comments on an asymptotic modal observer for linear autonomous time lag systems this note first shows that the proof of theorem1 in the above mentioned paper is incomplete and that the state observer scheme results in steady state error alternatively a memoryless state observer for linear time delay systems is provided

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

traveltime inversion for dipping layers large moveouts and arbitrary source and receiver positions the seismic traveltime inversion algorithm presented here requires few limiting assumptions and is computationally efficient the traveltime for a raypath in a layered earth uniform or linearly varying sound speeds in each layer can be written in closed form if the coordinates of the intersections between the raypath and the layer interfaces are known the traveltime on an actual raypath is minimal with respect to neighboring paths the raypath can thus be determined by equating the derivatives of the traveltimes with respect to the intersection coordinates to zero this approach can be applied to the inverse problem which seeks those layer parameter values which best fit the observed traveltimes from a cmp reflection gather the result is a system of equations in which the intersection coordinates and layer parameters are independent variables with the constraint that the traveltimes be minimal with respect to the coordinates when this systemis linearized with a step size constraint an efficient

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

a simple exact solution for the p sv wave conversion point via prestack migration the conversion point cp is the lateral position of the reflection point relative to the source receiver midpoint for a laterally homogeneous earth horizontal reflectors and the p sv converted wave c wave geometry depicted in figure 1 the source emits a p wave which propagates to the horizontal reflector where it reflects as an sv wave and then propagates to the receiver common conversion point ccp binning and stacking are tools that use cp modeling for processing c wave data applications include iterative statics and velocity analysis procedures acquisition design methods also use cp modeling see thomsen 1999 calculation of the cp is complicated unfortunately and exact solutions exist only for homogeneous and isotropic media thomsen 1999 developed asymptotic expansions that approximate the cp in vertically inhomogeneous isotropic and anisotropic media yuan and li 2001 extended these expressions improving their accuracy for large source receiver offsets exact solutions for the cp exist only in simple media and that is the subject of this paper tessmer and behle 1988 used fermat9s principle to find a quartic equation for the exact cp in homogeneous isotropic media i show that the equation for the c wave prestack migration impulse response yields a cubic equation for the exact cp in homogeneous isotropic media analytically solving the cubic equation is less cumbersome than analytically solving the quartic equation and i have obtained the exact solution of the new cubic equation in simple trigonometric form the tessmer and behle 1998 quartic cp equation depends upon the reflector depth whereas the new cubic cp equation depends instead upon the two way reflection traveltime of the c wave thus the new result directly specifies the cp for each sample of a prestack seismic trace assuming reflectors are horizontal

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

solution to morgan s problem a necessary and sufficient condition is presented for the solution of the row by row decoupling problem known as morgan s problem in the general case that is without any restrictive assumption added to the system to the feedback law u fx gy g may be noninvertible this is a structural condition in terms of invariant lists of integers which are easily computable from a given state realization of the system these integers are the infinite zero orders morse s list i sub 4 and the essential orders of the system which only depend on the input output behavior and morse s list i sub 2 of the system which depends on the choice of a particular state realization

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

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

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

series expansions for 1 πm and πm abstract in terms of the telescoping method we establish a general identity then it is used to derive several interesting series expansions for 1 m and m with m n

observers for discrete time systems with multiple delays a systematic approach to design observers for discrete time systems with multiple delays under general conditions is presented the main feature of the proposed technique is that necessary and sufficient conditions for asymptotic stability are derived while the observer s order is independent from the number of delays and is equal to the state dimension to illustrate the efficiency of the proposed technique two numerical examples are provided

multidimensional inverse scattering series internal multiple prediction in the coupled plane wave domain the inverse scattering series internal multiple prediction and attenuation algorithm predicts multiples using certain combinations of input seismic reflection data events which are computed in the wavenumber pseudo depth or plane wave vertical travel time i e p domains significant differences can arise in the algorithms output and computational expense depending on which domain is used many of these are traceable to the response of the algorithm to the users choice of the search limiting parameter the question of which domain is optimal can be addressed with benchmark synthetics the compactness of the input to the plane wave domain algorithm leads to the expectation that it will have a reduced computational expense also the lack of increase in the dominant period i e width of input events as the horizontal slowness increases leads to the expectation that it will respond well to a constant both of these expectations are borne out with a 1 5d benchmark example a 2d plane wave pred

an asymptotic modal observer for linear autonomous time lag systems a full state vector observer is derived for a class of linear autonomous time lag systems represented by differential equations the major features of the approach are 1 an observer is constructed just for the set of unstable or poorly damped modes of the delayed system once they have been determined and 2 the design is accomplished by using a partial fraction expansion separating the unstable part from the system and then applying well known finite dimensional state vector techniques a procedure for the construction is given along with some numerical results to illustrate the simplicity of the observer

