an energy conserving parabolic equation for elastic media the accuracy of the parabolic equation pe method is improved for range dependent elastic media the energy conserving elastic pe is a generalization of the energy conserving acoustic pe a range dependent waveguide is approximated by a sequence of range independent regions a linear approximation is derived for the nonlinear condition corresponding to conservation of compressional energy flux at the vertical interfaces between range independent regions the rotated elastic pe is applied to generate reference solutions for demonstrating accuracy the energy conserving elastic pe is applied to solve problems involving the propagation of an interface wave along a sloping ocean bottom and a problem involving mode cutoff and coupling into shear wave beams in the ocean bottom the latter problem is analogous to a cutoff problem that was first solved with the acoustic pe f b jensen and w a kuperman j acoust soc am 67 1564 1566 1980

treatment of a sloping fluid solid interface and sediment layering with the seismo acoustic parabolic equation the parabolic equation method is extended to handle problems in seismo acoustics that have multiple fluid and solid layers continuous depth dependence within layers and sloping interfaces between layers the medium is approximated in terms of a series of range independent regions and a single scattering approximation is used to compute transmitted fields across the vertical interfaces between regions the approach is implemented in terms of a set of dependent variables that is well suited to piecewise continuous depth dependence in the elastic parameters but one of the fluid solid interface conditions in that formulation involves a second derivative that complicates the treatment of sloping interfaces this issue is resolved by using a non centered four point difference formula for the second derivative the approach is implemented using a matrix decomposition that is efficient when the parameters of the medium have a general dependence within the upper layers of the sediment but only depend on depth in the water column and deep within the sediment

fast and explicit wentzel kramers brillouin mode sum for the bottom interacting field including leaky modes a fast and explicit method is presented for calculating the bottom interacting field by means of a mode sum using the wentzel kramers brillouin approximation for the eigenfunctions and the tindle weston formula for eigenvalue spacing both trapped and leaky modes are automatically included requiring no numerical inversions for either eigenfunctions or eigenvalues examples are provided which demonstrate the speed and accuracy of the approach by comparison with reference solutions the importance of leaky modes especially in range dependent environments with variable water depth is demonstrated a simple criterion for neglecting leaky modes is developed

bispectrum estimation a parametric approach higher order spectra contain information about random processes that is not contained in the ordinary power spectrum such as the degree of nonlinearity and deviations from normality estimation of the bispectrum which is a third order spectrum has been applied in various fields to obtain information regarding quadratic phase coupling among harmonic components and non gaussianness of processes existing methods of bispectrum estimation are patterned after the conventional methods of power spectrum estimation which are known to possess certain limitations the paper proposes a parametric approach to bispectrum estimation based on a non gaussian white noise driven autoregressive ar model the ar parameter estimates are obtained by solving the third order recursion equations which may be toeplitz in form but not symmetric it is shown that the method provides bispectral estimates that are far superior to the conventional estimates in terms of bispectral fidelity and that in the case of detecting phase coupling among sinusoids the method provides significantly better resolution

cocyclic hadamard matrices from forms over finite frobenius rings abstract we give a construction of cocyclic butson hadamard matrices from bilinear forms over finite frobenius rings which generalizes previous constructions we then prove a classification result showing that the resulting matrices depend only on the additive group of the underlying module of the form

generalized bremmer series with rational approximation for the scattering of waves in inhomogeneous media the bremmer series solution of the wave equation in generally inhomogeneous media requires the introduction of pseudodifferential operators in this paper sparse matrix representations of these pseudodifferential operators are derived the authors focus on designing sparse matrices keeping the accuracy high at the cost of ignoring any critical scattering angle phenomena such matrix representations follow from rational approximations of the vertical slowness and the transverse laplace operator symbols and of the vertical derivative as they appear in the parabolic equation method sparse matrix representations lead to a fast algorithm an optimization procedure is followed to minimize the errors in the high frequency limit for a given discretization rate the bremmer series solver consists of three steps directional decomposition into up and downgoing waves one way propagation and interaction of the counterpropagating constituents each of these steps is represented by a sparse matrix equation the resulting algorithm provides an improvement of the parabolic equation method in particular for transient wave phenomena and extends the latter method systematically for backscattered waves

