the adiabatic mode parabolic equation an efficient method is developed for solving problems involving three dimensional acoustic propagation in a thin spherical shell the method is based on the adiabatic mode approximation which assumes that energy coupling between modes is negligible and the parabolic equation pe method which efficiently handles spatial variations in the acoustic properties the pe method is used to solve two dimensional wave equations for the adiabatic mode coefficients over latitude and longitude due to this reduction in dimension the adiabatic mode pe should be practical for solving global scale ocean acoustics problems these types of problems which are currently of great interest heaney et al j acoust soc am 90 2586 2594 1991 baggeroer and munk phys today 45 9 22 30 1992 are too large to solve with other existing three dimensional models

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

multiterminal source coding with high resolution we consider separate encoding and joint decoding of correlated continuous information sources subject to a difference distortion measure we first derive a multiterminal extension of the shannon lower bound for the rate region then we show that this shannon outer bound is asymptotically tight for small distortions these results imply that the loss in the sum of the coding rates due to the separation of the encoders vanishes in the limit of high resolution furthermore lattice quantizers followed by slepian wolf lossless encoding are asymptotically optimal we also investigate the high resolution rate region in the remote coding case where the encoders observe only noisy versions of the sources for the quadratic gaussian case we establish a separation result to the effect that multiterminal coding aimed at reconstructing the noisy sources subject to the rate constraints followed by estimation of the remote sources from these reconstructions is optimal under certain regularity conditions on the structure of the coding scheme

indirect and direct gaussian distributed source coding problems we consider the distributed source coding system of l correlated gaussian sources y l l 1 2 ldots l which are noisy observations of correlated gaussian remote sources x k k 1 2 ldots k we assume that y l t y 1 y 2 ldots y l is an observation of the source vector x k t x 1 x 2 ldots x k having the form y l ax k n l where a is a l times k matrix and n l t n 1 n 2 ldots n l is a vector of l independent gaussian random variables also independent of x k in this system l correlated gaussian observations are separately compressed by l encoders and sent to the information processing center we study the remote source coding problem where the decoder at the center attempts to reconstruct the remote source x k we consider three distortion criteria based on the covariance matrix of the estimation error on x k for each of those three criteria we derive explicit inner and outer bounds of the rate distortion region next in the case of k l and a i l we study the multiterminal source coding problem where the decoder wishes to reconstruct the observation y l x l n l to investigate this problem we shall establish a result that provides a strong connection between the remote source coding problem and multiterminal source coding problem using this result we derive several new partial solutions to the multiterminal source coding problem

exact and operator rational approximate solutions of the helmholtz weyl composition equation in underwater acoustics the quadratic profile the application of phase space and path integral methods to both mathematical and computational direct and inverse wave propagation modeling at the level of the scalar one way helmholtz equation is briefly reviewed the role of operator symbols is stressed and their properties briefly discussed the construction of the operator symbol requires the exact or approximate solution of the helmholtz weyl composition equation in the weyl pseudodifferential operator calculus the exact symbols for several quadratic profiles are presented and briefly analyzed the symbols exactly corresponding to a family of operator rational approximations are also presented for the quadratic case these results are used to illustrate several points pertinent to wide angle propagation modeling and the refractive index profile reconstruction problem in underwater acoustics

the capacity region of the gaussian multiple input multiple output broadcast channel the gaussian multiple input multiple output mimo broadcast channel bc is considered the dirty paper coding dpc rate region is shown to coincide with the capacity region to that end a new notion of an enhanced broadcast channel is introduced and is used jointly with the entropy power inequality to show that a superposition of gaussian codes is optimal for the degraded vector broadcast channel and that dpc is optimal for the nondegraded case furthermore the capacity region is characterized under a wide range of input constraints accounting as special cases for the total power and the per antenna power constraints

