a transformation of the environment eliminates parabolic equation phase errors it is shown that phase errors associated with the standard parabolic wave equation can be eliminated in a range independent environment by appropriately transforming the environment solutions to the standard parabolic equation pe in the transformed environment are close to solutions to the helmholtz equation in the original environment in the sense that ray and asymptotic mode contributions to the pe wave fields have no phase errors relative to their helmholtz equation counterparts although phase errors are eliminated in the pe wave fields complete asymptotic equivalence to the helmholtz equation wave fields is not attained ray and mode based wave field expansions are used to illustrate similarities and differences between pe and helmholtz equation wave fields numerical simulations show excellent agreement between pe wave fields in the transformed environment and mode based solutions to the helmholtz equation in the true environment

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

transparent boundary conditions for a wide angle approximation of the one way helmholtz equation we present nonlocal discrete transparent boundary conditions for a fourth order wide angle approximation of the two dimensional helmholtz equation the boundary conditions are exact in the sense that they supply the same discrete solution on a bounded interior domain as would be obtained by considering the problem on the entire unbounded domain with zero boundary conditions at infinity the proposed algorithm results in an unconditionally stable propagation method numerical examples from optics illustrate the efficiency of our approach

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

computationally efficient parabolic equation solutions to seismo acoustic problems involving thin or low shear elastic layers shallow water environments typically include sediments containing thin or low shear layers numerical treatments of these types of layers require finer depth grid spacing than is needed elsewhere in the domain thin layers require finer grids to fully sample effects due to elasticity within the layer as shear wave speeds approach zero the governing system becomes singular and fine grid spacing becomes necessary to obtain converged solutions in this paper a seismo acoustic parabolic equation solution is derived utilizing modified difference formulas using galerkin s method to allow for variable grid spacing in depth propagation results are shown for environments containing thin layers and low shear layers

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

inverse maximal eigenvalues problems for leslie and doubly leslie matrices abstract in this paper we deal with leslie and doubly leslie matrices of order n in particular with the companion and doubly companion matrices we study three inverse eigenvalues problems which consist of constructing these matrices from the maximal eigenvalues of its all leading principal submatrices for leslie and doubly companion matrices an eigenvector associated with the maximal eigenvalue of the matrix is additionally considered and for the doubly leslie matrix also an eigenvector associated with the maximal eigenvalue of leading principal submatrix of order n 1 is required we give necessary and sufficient conditions for the existence of a leslie matrix and a companion matrix and sufficient conditions for the existence of a doubly leslie matrix and a doubly companion matrix our results are constructive and generate an algorithmic procedure to construct these special kinds of matrices

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

relationship between time reversal and linear equalization in digital communications iterative time reversal has been suggested as both an efficient method of creating a spatio temporal focus and for use in telecommunications as a form of equalization in this paper the equivalence of a passive i e via computation iterative time reversal to the moore penrose pseudo inverse of the propagation matrix is shown in the context of communications however any received signal is corrupted by noise therefore a regularization term is introduced to the iterative equations causing convergence to the canonical minimum mean squared error linear equalizer hence a relationship between time reversal and equalization is demonstrated

generalized inverse eigenvalue problem for matrices whose graph is a path abstract in this paper we analyse a special generalized inverse eigenvalue problem a n x b n x for the pair a n b n of matrices each of whose graph is a path on n vertices by investigating the leading principal minors of the matrix a n b n from the given data consisting of b n a matrix a k k n whose graph is a path on k vertices two column vectors x 2 y 2 r n k and distinct real numbers and we construct a n and two column vectors x 1 y 1 r k such that a k is the leading principal sub matrix of a n and x y are the eigenpairs of a n b n where x x 1 t x 2 t t and y y 1 t y 2 t t further numerical examples are also given to demonstrate the applicability of the results developed here

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

preview control for impulse free continuous time descriptor systems this paper studies the preview control problem of impulse free linear continuous time descriptor systems the system is first decomposed into a normal system i e slow subsystem and an algebraic equation set by restricted equivalent linear transformation then applying the method of preview control theory to the slow subsystem by taking derivatives on both the error vector and the state function and with the error vector being a part of the new state vector the augmented system is constructed and the tracking problem is transformed into a regulation problem according to preview control theory the controller of the augmented system can be obtained and the control input of the original descriptor system with preview function can be acquired by integrating on the controller of the augmented system both the stabilisability and detectability of the augmented system are discussed numerical simulation verifies the presented results