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

nonlocal boundary conditions for finite difference parabolic equation solvers theoretical numerical models of underwater sound propagation usually incorporate a downgoing radiation condition on the transverse component of the acoustic field most one way or parabolic equation pe solvers approximate this radiation condition by appending an absorbing layer to the computational mesh and setting the field to zero at the base of this layer papadakis et al j acoust soc am 92 2030 2038 1992 replaced such approximate treatments with a nonlocal boundary condition nlbc that exactly transforms the semi infinite pe problem to an equivalent one in a bounded domain papadakis approach requires the evaluation of a spectral wave number integral whose integrand is inversely proportional to the impedance of the subbottom medium in this paper an alternate procedure is analyzed for obtaining nlbcs directly from the z space crank nicolson solvers for both the tappert and claerbout pes formulas for the field at range r r are derived in terms of the known field at the previously c

structure and stability of wave theoretic kernels in the ocean wave theoretic modeling can be applied to obtain travel time sensitivity kernels tsks representing the amount ray travel times are affected by sound speed variations anywhere in the medium this work explores the spatial frequency content of the tsk compared to expected ocean variability it also examines the stability of the tsk in environments that produce strong sensitivity of ray paths to initial conditions the conclusion is that the linear tsk model is an effective predictor of travel time changes and that the rays perform nearly as well as the full wave kernel the tsk is examined in physical space and in wavenumber space and it is found that this is the key to understanding how the travel time reacts to ocean perturbations there are minimum vertical and horizontal length scales of ocean perturbations that are required for the travel time to be affected the result is that the correspondence between true travel times and those calculated from the kernel is high for large scale perturbations and somewhat less for the small scales this demonstrates the validity of ray based inversion of travel time observations for the cases under study

finite difference schemes for the parabolic equation in a variable depth environment with a rigid bottom boundary condition we consider a linear schrodinger type partial differential equation the parabolic equation of underwater acoustics in a layer of water bounded below by a rigid bottom of variable topography using a change of depth variable technique we transform the problem into one with horizontal bottom for which we establish an a priori h1 estimate and prove an optimal order error bound in the maximum norm for a crank nicolson type finite difference approximation of its solution we also consider the same problem with an alternative rigid bottom boundary condition due to abrahamsson and kreiss and prove again a priori h1 estimates and optimal order error bounds for a crank nicolson scheme

nonlinear wide angle paraxial acoustic propagation in shallow water channels a time domain model that describes wide angle paraxial propagation of acoustic pulses in shallow water is developed this model incorporates weak nonlinear effects and depth variability in both ambient density and sound speed derivations of paraxial approximations are based on an iterative approach in which the wide angle approximation is obtained by using a narrow angle equation to approximate the second range derivative in the two way equation scaling arguments are used to obtain a more tractable simplification of the equation the wide angle equation is solved numerically by splitting into components representing distinct physical processes and using a modified version of the time domain parabolic equation tdpe code m d collins j acoust soc am 84 2114 2125 1988 a high order upwind flux correction method is modified to handle the nonlinear component numerical results are presented for adiabatic propagation in a shallow isospeed channel it is demonstrated that nonlinear effects are significant even at small ranges if the peak source pressure is high enough both nonlinear and wide angle effects are illustrated and their differences are discussed

a mapping approach for handling sloping interfaces a mapping approach for handling sloping interfaces in parabolic equation solutions is developed and tested at each range the medium is rigidly translated vertically so that a sloping interface becomes horizontal to simplify the approach the slope is assumed to be small and the extra terms that arise in the wave equation under the mapping are neglected the effects of these terms can be approximately accounted for by applying a leading order correction to the phase the mapping introduces variations in topography which are relatively easy to handle for the case of a pressure release boundary condition the accuracy of the approach is demonstrated for problems involving fluid sediments the approach should also be accurate for problems involving elastic sediments and should be useful for solving three dimensional problems involving variable topography

a wide angle and high mach number parabolic equation various parabolic equations for advected acoustic waves have been derived based on the assumptions of small mach number and narrow propagation angles which are of limited validity in atmospheric acoustics a parabolic equation solution that does not require these assumptions is derived in the weak shear limit which is appropriate for frequencies of about 0 1 hz and above for atmospheric acoustics when the variables are scaled appropriately in this limit terms involving derivatives of the sound speed density and wind speed are small but can have significant cumulative effects to obtain a solution that is valid at large distances from the source it is necessary to account for linear terms in the first derivatives of these quantities a d pierce j acoust soc am 87 2292 2299 1990 this approach is used to obtain a scalar wave equation for advected waves since this equation contains two depth operators that do not commute with each other it does not readily factor into outgoing and incomin

a two way parabolic equation for a fluid elastic waveguide the parabolic equation pe method is applied to acoustic propagation in a fluid above an elastic layer two way pes have previously been developed for purely fluid or elastic waveguides m d collins j acoust soc am 93 1815 1825 1993 these methods divide a range dependent environment into a sequence of range independent regions and apply the single scattering method to obtain the transmitted and reflected fields at each interface recently these ideas have been extended to mixed media such as fluid poro acoustic waveguides improved pade approximations for the interface conditions are useful in stabilizing the iteration formula for the transmitted and reflected fields f a milinazzo et al j acoust soc am 101 760 766 1997 in this paper a fluid elastic waveguide is examined and examples of the coupling between the fluid compressional wave and elastic waves at vertical interfaces are presented work supported by onr