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

on the sum rate of gaussian multiterminal source coding new proofs and results we show that the lower bound on the sum rate of the direct and indirect gaussian multiterminal source coding problems can be derived in a unified manner by exploiting the semidefinite partial order of the distortion covariance matrices associated with the minimum mean squared error mmse estimation and the so called reduced optimal linear estimation through which an intimate connection between the lower bound and the berger tung upper bound is revealed we give a new proof of the minimum sum rate of the indirect gaussian multiterminal source coding problem i e the gaussian ceo problem for the direct gaussian multiterminal source coding problem we derive a general lower bound on the sum rate and establish a set of sufficient conditions under which the lower bound coincides with the berger tung upper bound we show that the sufficient conditions are satisfied for a class of sources and distortion constraints in particular they hold for arbitrary positive definite source covariance matrices in the high resolution regime in contrast with the existing proofs the new method does not rely on shannon s entropy power inequality

the worst additive noise under a covariance constraint the maximum entropy noise under a lag p autocorrelation constraint is known by burg s theorem to be the pth order gauss markov process satisfying these constraints the question is what is the worst additive noise for a communication channel given these constraints is it the maximum entropy noise the problem becomes one of extremizing the mutual information over all noise processes with covariances satisfying the correlation constraints r sub 0 r sub p for high signal powers the worst additive noise is gauss markov of order p as expected but for low powers the worst additive noise is gaussian with a covariance matrix in a convex set which depends on the signal power

vector gaussian two terminal source coding we derive a lower bound on each supporting line of the rate region of the vector gaussian two terminal ceo problem which is a special case of the indirect vector gaussian two terminal source coding problem the key technical ingredient is a new extremal inequality it is shown that the lower bound coincides with the berger tung upper bound in the high resolution regime similar results are derived for the direct vector gaussian two terminal source coding problem

on the sum rate loss of quadratic gaussian multiterminal source coding this work studies the sum rate loss of quadratic gaussian multiterminal source coding i e the difference between the minimum sum rates of distributed encoding and joint encoding both with joint decoding of correlated gaussian sources subject to mse distortion constraints on individual sources it is shown that under the nondegraded assumption i e all target distortions are simultaneously achievable by a gaussian berger tung scheme the supremum of the sum rate loss of distributed encoding over joint encoding of l jointly gaussian sources increases almost linearly in the number of sources l with an asymptotic slope of 0 1083 bit per sample per source as l goes to infinity this result is obtained even though we currently do not have the full knowledge of the minimum sum rate for the distributed encoding case the main idea is to upper bound the minimum sum rate of multiterminal source coding by that achieved by parallel gaussian test channels while lower bounding the minimum sum rate of joint encoding by a reverse water filling solution to a relaxed joint encoding problem of the same set of gaussian sources with a sum distortion constraint that equals the sum of the individual target distortions we show that under the nondegraded assumption the supremum difference between the upper bound for distributed encoding and the lower bound for joint encoding is achieved in the bi eigen equal variance with equal distortion case in which both bounds are known to be tight

parabolic approximations to the time independent elastic wave equation a splitting matrix method is used to derive two parabolic approximation partial differential equations to the three dimensional linear elastic wave equation in isotropic inhomogeneous media the derivation is valid for media whose lame parameters vary slowly on the length scale of the wavelength of the elastic waves next an integral form of the full wave equation is derived based on the splitting matrix and the parabolic approximation solution iteration of this equation gives a three dimensional vector valued series which generalizes the one dimensional bremmer series which was used in the study of second order ordinary differential equations these results are expected to have applications to geophysical modeling and nondestructive evaluation

the gaussian many help one distributed source coding problem jointly gaussian memoryless sources are observed at n distinct terminals the goal is to efficiently encode the observations in a distributed fashion so as to enable reconstruction of any one of the observations say the first one at the decoder subject to a quadratic fidelity criterion our main result is a precise characterization of the rate distortion region when the covariance matrix of the sources satisfies a tree structure condition in this situation a natural analog digital separation scheme optimally trades off the distributed quantization rate tuples and the distortion in the reconstruction each encoder consists of a point to point gaussian vector quantizer followed by a slepian wolf binning encoder we also provide a partial converse that suggests that the tree structure condition is fundamental