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

perfectly matched layer truncation for parabolic wave equation models solutions of the parabolic wave equation are extended to complex values of the height variable the fast decay of the solutions in certain directions of the complex plane is used to improve parabolic wave equation solvers the solution is marched on a suitably chosen contour in the complex plane rather than on the real line this interpretation of berenger s perfectly matched layer allows extremely efficient truncation of the computational domain with a split step fourier algorithm the theory is developed for both the wide angle and narrow angle parabolic wave equations and numerical results are presented

impedance matched absorbers for finite difference parabolic equation algorithms in this paper a perfectly matched layer pml absorber recently introduced into the electromagnetic propagation literature by berenger j comput phys 114 185 200 1994 is adapted for use with both paraxial and wide angle acoustic parabolic equations pes our procedure incorporates an imaginary component into the transverse coordinate that mimics the introduction of a fictitious absorber on the edge of the computational grid use of such an impedance matched layer can significantly reduce spurious reflections compared to physical absorbing layer methods and thus allows a smaller number of boundary points to be employed in pe calculations numerical results obtained with several higher order propagator approximations confirm that such impedance matched absorbers efficiently eliminate reflections

optimal preview control for discrete time descriptor causal systems in a multirate setting the optimal preview control problem is proposed for discrete time descriptor causal systems in a multirate setting first according to the characteristics of causal systems the descriptor system is transformed into an algebraic equation and a normal system represented by a dynamic equation second by using the discrete lifting technique the normal multirate system is converted to a single rate system then by making use of the standard linear qudratic lq preview control method we construct the augmented error system and the optimal preview control law of the augmented error system is obtained the control law for the augmented system is then transformed to the optimal preview control law of the descriptor causal system in a multirate setting the stabilisability and detectability of the lifted single rate system and the augmented error system are discussed in detail the effectiveness of the proposed method is shown by simulation

left and right inverse eigenpairs problem of generalized centrosymmetric matrices and its optimal approximation problem in this paper the left and right inverse eigenpairs problem for generalized centrosymmetric matrices is considered we obtain the necessary and sufficient conditions for the solvability of the problem and we present the general expression of the solution the related optimal approximation problem to a given matrix over the solution set is solved in addition a numerical algorithm and examples to solve the problem are given

examination of coherent surface reflection coefficient csrc approximations in shallow water propagation the parabolic wave equation pe code of rosenberg j acoust soc am 105 144 153 1999 is used as a benchmark to study acoustic propagation in an ocean waveguide with a rough air water interface the pe results allow a close examination of the ability of a ray code i e gaussian ray bundle grab to accurately estimate coherent field propagation using a coherent reflection coefficient derived from scattering theory comparison with pe implies that the beckmann spizzichino model as given within the grab software package does not give accurate predictions of the coherent field at long ranges three other coherent reflection coefficient approximations are tested the perturbation the small slope and the kirchhoff approximations the small slope approximation is the most accurate of the models tested however the kirchhoff approximation is perhaps accurate enough for some purposes and would be simpler to implement as a module within grab

optimal focusing by spatio temporal inverse filter i basic principles a focusing technique based on the inversion of the propagation operator relating an array of transducers to a set of control points inside a medium was proposed in previous work tanter et al j acoust soc am 108 223 234 2000 and is extended here to the time domain as the inversion of the propagation operator is achieved both in space and time this technique allows calculation of the set of temporal signals to be emitted by each element of the array in order to optimally focus on a chosen control point this broadband inversion process takes advantage of the singular value decomposition of the propagation operator in the fourier domain the physical meaning of this decomposition is explained in a homogeneous medium in particular a definition of the number of degrees of freedom necessary to define the acoustic field generated by an array of limited aperture in a focal plane of limited extent is given this number corresponds to the number of independent signals that can be created in the focal area both in space and time in this paper this broadband inverse focusing technique is compared in homogeneous media with the classical focusing achieved by simple geometrical considerations but also with time reversal focusing it is shown that even in a simple medium slight differences appear between these three focusing strategies in the companion paper aubry et al j acoust soc am 110 48 58 2001 the three focusing techniques are compared in heterogeneous absorbing or complex media where classical focusing is strongly degraded the strong improvement achieved by the spatio temporal inverse filter technique emphasizes the great potential of multiple channel systems having the ability to apply completely different signal waveforms on each transducer of the array the application of this focusing technique could be of great interest in various ultrasonic fields such as medical imaging nondestructive testing and underwater acoustics