the one way wave equation and its invariance properties the harmonic wave equation in inhomogeneous media is exactly split into coupled first order equations with respect to a principal direction of propagation according to the bremmer scheme the resulting one way wave equation is shown not to conserve energy flux for dimensions two and three against the general belief in one way wave propagation or parabolic equation literature conservation of energy flux is only ensured in the high frequency limit on the other hand a simple invariant is found that may be seen as a generalization of the snell law to arbitrary non stratified media similarly the reciprocity property is not fully ensured in general and the time reversal symmetry is ensured for propagating fields besides in the one way wave equation the additional term to the standard parabolic equation is shown to strengthen mode coupling the analysis encompasses the evanescent waves

an improved numerical scheme for the sine gordon equation in 2 1 dimensions a rational approximant of order 4 which is applied to a three time level recurrence relation is used to transform the initial boundary value problem associated with the two dimensional sine gordon sg equation arising in the josephson junctions problem the resulting non linear system which is analyzed for stability is solved using an appropriate predictor corrector p c scheme in which an explicit scheme of order 2 is used as predictor for the implementation of the corrector in order to avoid extended matrix evaluations an auxiliary vector was successfully introduced in this p c scheme a modification in the corrector has been proposed according to which the already evaluated corrected values are considered the behavior of this p c scheme is tested numerically on line and ring solitons known from the bibliography regarding the sg equation and conclusions for both undamped and damped problems are derived copyright 2008 john wiley sons ltd

rational square root approximations for parabolic equation algorithms in this article stable pade approximations to the function 1 z are derived by choosing a branch cut in the negative half plane the pade coefficients are complex and may be derived analytically to arbitrary order from the corresponding real coefficients associated with the principal branch defined by z 1 i z 0 i z denotes the imaginary part of z the characteristics of the corresponding square root approximation are illustrated for various segments of the complex plane in particular for waveguide problems it is shown that an increasingly accurate representation may be obtained of both the evanescent part of the mode spectrum for the acoustic case and the complex mode spectrum for the elastic case an elastic parabolic equation algorithm is used to illustrate the application of the new pade approximations to a realistic ocean environment including elasticity in the ocean bottom

range dependent matched field inversion of swellex 96 data using the downhill simplex algorithm matched field inversion mfi techniques have been applied for effective and efficient estimation of geoacoustic parameters of the ocean bottom this paper presents a new tomographic mfi method for use in range dependent environments the mfi correlates modeled data with measured data and uses a search algorithm to determine model parameter values that maximize the correlator in the present method a parabolic equation propagation model is used to compute replica fields to account for mode coupling in the environment the search algorithm is a two stage hybrid method that combines an initial global component and a final local component the first stage employs a random search to determine n 1 parameter sets with the best correlations where n is the number of parameters being determined in the next stage the n 1 sets are used as inputs to the local downhill simplex algorithm the algorithm is shown to perform well for simulated vertical line array data for an environment representative of the swellex 96 experimental site the inversion technique is then applied to measured data obtained from a radial track in swellex 96 the geometric parameters of the experiment and dominant geoacoustic parameters were successfully recovered for data with relatively low signal to noise ratio

a two way parabolic equation for acoustic backscattering in the ocean the parabolic equation pe method is generalized to handle backscattered acoustic energy in the ocean the two way pe is based on the single scattering approximation and the approach of two way coupled modes in which range dependent environments are approximated by a sequence of range independent regions at the vertical boundaries between regions the solution of the two way pe is required to satisfy two continuity conditions the range derivative in one of the conditions is replaced by a higher order pe depth operator the reflected and transmitted fields that satisfy these conditions are computed with an efficient iteration scheme the outgoing and incoming fields are propagated by two way range marching the two way pe which is presently implemented for two dimensional problems is a practical method for solving large scale reverberation problems the accuracy of the two way pe is demonstrated by comparisons with reference solutions the two way pe is applied to simulate the localization of a source of backscattering using the method of back propagation

scalar electromagnetic propagation modelling using parabolic equations and the split step pade approximation this paper presents a solution of 2d electromagnetic wave propagation problems in complicated terrestrial domains scalar 2d parabolic approximations are derived from maxwell s equations for both vertical and horizontal polarization the parabolic equations are then solved using a new technique involving pade rational function approximations of the macroscopic operator this method allows for larger than normal range stepping speeding up computational time significantly the pade approximation and the numerical implementation are fully discussed the discontinuity at the earth s surface is handled directly by using classical continuity conditions and deriving exact interface conditions for linking the fields in the atmosphere to those in the terrain the interface conditions are then implemented using the concept of virtual points preliminary benchmark tests show the interface treatment to work well finally several example runs are presented illustrating results

applications and time domain solution of higher order parabolic equations in underwater acoustics a higher order parabolic equation and the corresponding higher order time domain parabolic equation are derived from a pade series and solved numerically these models provide accurate solutions for problems involving very wide angle propagation e g propagation in the nearfield or over a hard ocean bottom propagation in domains in which sound speed variations are large e g propagation in deep water deep within the ocean bottom in high speed ocean bottoms and possibly of different wave types and propagation out to very long ranges the possibility of modeling elastic wave propagation with a similar approach is considered