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

the traveltime equation tau p mapping and inversion of common midpoint data seismic traveltime data can be described in terms of instantaneous slope dt dx and intercept time with this parameterization the most general form of the traveltime equation is developed and found to be valid for reflections and refractions all source receiver offsets and all commonly used experimental geometries for common source receiver experiments the instantaneous slope of the traveltime data represents the true horizontal ray parameter p and the intercept time is the sum along the ray of thickness vertical slowness products referenced to the fixed surface station for common midpoint cmp experiments we show that the instantaneous slope is the average slowness of upgoing and downgoing rays at the surface and the intercept time is again the sum of thickness vertical slowness products but referenced to the common midpoint this inherent averaging immediately explains the superiority of the cmp experiment in the presence of dip in addition to providing a general traveltime equation w

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

a memoryless state observer for discrete time delay systems this paper presents a reduced order memoryless state observer with a spl gamma stability margin 0 spl gamma 1 for linear discrete time systems with multiple delays in both the state and control vectors the dynamics of the observer are derived from a set of unstable and or poorly damped eigenvalues of the system a simple and systematic design method is presented a numerical example is given to illustrate the properties of the new observer and its design method

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

adaptive robust state observers for a class of uncertain nonlinear dynamical systems with delayed state perturbations the problem of adaptive robust state observer design is considered for a class of uncertain nonlinear systems with delayed state perturbations it is supposed that the upper bound of the nonlinearity and uncertainty including delayed states is a linear function of some parameters which are still assumed to be unknown here it is not required that the nonlinear terms including delayed states are linear norm bounded in the states an improved adaptation law with sigma modification is employed to estimate the unknown parameters then by making use of the updated values of these unknown parameters a class of memoryless adaptive robust state observers is proposed for uncertain nonlinear time delay systems it is also shown that by employing the proposed adaptive robust state observer the observation error between the observer state estimate and the true state can converge asymptotically to zero in the presence of significant uncertainties and time delays finally a numerical example is given to demonstrate the validity of the results

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

π formulas with free parameters abstract in terms of the hypergeometric method we establish ten general formulas with free parameters which include several known results as special cases

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

time delay systems design of delay free and low order observers this note provides a comprehensive treatment on the design of functional observers for linear systems having a time varying delay in the state variables the designed observers possess attractive features of being low order and delay free and hence they are cost effective and easy to implement existence conditions are derived and a design procedure for finding low order observers is given

a summation formula and ramanujan type series abstract using some properties of the general rising shifted factorial and the gamma function we derive a variant form of dougall s f 4 5 summation for the classical hypergeometric functions this variant form allows us to derive easily many ramanujan type series for 1 and ramanujan type series for some other constants

hypergeometric series summations and π formulas abstract by two slater s hypergeometric series identities we establish two general summation formulas with six free parameters specializing certain parameters in these formulas we obtain a list of ramanujan type series formulas for 2 and 3

observers design for linear time delay systems abstract in this contribution we propose a simple and useful approach to design observers for discrete time systems with delays in the state and output variables the main feature is that the necessary and sufficient conditions for the existence of such observer are derived the stability analysis is performed by the lyapunov approach where the obtained conditions are expressed in terms of a modified riccati equation numerical examples are provided to show efficiency of the proposed observer

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

decoupling without prediction of linear systems with delays a structural approach we exhibit using a new structural approach specific for linear systems with delays a necessary and sufficient condition for static row by row decoupling without prediction moreover the decoupling problem is analyzed through two different concepts the total one without any time assumption and the partial one in a temporal mean which can be considered if the total one has no solution

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