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

experimental demonstration of the utility of pressure sensitivity kernels in time reversal pressure sensitivity kernels were recently applied to time reversal acoustics in an attempt to explain the enhanced stability of the time reversal focal spot raghukumar et al j acoust soc am 124 98 112 2008 the theoretical framework developed was also used to derive optimized source functions closely related to the inverse filter the use of these optimized source functions results in an inverse filter like focal spot which is more robust to medium sound speed fluctuations than both time reversal and the inverse filter in this paper the theory is applied to experimental data gathered during the focused acoustic fields experiment conducted in 2005 north of elba island in italy sensitivity kernels are calculated using a range independent sound speed profile for a geometry identical to that used in the experiment and path sensitivities are identified with observed arrivals the validity of the kernels in tracking time evolving green s functions is studied along with limitations that resul

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

traffic noise prediction with the parabolic equation method validation of a split step pade approach in complex environments this study deals with sound propagation in typical traffic noise conditions the numerical results are obtained through the split step pade method and the discrete random fourier modes technique these are first evaluated qualitatively by color contour maps showing noise propagation diffraction by an impedance discontinuity or a screen edge and scattering by atmospheric turbulence next our numerical results are quantitatively validated by comparison with analytical models and other parabolic equation models for all the atmospheric conditions and geometrical configurations available in literature the agreement between the different methods is very good except for some cases involving the atmospheric turbulence however in those particular cases the split step pade results are shown to be more consistent with physical theory finally our method seems to be very powerful and reliable for traffic noise prediction

rytov approximation of tomographic receptions in weakly range dependent ocean environments an approximation method based on the born and rytov approximation is presented for the wave theoretic prediction of acoustic arrival patterns associated with long range pulse propagation in weakly range dependent ocean environments the environment is considered as a perturbation of a range independent background state and normal mode theory is used for the representation of the background green s function using the born and rytov approximations the perturbed green s function corresponding to the range dependent environment is expressed for each frequency within the source bandwidth in terms of the background green s function and the medium sound speed perturbation the actual arrival pattern in the time domain is then computed through the inverse fourier transform using the normal mode representation closed form expressions for the first and second born and rytov approximations are derived generalizing previous range independent results and indicating that the effect of range dependence on the

a higher order on surface radiation condition derived from an analytic representation of a dirichlet to neumann map on surface radiation conditions are useful for obtaining approximate solutions to scattering problems involving compact obstacles an analytic representation of the dirichlet to neumann map for a circle is derived and used to construct a higher order on surface radiation condition for a generally convex perfectly conducting body in two dimensions this approach is based on a hankel function in which a tangential operator appears in the index in the high frequency limit this analytic representation approaches the square root of a differential operator which commonly arises in the application of parabolic equation techniques to propagation problems treating the scattered field propagation angle relative to the surface normal and the surface curvature as independent parameters the representation is fit to a rational function to provide an accurate and efficient on surface radiation condition that is tested for various examples

real time inverse filter focusing through iterative time reversal in order to achieve an optimal focusing through heterogeneous media we need to build the inverse filter of the propagation operator time reversal is an easy and robust way to achieve such an inverse filter in nondissipative media however as soon as losses appear in the medium time reversal is not equivalent to the inverse filter anymore consequently it does not produce the optimal focusing and beam degradations may appear in such cases we showed in previous works that the optimal focusing can be recovered by using the so called spatiotemporal inverse filter technique this process requires the presence of a complete set of receivers inside the medium it allows one to reach the optimal focusing even in extreme situations such as ultrasonic focusing through human skull or audible sound focusing in strongly reverberant rooms but this technique is time consuming and implied fastidious numerical calculations in this paper we propose a new way to process this inverse filter focusing technique in real time and without any calculation the new process is based on iterative time reversal process contrary to the classical inverse filter technique this iteration does not require any computation and achieves the inverse filter in an experimental way using wave propagation instead of computational power the convergence from time reversal to inverse filter during the iterative process is theoretically explained finally the feasibility of this iterative technique is experimentally demonstrated for ultrasound applications