treatment of ice cover and other thin elastic layers with the parabolic equation method the parabolic equation method is extended to handle problems involving ice cover and other thin elastic layers parabolic equation solutions are based on rational approximations that are designed using accuracy constraints to ensure that the propagating modes are handled properly and stability constrains to ensure that the non propagating modes are annihilated the non propagating modes are especially problematic for problems involving thin elastic layers it is demonstrated that stable results may be obtained for such problems by using rotated rational approximations milinazzo zala and brooke j acoust soc am 101 760 766 1997 and generalizations of these approximations the approach is applied to problems involving ice cover with variable thickness and sediment layers that taper to zero thickness

the z sub 4 linearity of kerdock preparata goethals and related codes certain notorious nonlinear binary codes contain more codewords than any known linear code these include the codes constructed by nordstrom robinson 1967 kerdock 1972 preparata 1968 goethals 1974 and delsarte goethals 1975 it is shown here that all these codes can be very simply constructed as binary images under the gray map of linear codes over z sub 4 the integers mod 4 although this requires a slight modification of the preparata and goethals codes the construction implies that all these binary codes are distance invariant duality in the z sub 4 domain implies that the binary images have dual weight distributions the kerdock and preparata codes are duals over z sub 4 and the nordstrom robinson code is self dual which explains why their weight distributions are dual to each other the kerdock and preparata codes are z sub 4 analogues of first order reed muller and extended hamming codes respectively all these codes are extended cyclic codes over z sub 4 which greatly simplifies encoding and decoding an algebraic hard decision decoding algorithm is given for the preparata code and a hadamard transform soft decision decoding algorithm for the i kerdock code binary first and second order reed muller codes are also linear over z sub 4 but extended hamming codes of length n spl ges 32 and the golay code are not using z sub 4 linearity a new family of distance regular graphs are constructed on the cosets of the preparata code

green s function estimation using secondary sources in a shallow water environment this work provides a new way to measure the green s function between two points in an acoustic channel without emitting a pulse by any of the two points the green s function between a and b is obtained from a set of secondary sources in the guide by averaging either the correlation or the convolution of the signals received in a and b a theoretical approach based on mode propagation in a monochromatic regime is presented results are then extended to the time domain estimation of the green s function is performed numerically in a range independent and a range dependent environment application to discreet acoustic communications is discussed

two parabolic equations for propagation in layered poro elastic media parabolic equation methods for fluid and elastic media are extended to layered poro elastic media including some shallow water sediments a previous parabolic equation solution for one model of range independent poro elastic media collins et al j acoust soc am 98 1645 1656 1995 does not produce accurate solutions for environments with multiple poro elastic layers first a dependent variable formulation for parabolic equations used with elastic media is generalized to layered poro elastic media an improvement in accuracy is obtained using a second dependent variable formulation that conserves dependent variables across interfaces between horizontally stratified layers furthermore this formulation expresses conditions at interfaces using no depth derivatives higher than first order this feature should aid in treating range dependence because convenient matching across interfaces is possible with discretized derivatives of first order in contrast to second order

4 phase sequences with near optimum correlation properties two families of four phase sequences are constructed using irreducible polynomials over z sub 4 family a has period l 2 sup r 1 size l 2 and maximum nontrivial correlation magnitude c sub max

discrete non local boundary conditions for split step pade approximations of the one way helmholtz equation this paper deals with the efficient numerical solution of the two dimensional one way helmholtz equation posed on an unbounded domain in this case one has to introduce artificial boundary conditions to confine the computational domain the main topic of this work is the construction of the so called discrete transparent boundary conditions for state of the art parabolic equation methods namely a split step discretization of the high order parabolic approximation and the split step pade algorithm of collins finally several numerical examples arising in optics and underwater acoustics illustrate the efficiency and accuracy of our approach

higher order pade approximations for accurate and stable elastic parabolic equations with application to interface wave propagation higher order pade approximations are applied to derive accurate and stable parabolic equations for sound propagation in oceans bounded below by an elastic bottom or bounded above by ice cover accuracy is achieved by placing constraints on the derivatives of the pade approximations at the point corresponding to the reference wave number stability is achieved by requiring that the pade approximations map part of the lower left quadrant of the complex plane into the upper half of the complex plane elastic parabolic equations based on these pade series can handle problems involving compressional shear and interface waves very wide propagation angles and large depth variations and weak range variations in the seismoacoustic parameters a finite difference spectral solution is developed for generating reference solutions and starting fields the rotated elastic parabolic equation is used to investigate the accuracy of the elastic parabolic equation for range dependent problems