rate region of the quadratic gaussian two encoder source coding problem we determine the rate region of the quadratic gaussian two encoder source coding problem this rate region is achieved by a simple architecture that separates the analog and digital aspects of the compression furthermore this architecture requires higher rates to send a gaussian source than it does to send any other source with the same covariance our techniques can also be used to determine the sum rate of some generalizations of this classical problem our approach involves coupling the problem to a quadratic gaussian ldquoceo problem rdquo

time domain formulation for pulse propagation including nonlinear behavior at a caustic this paper derives a simple three dimensional time domain formulation for small angle propagation of finite amplitude acoustic pulses and weak shocks in a medium that can be range dependent the formulation is closely related to a full wave model and does not contain the artificial singularities of ray based models at caustics the simplicity of the formulation suggests that it may be of value for broadband linear as well as nonlinear propagation two derivations are given one heuristic and the other a formal series expansion from the fluid equations the result is a first order nonlinear progressive wave equation npe cast in a wave following frame of reference the npe is shown to be the nonlinear time domain counterpart of the frequency domain parabolic wave equation pe the npe gives a natural separation of terms governing refraction spreading and nonlinear steepening numerical calculations not involving fourier synthesis are presented using the npe to simulate the following cases a broadband linear pulse propagation in a waveguide b the development of an initially smooth nonlinear pulse into an n wave and c the behavior of a weak shock at a caustic cases a and b are compared with analytic solutions

three dimensional time domain paraxial approximations for ocean acoustic wave propagation one way narrow and wide angle three dimensional time domain paraxial approximations to the wave equation are developed to model acoustic propagation the approximate equations are designed to be appropriate for ocean applications including pulse propagation with volume attenuation and variable density first differential equations for acoustic pressure are obtained from a thermodynamic model which incorporates attenuation due to the presence of acoustically absorbing chemical species the method of multiple scales is then used to generate partial differential equations with paraxial characteristics comparisons are made to corresponding previous results results from an operator formalism using pade approximants are subsequently compared with the corresponding multiscaling results appropriate boundary initial and interface conditions are described for the model equations stability of the three dimensional narrow angle approximation is demonstrated by means of an energy integral analytical expressi

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

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

a higher order parabolic equation for wave propagation in an ocean overlying an elastic bottom a higher order elastic parabolic equation hepe is derived for wave propagation in depth dependent and weakly range dependent fluid solid media galerkin s method is used to discretize the depth operators in the hepe within layers in which depth variations in the lame constants and density are continuous discontinuities in material properties are handled with centered differences for the interface conditions between layers the numerical solution of the hepe also involves the method of alternating directions and crank nicolson integration the hepe is applied to underwater wave propagation problems involving a water column over an elastic bottom including a weakly range dependent problem the accuracy of the hepe is demonstrated with benchmark calculations

on non negative quasiconvex functions with unbounded zero sets we construct nontrivial non negative quasiconvex functions denned on m 2 2 with p th order growth such that the zero sets of the functions are lipschitz graphs of mappings from subsets of a fixed two dimensional subspace to its orthogonal complement we assume that the graphs do not have rank one connections with the lipschitz constants sufficiently small in particular we are able to construct quasiconvex functions which are homogeneous of degree p p 1 and conjugating invariant

a new sufficient condition for sum rate tightness in quadratic gaussian multiterminal source coding this paper considers the quadratic gaussian multiterminal mt source coding problem and provides a new sufficient condition for the berger tung bt sum rate bound to be tight the converse proof utilizes a set of virtual remote sources given which the observed sources are block independent with a maximum block size of 2 the given mt source coding problem is then related to a set of two terminal problems with matrix distortion constraints for which a new lower bound on the sum rate is given by formulating a convex optimization problem over all distortion matrices a sufficient condition is derived for the optimal bt scheme to satisfy the subgradient based karush kuhn tucker condition the subset of the quadratic gaussian mt problem satisfying our new sufficient condition subsumes all previously known tight cases and our proof technique opens a new direction for more general partial solutions