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

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

quantitative imaging using a time domain eigenfunction method an inverse scattering method that uses eigenfunctions of a scattering operator at a single frequency is extended to include the full range of frequencies present in the incident pulse waveform the resulting so called time domain eigenfunction method is shown to yield a modulated version of the scattering potential the potential is recovered by a demodulation process using cross correlation with a reference including an adaptive delay in the reference is shown to compensate partially for the linearization of the born approximation and to extend its valid range the k space window of the time domain solution is expressed in terms of the incident waveform and shown to be smoother than that of a single frequency solution the time domain method is examined using both calculated and measured data in the calculations an exact solution for scattering from one or multiple nonconcentric cylinders is used to obtain the scattered field in the measurements a novel ring transducer system was employed to obtain the incident and total fields the results of simulations and experiments show that the method is robust and accurate for the size of objects considered and that the point resolution approaches one half the wavelength at the pulse center frequency

one way large range step methods for helmholtz waveguides a useful approach for long range computation of the helmholtz equation in a waveguide is to re formulate it as the operator differential riccati equation for the dirichlet to neumann map for waveguides with slow range dependence the piecewise range independent approximation is used to derive a second order range stepping method for this one way re formulation the range marching formulas are exact for each range independent piece and a large range step is possible if the range dependence is gradual based on a fourth order conservative exponential method for linear evolution equations a fourth order method that admits even larger range steps is developed for the one way re formulation numerical examples are used to demonstrate the improved accuracy of the fourth order method

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

the inverse eigenvalue problem for symmetric quasi anti bidiagonal matrices in this paper we construct the symmetric quasi anti bidiagonal matrix that its eigenvalues are given and show that the problem is also equivalent to the inverse eigenvalue problem for a certain symmetric tridiagonal matrix which has the same eigenvalues not only elements of the tridiagonal matrix come from quasi anti bidiagonal matrix but also the places of elements exchange based on some conditions

a scaled mapping parabolic equation for sloping range dependent environments parabolic equation solutions use various techniques for approximating range dependent interfaces one is a mapping approach m d collins et al j acoust soc am 107 1937 1942 2000 where at each range the domain is vertically translated so that sloping bathymetry becomes horizontal and range dependence is transferred to the upper surface in this paper a scaled mapping is suggested where the domain is vertically distorted so that both the bathymetry and upper surface are horizontal accuracy is demonstrated for problems involving fluid sediments generalizations of the approach should be useful for environments with layer thicknesses that vary with range

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

the flattened surface parabolic equation a simple surface flattening transformation is applied to the problem of underwater acoustic scattering from an irregular sea surface the flattened surface elliptic wave equation is factored to obtain a parabolic equation that has the same high angle capability as recent parabolic equation models the flattened surface parabolic equation contains an additional surface slope term but includes forward scattering from the irregular sea surface the transformation creates additional volume irregularities and irregularities in the water sediment interface that the range and depth mesh need to resolve the surface flattening transformation is easy to implement in a parabolic equation model it compares very well with results obtained using boundary integral equations an example is given of scattering at 400 hz from a sea surface with wave heights of 3 m or nearly a wavelength

optimal tracking performance preview control and exponential signals we study tracking performance limitation problems two issues are addressed concerning how earlier results developed elsewhere may be extended to more general classes of reference signals and how tracking performance may be further improved beyond that offered by feedback control toward these issues we consider exponentially increasing reference inputs and examine the use of preview control for tracking using an optimal interpolation approach we develop analytical expressions and a conceptual insight which will aid in the understanding of these issues to this effect we derive explicit expressions for the optimal tracking error either as exact solutions or bounds it is found that for the exponential signals the earlier results can be directly extended and similar conclusive statements can be drawn it is also shown that in general preview can be used to advantage for improving tracking performance especially in dealing with the effect resulted from plant nonminimum phase zeros