a single scattering correction for large contrasts in elastic layers the single scattering solution is implemented in a formulation that makes it possible to accurately handle solid solid interfaces with the parabolic equation method problems involving large contrasts across sloping stratigraphy can be handled by subdividing a vertical interface into a series of two or more scattering problems the approach can handle complex layering and is applicable to a large class of seismic problems the solution of the scattering problem is based on an iteration formula which has improved convergence in the new formulation and the transverse operator of the parabolic wave equation which is implemented efficiently in terms of banded matrices accurate solutions can often be obtained by using only one iteration

dispersive waves and pml performance for large stencil differencing in the parabolic equation the discrete reflection absorption performance of perfectly matched layer pml boundary conditions is a function of the field discretization scheme throughout the computational domain and the form of the complex coordinate extension that characterizes the pml region most previous work analyzing pml reflection performance has assumed use of low order second or fourth small stencil three or five point spatial discretizations whose dispersion is poor outside a narrow low wavenumber band large stencil differencing schemes regardless of order are easily created with physically realistic wide band dispersive characteristics but the effect their adoption has on pml performance has not been analyzed previously in the context of a semi discrete approximation to the parabolic paraxial equation this work derives the analytic form of the discrete pml reflection coefficient for an arbitrary order arbitrary stencil interior differencing scheme coupled to an arbitrary order arbitrary stencil pml region this result is then used to show that the absorption performance of the pml regions is largely insensitive to changes in interior differencing as a consequence the benefits of superior dispersion characteristics provided by use of large stencil differencing in the interior physical domain can be had without incurring discrete reflection penalties at the pml

translates of linear codes over z sub 4 we give a method to compute the complete weight distribution of translates of linear codes over z sub 4 the method follows known ideas that have already been used successfully by others for hamming weight distributions for the particular case of quaternary preparata codes we obtain that the number of distinct complete weights for the dual preparata codes and the number of distinct complete coset weight enumerators for the preparata codes are both equal to ten independent of the code length

a three dimensional parabolic equation model of sound propagation using higher order operator splitting and pade approximants an alternating direction implicit adi three dimensional fluid parabolic equation solution method with enhanced accuracy is presented the method uses a square root helmholtz operator splitting algorithm that retains cross multiplied operator terms that have been previously neglected with these higher order cross terms the valid angular range of the parabolic equation solution is improved the method is tested for accuracy against an image solution in an idealized wedge problem computational efficiency improvements resulting from the adi discretization are also discussed

generalization of the rotated parabolic equation to variable slopes the rotated parabolic equation j acoust soc am 87 1035 1037 1990 is generalized to problems involving ocean sediment interfaces of variable slope the approach is based on approximating a variable slope in terms of a series of constant slope regions the original rotated parabolic equation algorithm is used to march the field through each region an interpolation extrapolation approach is used to generate a starting field at the beginning of each region beyond the one containing the source for the elastic case a series of operators is applied to rotate the dependent variable vector along with the coordinate system the variable rotated parabolic equation should provide accurate solutions to a large class of range dependent seismo acoustics problems for the fluid case the accuracy of the approach is confirmed through comparisons with reference solutions for the elastic case variable rotated parabolic equation solutions are compared with energy conserving and mapping solutions

on the effect of error correlation on matched field geoacoustic inversion the effect of correlated data errors on matched field geoacoustic inversion for vertical array data is examined the correlated errors stem from our inability to model the inhomogeneities in the environment resulting in an additional error term beyond ambient noise simulated data with these correlated errors are generated and then inverted with or without using the proper covariance matrix results show that the correlated error has a negative impact on geoacoustic parameter estimation if not accounted for properly

numerical solutions of range dependent benchmark problems in ocean acoustics accurate numerical solutions are presented for benchmark problems associated with propagation in a wedge with different bottom boundary conditions as well as in a plane parallel waveguide with changing profile with range the benchmark problems are solved with three different general purpose numerical codes 1 a coupled mode code providing a full spectrum two way backscatter included solution of the wave equation 2 a finite difference pe code providing a wide angle solution of the one way parabolic approximation of the wave equation and 3 a split step pe code providing a more narrow angle solution of the one way parabolic approximation of the wave equation only the two way coupled mode results can be considered accurate solutions of the benchmark problems

a comparison of transparent boundary conditions for the fresnel equation abstract we consider two numerical transparent boundary conditions that have been previously introduced in the literature the first condition bpp was proposed by baskakov and popov 1991 wave motion 14 121 128 and papadakis et al 1992 j acoust soc am 92 2030 2038 while the second sdy is that of schmidt and deuflhard 1995 comput math appl 29 53 76 and schmidt and yevick 1997 j comput phys 134 96 107 the latter procedure is explicitly tailored to the form of the underlying numerical propagation scheme and is therefore unconditionally stable and highly precise here we present a new derivation of the sdy approach as a result of this analysis we obtain a simple modification of the bpp method that guarantees accuracy and stability for long propagation step lengths

