroelasticity for inhomogeneous cylinder of finite length abstract in the present paper static bending problem of the electroelasticity for an inhomogeneous cylinder of finite length with sliding fixed end supports is investigated the given boundary value problem is reduced to a system of 12 k k 1 2 integro differential equations expressions for the components characterizing the state of stress for the inhomogeneous cylinder are presented based on the developed analytical algorithm extensive numerical investigations associated with the stress analysis of an inhomogeneous piezoceramic cylinder have been conducted the results of these investigations are illustrated graphically demonstrating the stress distributions in piezoceramic circular and elliptical cylinders with inclusions of various geometries

the automorphism group of a shift of finite type let xt at be a shift of finite type and g aut vt denote the group of homeomorphisms of xt commuting with t we investigate the algebraic properties of the countable group g and the dynamics of its action on xt and associated spaces using marker constructions we show g contains many groups such as the free group on two generators however g is residually finite so does not contain divisible groups or the infinite symmetric group the doubly exponential growth rate of the number of automorphisms depending on n coordinates leads to a new and nontrivial topological invariant of crt whose exact value is not known we prove that modulo a few points of low period g acts transitively on the set of points with least at period n using padic analysis we generalize to most finite type shifts a result of boyle and krieger that the gyration function of a full shift has infinite order the action of g on the dimension group of at is investigated we show there are no proper infinite compact g invariant sets we give a complete characterization of the g orbit closure of a continuous probability measure and deduce that the only continuous g invariant measure is that of maximal entropy examples questions and problems complement our analysis and we conclude with a brief survey of some remaining open problems

a theory of normal moveout three geophysical principles are shown to be sufficient to determine the most general practical normal moveout nmo equation the principles are reciprocity in a common midpoint cmp gather finite slowness and exact constant velocity limit the resulting equation is the shifted hyperbola nmo equation that has three parameters comparisons at both near and far offsets between the shifted hyperbola nmo equation and the results for layered media assign geophysical meaning to the parameters two of the parameters zero offset time and nmo velocity are constants and control the very near offset behavior the third parameter is dimensionless and controls the far offset behavior of the nmo curve but it may be a function of offset so as to exactly fit any traveltime curve the parameters may be found by a linear least squares fit to data the theory applies to all offsets for nonturning wave reflections in an isotropic earth for both p waves and converted p sv waves

low frequency layer induced anisotropy abstractit is often useful to compare velocities estimated from seismic experiments with those measured by well logs at much smaller scales and higher frequencies in the presence of fine layering seismic velocities and anisotropy are scale and frequency dependent and upscaling of well logs is necessary to allow comparison when upscaling well log data measured at sonic frequencies we assumed a layered medium with layer thickness given by logging step standard upscaling gives the exact solution for effective properties of a layered medium assuming that all constituents of the medium are linearly elastic and there are no anelastic energy losses this method is static meaning that the solution is obtained in the zero frequency limit we expand upscaling to low frequencies and propose to use a seismic wavelet estimate for weighted averaging of effective properties by expanding the logarithm of the propagator matrix computed from the stack of horizontal transversely isotropic layers with a vertical symm

processing of anisotropic data in the τ p domain ii common conversion point sorting common conversion point ccp sorting of p sv converted wave data is conventionally done by first sorting data into common asymptotic conversion point cacp gathers and then computing the involved ccp shifts from analytic approximations i explore an alternative method where the latter step is replaced by an entirely data driven approach moveout curves of correlated p p and p sv reflections in collocated cmp and cacp gathers are first scanned for points of equal slowness a common source slowness indicates that the downgoing branches of the p p and p sv waves overlap if the conversion occurs at the reflecting interface the p sv conversion point is then assumed to be situated underneath the associated p p wave midpoint a migration of amplitudes from cacp to ccp gathers is straightforward once the exact ccp position is known this data driven approach requires kinematic information only and is exact for laterally homogeneous media with arbitrary strength of anisotropy if horizontal symmetry planes are present at all depths both time offset and p domain implementations are possible although the latter are preferred

nonhyperbolic moveout analysis in vti media using rational interpolation anisotropic velocity analysis using qp waves in transversely isotropic media with a vertical symmetry axis vti usually is done by inferring the anellipticity parameter and the normal moveout velocity vnmo from the nonhyperbolic character of the moveout several approximations explicit in these parameters exist with varying degrees of accuracy here we present a rational interpolation approach to nonhyperbolic moveout analysis in the t x domain this method has no additional computational overhead compared to using expressions explicit in and vnmo the lack of such overhead stems from the observation that for fixed and zero offset two way traveltime t0 the moveout curve for different values of vnmo can be calculated by simple stretching of the offset axis this observation is based on the assumptions that the traveltimes of qp waves in transversely isotropic media mainly depend on and vnmo and that the shear wave velocity along the symmetry axis has a negligibleinfluence on these traveltim

global optimization of generalized nonhyperbolic moveout approximation for long offset normal moveout the approximation of normal moveout is essential for estimating the anisotropy parameters of anisotropic media the generalized nonhyperbolic moveout approximation gma brings a considerable improvement in accuracy compared to known analytical approximations however this is still prone to relatively large errors in the presence of relatively long offsets and large anisotropic parameters which would degrade the inversion accuracy due to error accumulation in velocity analysis we optimize the constant coefficients for all possible groupings of the anellipticity parameter and the offset to depth ratio within practical ranges theoretical analyses and numerical experiments show that the traditional optimization scheme using 2 norm solved by the least squares method could not provide an error constrained result in addition a direct optimization without constant coefficient extension would not lead to a satisfactory accuracy improvement we construct the objective function using the maximum norm and so

the sums of powers theorem for commuting block maps a block map is a map f 0 1 0 1 for some n 2 1 a block map f induces an endomorphismfto of the full 2 shift x a composition of block maps is defined in such a way that f o g o fo o go in this paper some recent results concerning the set g i g o f f o g for certain types of block maps f are extended it has been shown 2 theorem 3 4 that the endomorphisms of the shift dynamical system over 0 1 can be studied via block maps an n block is a sequence b bn where b1 e 0 11 for 1 i n and an n block map is a function from the set of n blocks to the set 0 11 two endomorphisms commute if and only if the corresponding block maps commute there is a 1 1 relationship between n block maps and polynomial functions in n variables over z2 2 theorem 19 1 the commuting block maps problem has been solved in 1 for linear block maps and for block maps with polynomials of the form