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

estimating parameter uncertainties in matched field inversion by a neighborhood approximation algorithm in bayesian inversion the solution is characterized by its posterior probability density ppd a fast gibbs sampler fgs has been developed to estimate the multi dimensional integrals of the ppd which requires solving the forward models many times and leads to intensive computation for multi frequency or range dependent inversion cases this paper presents an alternative approach based on a neighborhood approximation bayes nab algorithm for lower dimension geoacoustic inversion the nab can approximate the ppd very well for higher dimensional problems and sensitive parameters however the nab algorithm has difficulty estimating the ppd accurately with limited model samples according to the preliminary ppd estimation from the nab this paper developed a multi step inversion scheme which adjusts the parameter search intervals flexibly in order to improve the approximation accuracy of the nab and obtain more complete parameter uncertainties the prominent feature of the nab is to approximate the ppd by incorporating all models for which the forward problem has been solved into the appraisal stage comparison of the fgs and nab for noisy synthetic benchmark test cases and mediterranean real data indicates that the nab provides reasonable estimates of the ppd moments while requiring significantly less computation time

target scattering calculations with the parabolic equation method the parabolic equation technique is used to solve the helmholtz equation in the presence of scatterers of arbitrary shape in two and three dimensions the scattered field is computed directly using non homogeneous boundary conditions on the scattering object to represent the incident field effectively this decouples the pe paraxial direction from the direction of incidence for convex objects the whole range of scattering angles can be covered with a small number of narrow angle calculations finite difference implementations involve tridiagonal matrices in two dimensions and more general sparse matrices in three dimensions the resulting codes can be used to solve scattering problems for objects ranging in size from a few wavelengths to hundreds of wavelengths the method has been tested against analytical solutions for soft and rigid circular cylinders in 2d and soft and rigid spheres in 3d showing good agreement at all scattering angles

design of an optimal controller for a discrete time system subject to previewable demand this paper is concerned with a method of designing a type one servomechanism for a discrete time system subject to a time varying demand and an unmeasurable constant disturbance it is assumed that the time varying demand is previewable in the sense that some finite future as well as present and past values of the demands are available at each time a controller with state feedback plus integral and preview actions is derived by applying a linear quadratic integral lqi technique due to tomizuka and rosenthal 1979 it is shown under the stabilizability and detectability conditions that the closed loop system achieves a complete regulation in the presence of small perturbations in system parameters eliminating the effect of disturbance an example of power plant control is presented to show the flexibility of the design method and the effectiveness of the preview action for improving the transient responses of the closed loop system

nonuniform depth grids in parabolic equation solutions the parabolic wave equation is solved using a finite difference solution in depth that involves a nonuniform grid the depth operator is discretized using galerkin s method with asymmetric hat functions examples are presented to illustrate that this approach can be used to improve efficiency for problems in ocean acoustics and seismo acoustics for shallow water problems accuracy is sensitive to the precise placement of the ocean bottom interface this issue is often addressed with the inefficient approach of using a fine grid spacing over all depth efficiency may be improved by using a relatively coarse grid with nonuniform sampling to precisely position the interface efficiency may also be improved by reducing the sampling in the sediment and in an absorbing layer that is used to truncate the computational domain nonuniform sampling may also be used to improve the implementation of a single scattering approximation for sloping fluid solid interfaces

ray convergence in a flux like propagation formulation the energy flux formulation of waveguide propagation is closely related to the incoherent mode sum and its simplicity has led to development of efficient computational algorithms for reverberation and target echo strength but it lacks the effects of convergence or modal interference by starting with the coherent mode sum and rejecting the most rapid interference but retaining beats on a scale of a ray cycle distance it is shown that convergence can be included in a hybrid formulation requiring minimal extra computation three solutions are offered by evaluating the modal intensity cross terms using taylor expansions in the most efficient approach the double summation of the cross terms is reduced to a single numerical sum by solving the other summation analytically the other two solutions are a local range average and a local depth average favorable comparisons are made between these three solutions and the wave model orca with and without spatial averaging in an upward refracting duct as a by product it is shown that the running range average is very close to the mode solution excluding its fringes given a relation between averaging window size and effective number of modes which in turn is related to the waveguide invariant