arma bispectrum approach to nonminimum phase system identification an identification procedure is proposed for a nongaussian white noise driven linear time invariant nonminimum phase fir finite impulse response system the method is based on parametric modeling of the third moments of the output sequence and uses causal and anticausal autoregressive moving average arma models the magnitude and phase response of the system are expressed in terms of the ar parameters of the arma models in particular the ar part of the causal arma model captures the minimum phase component of the system and the ar part of the anticausal arma captures the maximum phase component both sets of parameters are obtained by solving overdetermined linear systems of equations a model order selection criterion based on third order moments is proposed the arma bispectrum approach is compared to more conventional approaches for magnitude and phase reconstruction it is demonstrated that the proposed identification procedure exhibits improved modeling performance the method does not require knowledge of the non gaussian noise distribution

single scattering parabolic equation solutions for elastic media propagation including rayleigh waves the parabolic equation method with a single scattering correction allows for accurate modeling of range dependent environments in elastic layered media for problems with large contrasts accuracy and efficiency are gained by subdividing vertical interfaces into a series of two or more single scattering problems this approach generates several computational parameters such as the number of interface slices an iteration convergence parameter and the number of iterations n for convergence using a narrow angle approximation the choices of n 1 and 2 give accurate solutions analogous results from the narrow angle approximation extend to environments with larger variations when slices are used as needed at vertical interfaces the approach is applied to a generic ocean waveguide that includes the generation of a rayleigh interface wave results are presented in both frequency and time domains

a single scattering correction for the seismo acoustic parabolic equation an efficient single scattering correction that does not require iterations is derived and tested for the seismo acoustic parabolic equation the approach is applicable to problems involving gradual range dependence in a waveguide with fluid and solid layers including the key case of a sloping fluid solid interface the single scattering correction is asymptotically equivalent to a special case of a single scattering correction for problems that only have solid layers kusel et al j acoust soc am 121 808 813 2007 the single scattering correction has a simple interpretation conservation of interface conditions in an average sense that facilitated its generalization to problems involving fluid layers promising results are obtained for problems in which the ocean bottom interface has a small slope

a numerical method for ocean acoustic normal modes the method of normal modes is frequently used to solve acoustic propagation problems in stratified oceans the propagation numbers for the modes are the eigenvalues of the boundary value problem to determine the depth dependent normal modes errors in the numerical determination of these eigenvalues appear as phase shifts in the range dependence of the acoustic field such errors can severely degrade the accuracy of the normal mode representation particularly at long ranges in this paper we present a fast finite difference method to accurately determine these propagation numbers and the corresponding normal modes it consists of a combination of well known numerical procedures such as sturm sequences the bisection method newton s and brent s methods richardson extrapolation and inverse iteration we also introduce a modified richardson extrapolation procedure that substantially increases the speed and accuracy of the computation

rational operators for filtering the parabolic equation pe method has been applied to approximate the wave equation construct radiation conditions solve scattering problems construct initial conditions and derive energy flux conditions a new application of the pe method filters based on rational approximations of depth operators is described and tested the rational approximation is designed to act as the identity operator on the desired part of the spectrum of the depth separated wave equation and to annihilate other parts of the spectrum this approach does not require explicit knowledge of the spectrum the applications of the filter include directly solving eigenvalue problems annihilating components of the wave number spectrum generating initial conditions at the source range and eliminating gibbs oscillations that arise in energy conserving pe solutions

the rational approximation to the acoustic wave equation with bottom interaction the rational linear approximation to the wave equation is a full wave approach to modeling range dependent ocean acoustic propagation with bottom interaction it is a one way wave equation which gives an accurate treatment of high angle propagation to angles of about 40 with respect to the horizontal reflection from sound speed and density discontinuities is treated using the natural wave equation matching conditions bathymetry is allowed to vary in range a tridiagonal implicit finite difference solution of this equation has been implemented it has several advantages over the tridiagonal crank nicholson solution of the parabolic equation it more accurately models high angles of propagation treats attenuation as a function of path length rather than range and models range dependent bathymetry in a way that suits the form of the one way wave equation the numerical methods are fourth order accurate in depth the resulting implicit range step is still in the simple tridiagonal form

the problem of energy conservation in one way models it is shown that the standard stair step representation of a sloping bottom may result in significant prediction errors in fact current parabolic equation implementations are not energy conserving the problem is shown to derive from the approximate treatment of the interface conditions at vertical boundaries along the stair steps several improved interface conditions are proposed

complex pade approximants for wide angle acoustic propagators procedures for constructing wide angle propagators that attenuate evanescent modes in parabolic equation pe models of underwater sound propagation are examined in this context two new split step pade approximations to exponentiated square root operators are proposed the first method involves a propagator derived from a pade approximant of a square root operator that has been rotated in the complex plane while the propagator of the second method is obtained from a complex coefficient rational approximation of the square root operator numerical evidence confirms that these proposed methods yield improved behavior compared to previously suggested techniques in selected areas of the spectral domain