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

a two way parabolic equation that accounts for multiple scattering a two way parabolic equation that accounts for multiple scattering is derived and tested a range dependent medium is divided into a sequence of range independent regions the field is decomposed into outgoing and incoming fields in each region the conditions between vertical interfaces are implemented using rational approximations for the square root of an operator rational approximations are also used to relate fields between neighboring interfaces an iteration scheme is used to solve for the outgoing and incoming fields at the vertical interfaces the approach is useful for solving problems involving scattering from waveguide features and compact objects

a variational data assimilation system for the range dependent acoustic model using the representer method theoretical derivations this study presents the theoretical framework for variational data assimilation of acoustic pressure observations into an acoustic propagation model namely the range dependent acoustic model ram ram uses the split step pade algorithm to solve the parabolic equation the assimilation consists of minimizing a weighted least squares cost function that includes discrepancies between the model solution and the observations the minimization process which uses the principle of variations requires the derivation of the tangent linear and adjoint models of the ram the mathematical derivations are presented here and for the sake of brevity a companion study presents the numerical implementation and results from the assimilation simulated acoustic pressure observations

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

greene approximation wide angle parabolic equation for radio propagation this paper presents the greene second order parabolic equation pe solution to model radio propagation over irregular terrain this solution is implemented by finite difference and shift map technique the second order finite difference approach performs well for slopes up to about 15 and discontinuous slope changes up to about 30 which is better than can be achieved using the split step fourier approach the results prove that the greene pe solution has greater propagation angles than does the claerbout pe solution thus the greene second order solution can certainly give better results for large angle propagation over complicated terrain boundaries besides we treat the important problem of accuracy of the different approximations of the pe based propagation models and derive the general pe solutions with respect to the terrain slope for the different approximations it is useful to have collected in one paper the most important pe approximations for radio propagation

transducer sensitivity compensation using diagonal preconditioning for time reversal and tikhonov inverse filtering in acoustic systems filters are commonly used in sound reproduction and communication systems as a means of compensating for the response of the electro acoustic plant two commonly used filter designs in the field of acoustics are the time reversal filter and the tikhonov inverse filter in this paper the influence of transducer sensitivities on the performance of these filters is examined it is shown that the sensitivity of the transducers can negatively affect the performance of the resulting filter to compensate for the decrease in performance diagonal preconditioning can be implemented in the system it is shown that by using diagonal matrices which minimize the condition number of the system the loss in performance arising from unbalanced sensitivities is minimized this paper proposes an algorithm to find such a set of diagonal matrices and results are presented showing the improvements in performance arising from the modified filter design

linear parameterized inverse eigenvalue problem of bisymmetric matrices abstract in this paper we consider the linear parameterized inverse eigenvalue problem of bisymmetric matrices which is described as follows problem i given real bisymmetric matrices b 0 b 1 b 2 b n and real numbers 1 2 n find values of c c 1 c 2 c n t such that the eigenvalues of the matrix b c b 0 c 1 b 1 c 2 b 2 c n b n are precisely 1 2 n we first establish the structure of bisymmetric matrices based on the newton type iterative method we then turn problem i into the inverse eigenvalue problems of symmetric matrices with small sizes and solve these problems finally we present numerical examples to illustrate our results

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

discrete transparent boundary conditions for wide angle parabolic equations in underwater acoustics this paper is concerned with transparent boundary conditions tbcs for wide angle parabolic equations wapes in the application to underwater acoustics assuming cylindrical symmetry existing discretizations of these tbcs introduce slight numerical reflections at this artificial boundary and also render the overall crank nicolson finite difference method only conditionally stable here a novel discrete tbc is derived from the fully discretized whole space problem that is reflection free and yields an unconditionally stable scheme while we shall assume a uniform discretization in range the interior depth discretization i e in the water column may be nonuniform and we shall discuss strategies for the best exterior discretization i e in the sea bottom the superiority of the new discrete tbc over existing discretizations is illustrated on several benchmark problems in the literature different wapes or wape and the standard parabolic equation have been coupled in the water and the sea bottom we analyze under which conditions this yields a hybrid model that is conservative for the acoustic field

two filter smoothing formulae by diagonalization of the hamiltonian equations we present a new approach to two filter smoothing formulae via diagonalization of the general time variant hamiltonian equations of the linear estimation problem this approach shows the special role of the famous mayno fraser two filter formulae and also provides insight into certaini nvariance properties of backwards kalman filter estimates

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

tracking sensitivity and achievable h performance in preview control preview control has been explored in the context of linear and nonlinear inverse tracking controllers preview of the reference trajectory has been shown to give significant performance advantages particularly for plants that do not possess a stable inverse without preview it has been known that lack of a stable inverse imposes inherent limitations on the achievable tracking performance of a control loop this has been explored in the context of time and frequency domain integral constraints in this paper we explore the extension of these performance limitations to the preview control case in particular we consider the infimal achievable h performance and poisson sensitivity integral results we also briefly discuss tracking sensitivity properties of preview control

overdetermined least squares aberration estimates using common midpoint signals as medical ultrasound imaging moves to larger apertures and higher frequencies tissue sound speed variations continue to limit resolution in geophysical imaging a standard approach for estimating near surface aberrating delays is to analyze the time shifts between common midpoint signals this requires complete data echoes from every source receiver pair in the array unfocused common midpoint signals remain highly correlated in the presence of delay aberrations there is also tremendous redundancy in the data in medical ultrasound this technique has been impaired by the wide angle random scattering nature of tissue this has made it difficult to estimate azimuth dependent aberration profiles or to harness the full redundancy in the complete data prefiltering the data with two dimensional fan filters mitigates these problems permitting highly overdetermined least squares solutions for the aberration profiles at many steering angles in experiments with a tissue mimicking phantom target and silicone rubber aberrators at nonzero stand off distances from a one dimensional phased array this overdetermined fan filtering algorithm significantly outperformed other phase screen algorithms based on nearest neighbor cross correlation speckle brightness maximization and common midpoint signal analysis our results imply that there is still progress to be made in imaging with single valued focusing operators it also appears that the signal to noise penalty for using complete data sets is partially compensated by the overdetermined nature of the problem

exact discrete nonlocal boundary conditions for high order pade parabolic equations the nonlocal boundary conditions nlbcs for high order finite difference parabolic equations pes are obtained by z transformation of the discrete pe in a homogeneous medium the considered nlbcs include the free space radiation condition possibly with a density jump at the nlbc interface the nlbc at an arbitrary impedance interface and the nlbcs for sources and the starting field beyond the nlbc interface the derivation is presented for the multiterm pade pe model owwe one way wave equation but the developed technique is applicable to a broad class of finite difference pes the obtained nlbcs are exact for the given finite difference scheme they are not limited by the order of pade approximation or by the pe steps in range and depth the nlbc convolution coefficients are calculated by the numerical inverse z transformation the accuracy and performance of the algorithm are analyzed for several benchmark problems the solution is robust for the range steps over 25 wavelengths and or the approximations up to the tenth order the nlbcs are faster and more accurate for large steps because fewer previous range steps contribute to the convolution precomputation of the nlbc coefficients may be required in time demanding applications the results are compared with earlier proposed nlbcs for high order pes

on the construction of a jacobi matrix from its mixed type eigenpairs in this paper an inverse eigenvalue problem of constructing a jacobi matrix from its mixedtype eigenpairs is considered the necessary and sufficient conditions for the existence and uniqueness of the solutions are derived one numerical algorithm and two numerical examples are given

extremal inverse eigenvalue problem for bordered diagonal matrices abstract the following inverse eigenvalue problem was introduced and discussed in j peng x y hu l zhang two inverse eigenvalue problems for a special kind of matrices linear algebra appl 416 2006 336 347 to construct a real symmetric bordered diagonal matrix a from the minimal and maximal eigenvalues of all its leading principal submatrices however the given formulae in 4 theorem 1 to compute the matrix a may lead us to a matrix which does not satisfy the requirements of the problem in this paper we rediscuss the problem to give a sufficient condition for the existence of such a matrix and necessary and sufficient conditions for the existence of a nonnegative such a matrix results are constructive and generate an algorithmic procedure to construct the matrices