transparent boundary conditions for split step pade approximations of the one way helmholtz equation in this paper we generalize the nonlocal discrete transparent boundary condition introduced by f schmidt and p deuflhard 1995 comput math appl 29 53 76 and by f schmidt and d yevick 1997 j comput phys 134 96 107 to propagation methods based on arbitrary pade approximations of the two dimensional one way helmholtz equation our approach leads to a recursive formula for the coefficients appearing in the nonlocal condition which then yields an unconditionally stable propagation method

a split step pade solution for the parabolic equation method a split step pade solution is derived for the parabolic equation pe method higher order pade approximations are used to reduce both numerical errors and asymptotic errors e g phase errors due to wide angle propagation this approach is approximately two orders of magnitude faster than solutions based on pade approximations that account for asymptotic errors but not numerical errors in contrast to the split step fourier solution which achieves similar efficiency for some problems the split step pade solution is valid for problems involving very wide propagation angles large depth variations in the properties of the waveguide and elastic ocean bottoms the split step pade solution is practical for global scale problems

higher order paraxial wave equation approximations in heterogenous media a new family of paraxial wave equation approximations is derived these approximations are of higher order accuracy than the parabolic approximation and they can be applied to the same computationa

cocyclic simplex codes of type spl alpha over z sub 4 and z sub 2 sup s over the past decade cocycles have been used to construct hadamard and generalized hadamard matrices this in turn has led to the construction of codes self dual and others here we explore these ideas further to construct cocyclic complex and butson hadamard matrices and subsequently we use the matrices to construct simplex codes of type spl alpha over z 4 and z 2 sup s respectively

on the use of higher order azimuthal schemes in 3 d pe modeling in this paper the issue of using higher order finite difference schemes to handle the azimuthal derivative term in a three dimensional parabolic equation based model is addressed the three dimensional penetrable wedge benchmark problem is chosen to illustrate the accuracy and efficiency of the proposed schemes both point source and modal initializations of the pressure field are considered for each higher order finite difference scheme used in azimuth the convergence of the numerical solution with respect to the azimuth is investigated and the cpu times are given some comparisons with solutions obtained from another 3 d model j a fawcett j acoust soc am 93 2627 2632 1993 are presented the numerical simulations show that the use of a higher order scheme in azimuth allows one to reduce the required number of points in the azimuthal direction while still obtaining accurate solutions the higher order schemes have approximately the same efficiency as a fft based approach in fact may outperform it slightly however the finite difference approach has the advantage that it may be more flexible than the fft approach for various pe approximations

modeling rayleigh and stoneley waves and other interface and boundary effects with the parabolic equation an improved approach for handling boundaries interfaces and continuous depth dependence with the elastic parabolic equation is derived and benchmarked the approach is applied to model the propagation of rayleigh and stoneley waves depending on the choice of dependent variables the operator in the elastic wave equation may not factor or the treatment of interfaces may be difficult these problems are resolved by using a formulation in terms of the vertical displacement and the range derivative of the horizontal displacement these quantities are continuous across horizontal interfaces which permits the use of galerkin s method to discretize in depth this implementation extends the capability of the elastic parabolic equation to handle arbitrary depth dependence and should lead to improvements for range dependent problems

on higher order autospectra by maximum entropy method by r o plaisted and h g pena geophysics 48 1409 1410 october 1983 as stated in the short note by r o plaisted and h g pena they derived what they believed to be a maximum entropy mem representation for bispectra and they claimed that their result can be generalized to an nth order mem auto spectrum the purpose of this discussion is to point out the faults in the derivation and show that the claim of mem is not correct furthermore the parametric representation of the bispectrum that the authors actually attempted to derive is not new but can be found elsewhere huber et al 1971 plaisted and pena stated that the random variables et in their equation 1 can be assumed generally to be normally distributed with zero mean and variance e2 this is a serious mistake with this assumption the third order moment sequence of the xt process given by their equation 3 vanishes i e k l 0 for all k l and therefore the bispectrum of xt is identical to zero the random variables et in their equation 1 and hence xt have to be non gaussian for the b

a self starter for the parabolic equation method an efficient method is developed for generating accurate starting fields for the parabolic equation pe method for both fluid and solid media the self starter which is constructed by solving a one dimensional boundary value problem bvp involving the pe operator is more efficient than the normal mode starter which requires the solution of a large number of similar bvps the self starter depends on the depth dependent properties of the medium and satisfies all interface and boundary conditions since the self starter is based on higher order parabolic approximations it is accurate for problems involving wide propagation angles large depth variations in the properties of the medium low frequencies interface waves and the continuous spectrum