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

comparison of ocean acoustic horizontal coherence predicted by path integral approximations and parabolic equation simulation results a line integral approximation to the acoustic path integral has been used to generate predictions for the characteristic length scale of horizontal cross range coherence in long range ocean acoustic propagation these estimates utilize a single range independent sound speed profile and the mean variance as a function of depth of fractional sound speed perturbations due to internal waves the length scales predicted by the integral approximation have been compared to the values generated by parabolic equation simulations through multiple realizations of garrett munk internal waves one of the simulation environments approximates the slice89 experiment transmissions from a 250 hz source were simulated in a deep water transect to a maximum range of 1000km the second environment corresponds to one of the propagation paths in the north pacific acoustic laboratory npal experiment the source in this experiment was bottom mounted near kauai hawaii and the relevant receiver consisted of five vertical line

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

time reversal and the inverse filter to focus ultrasonic waves in an unknown inhomogeneous medium using a phased array one has to calculate the optimal set of signals to be applied on the transducers of the array in the case of time reversal mirrors one assumes that a source is available at the focus providing the green s function of this point in this paper the robustness of this time reversal method is investigated when loss of information breaks the time reversal invariance it arises in dissipative media or when the field radiated by the source is not entirely measured by the limited aperture of a time reversal mirror however in both cases linearity and reciprocity relations ensure time reversal to achieve a spatiotemporal matched filtering nevertheless though it provides robustness to this method no constraints are imposed on the field out of the focus and sidelobes may appear another approach consists of measuring the green s functions associated to the focus but also to neighboring points thus the whole information characterizing the medium is known and the inverse source problem can be solved a matrix formalism of the propagation operator is introduced to compare the time reversal and inverse filter techniques moreover experiments investigated in various media are presented to illustrate this comparison

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

two inverse eigenvalue problems for a special kind of matrices in this paper a special kind of matrices which are symmetric all elements are equal to zero except for the first row the first column and the diagonal elements and the elements of the first row are positive except for the first one are considered two inverse problems are discussed one is to construct one of this kind of matrices by the minimal and maximal eigenvalues of its all leading principal submatrices the other is to construct one of this kind of matrix by one of its eigenpair and eigenvalues of its all leading principal submatrices the necessary and sufficient conditions for the solvability of the two problems are derived furthermore corresponding numerical algorithms and some examples are given

design of an optimal servomechanism with preview action and its dual problem this paper presents a method for designing a type one servomechanism for a conlinuous time linear system subject to a previewable desired output and an unmeasurable step disturbance by applying the linear quadratic integral technique a state feedback controller with integral and preview actions is derived we show that the closed loop system is asymptotically stable and hence achieves a complete regulation under the stabilizability and detectability assumption we also show that the preview servomechanism problem is the dual of a smoothing problem for a linear stochastic system with a brownian motion input an example is included to illustrate the transient response and frequency characteristics of the closed loop system

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

inverse spectral problem for pseudo jacobi matrices with partial spectral data in this paper we study a special kind of matrices which are pseudo jacobi matrices four inverse eigenvalue problems are discussed and the necessary and sufficient conditions are presented under which these problems are solvable the corresponding algorithms and some examples are given numerical results demonstrate that these methods are of practical utility

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

a higher order tangent linear parabolic equation solution of three dimensional sound propagation a higher order square root operator splitting algorithm is employed to derive a tangent linear solution for the three dimensional parabolic wave equation due to small variations of the sound speed in the medium the solution shown in this paper unifies other solutions obtained from less accurate approximations examples of three dimensional acoustic ducts are presented to demonstrate the accuracy of the solution future work on the applications of associated adjoint models for acoustic inversions is proposed and discussed

a phase and space coherent direct imaging method a direct imaging algorithm for point and extended targets is presented the algorithm is based on a physical factorization of the response matrix of a transducer array the factorization is used to transform a passive target problem to an active source problem and to extract principal components tones in a phase consistent way the multitone imaging function can superpose multiple tones spatial diversity aperture of the array and frequencies bandwidth of the probing signal based on phase coherence the method is a direct imaging algorithm that is simple and efficient since no forward solver or iteration is needed robustness of the algorithm with respect to noise is demonstrated via numerical examples