energy conserving and reciprocal solutions for higher order parabolic equations conservation of sound energy in nonabsorbing media and acoustical reciprocity in stationary fluids at rest are the fundamental properties of the acoustic field preserving these properties in parabolic approximations is essential for accurate modeling of sound propagation in shallow water environments recently godin proposed a family of wide angle parabolic equations pe s for motionless fluid which are energy conserving reciprocal and allow improved description of mode coupling without adversely affecting phase accuracy for moving media a generalized claerbout pe was also derived its solutions preserve the acoustic energy in the absence of dissipation and satisfy the flow reversal theorem frt which is an extension of reciprocity principle to moving media the report presents a numerical scheme for solving the proposed pe s the algorithm preserves the conservational properties of the original pseudo differential equations implicit finite difference solutions satisfy discrete analogs of the e

wave number sampling at short range the self starter is an efficient approach for solving geoacoustic inverse problems involving a vertical array of receivers located on the order of ten wavelengths from a source r j cederberg and m d collins j acoust soc am 22 102 109 1997 with this approach accurate solutions can be obtained by sampling the wave number spectrum at approximately ten points there have been unsubstantiated claims that the spectral solution provides similar efficiency when implemented with an approach that involves perturbing the integration contour off the real line f b jensen et al computational ocean acoustics american institute of physics new york 1994 pp 231 240 it is demonstrated that this quadrature scheme breaks down at short ranges and that the self starter solution is about an order of magnitude faster these conclusions are based on comparisons of the rational approximations associated with the self starter and the quadrature scheme comparisons of acoustic fields for particular proble

a higher order energy conserving parabolic equqation for range dependent ocean depth sound speed and density outgoing solutions of the wave equation including parabolic equation pe and normal mode solutions are usually formulated so that pressure is continuous with range for range dependent problems the accuracy of normal mode solutions has been improved by conserving energy rather than maintaining continuity of pressure porter et al the problem of energy conservation in one way equations j acoust soc am 89 1058 1067 1991 this approach is applied to derive a higher order energy conserving pe that provides improved accuracy for problems involving large ocean bottom slopes and large range and depth variations in sound speed and density a special numerical approach and complex pade coefficients are applied to suppress gibbs oscillations the back propagated half space field an improved pe starter is applied to handle wide propagation angles reference solutions generated with a complex ray model and with the rotated pe are used for comparison

a new rough surface parabolic equation program for computing low frequency acoustic forward scattering from the ocean surface the parabolic equation acoustic propagation program ram has been extended to handle a rough air water interface treated as a series of stair steps because much finer vertical spacing may be needed to resolve the interface than to propagate a low frequency acoustic wave accurately in the rest of the domain an option to use one vertical spacing near the surface and a coarser one in the rest of the ocean has been implemented the necessary galerkin approximations of derivatives on an unequally spaced grid have been worked out using computer algebra the new program rrsfc is efficient enough to compute the frequency spectrum of the field scattered off a moving surface by treating the surface as a sequence of frozen surfaces a comparison of one such spectrum with actual ocean data is provided in order to make a more detailed assessment of the program both the depth dependent pressure for individual surfaces and the frequency spectrum for sequences of surfaces are compared with the results of numerically exact integral equation calculations for a few constant sound speed test cases

quantifying uncertainty in geoacoustic inversion ii application to broadband shallow water data this paper applies the new method of fast gibbs sampling fgs to estimate the uncertainties of seabed geoacoustic parameters in a broadband shallow water acoustic survey with the goal of interpreting the survey results and validating the method for experimental data fgs applies a bayesian approach to geoacoustic inversion based on sampling the posterior probability density to estimate marginal probability distributions and parameter covariances this requires knowledge of the statistical distribution of the data errors including both measurement and theory errors which is generally not available invoking the simplifying assumption of independent identically distributed gaussian errors allows a maximum likelihood estimate of the data variance and leads to a practical inversion algorithm however it is necessary to validate these assumptions i e to verify that the parameter uncertainties obtained represent meaningful estimates to this end fgs is applied to a geoacoustic experiment carried out at a site off the west coast of italy where previous acoustic and geophysical studies have been performed the parameter uncertainties estimated via fgs are validated by comparison with i the variability in the results of inverting multiple independent data sets collected during the experiment ii the results of fgs inversion of synthetic test cases designed to simulate the experiment and data errors and iii the available geophysical ground truth comparisons are carried out for a number of different source bandwidths ranges and levels of prior information and indicate that fgs provides reliable and stable uncertainty estimates for the geoacoustic inverse problem

the coupled mode parabolic equation the coupled mode parabolic equation pe is a generalization of the adiabatic mode pe that includes mode coupling terms it is practical to apply this approach to large scale problems involving coupling of energy between both modes and azimuths the solution is expressed in terms of the normal modes and mode coefficients which satisfy coupled horizontal wave equations the coupled mode pe may be solved efficiently with the splitting method the first step is equivalent to solving the adiabatic mode pe over one range step the second step involves the integration of the coupling term the coupling mode pe solution conserves energy which is an important aspect of a range dependent propagation model the derivation of the coupled mode pe which involves completing the square of an operator is related to the derivation of an adiabatic mode pe that accounts for ambient flow examples are presented to illustrate the accuracy of the coupled mode pe