distributed source coding of correlated gaussian remote sources we consider the distributed source coding system for l correlated memoryless gaussian remote sources specified with l correlated gaussian random variables xi i 1 2 l we deal with the case where each of l distributed encoders cannot directly observe the source xi but its noisy version yi xi ni here ni i 1 2 l are independent additive l gaussian noises also independent of xi i 1 2 l on this coding system the determination problem of the rate distortion region remains open in general in this paper we derive explicit outer and inner bounds of the rate distortion region we further find an explicit sufficient condition for those two bounds to match we also study the sum rate part of the rate distortion region when the correlation has some symmetrical property we derive an explicit upper bound of the sum rate part from the inner bounds of the rate distortion region on a lower bound we derive a new explicit bound by making full use of the symmetrical property of the correlation we further derive an explicit necessary and sufficient condition for the lower bound to coincide with the upper bound

a multiscale derivation of a new parabolic equation which includes density variations abstract a new parabolic equation is obtained from the acoustic equation by the multiscale method the new equation incorporates the effects of a variable ocean density the density can be smooth or piecewise smooth thus the new formulation alleviates the need for interfacial conditions when the density is stratified in a piecewise constant fashion it also reduces to the standard p e when the density is constant the new equation has the same conservation law as the p e a difference equation is presented which has a discrete version of the same law

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 generalized westervelt equation for nonlinear medical ultrasound a model equation is derived for nonlinear medical ultrasound unlike the existing models which use spatial coordinates material coordinates are used and hence a model for a heterogeneous medium is able to be derived the equation is a generalization of the westervelt equation and includes the nonlinearity relaxation and heterogeneity of soft tissue the validity of the generalized westervelt equation as a model equation for a piola kirchoff acoustic pressure and as an equation for the acoustic pressure is discussed in the second case it turns out that the model follows from two geometric approximations which are valid when the radius of curvature of the phase fronts is much larger than the particle displacements the model is exact for plane waves and includes arbitrary nonlinearity in the stress strain relation

the sum rate bound for a new class of quadratic gaussian multiterminal source coding problems in this paper we show tightness of the berger tung bt sum rate bound for a new class of quadratic gaussian multiterminal mt source coding problems dubbed bi eigen equal variance with equal distortion beev ed where the l l source covariance matrix has equal diagonal elements with two distinct eigenvalues and the l target distortions are equal let k k l be the number of repetitions of the larger eigenvalue the beev covariance structure allows us to connect k i i d virtual gaussian sources with the l given mt sources via an l k semiorthogonal transform whose rows have equal euclidean norm plus additive i i d gaussian noises resulting in the two sets of sources being mutually conditional i i d by relating the given mt source coding problem to a generalized gaussian ceo problem with the k virtual sources as remote sources and the l mt sources as observations we obtain a lower bound on the mt sum rate and show its achievability by bt schemes under the equal distortion constraints our beev ed class of quadratic gaussian mt source coding problems subsumes both the positive symmetric case considered by wagner et al and the negative symmetric case other examples including a subclass of sources with be circulant symmetric covariance matrices and equal distortion constraints are also provided to highlight tightness of the sum rate bound

distributed compression of linear functions partial sum rate tightness and gap to optimal sum rate we consider the problem of distributed compression of the difference z y1 cy2 of two jointly gaussian sources y1 and y2 under an mse distortion constraint d on z the rate region for this problem is unknown if the correlation coefficient and the weighting factor c satisfy c 0 inspired by ahlswede and han s scheme for the problem of distributed compression of the modulo 2 sum of two binary sources we first propose a hybrid random structured coding scheme that is capable of saving the sum rate over both the random quantize and bin qb coding scheme and krithivasan and pradhan s structured lattice coding scheme the main idea is to use a random coding component in the first layer to adjust the source correlation so that the structured coding component in the second layer can be more efficient with the outputs from the first layer as decoder side information we then provide a new sum rate lower bound for the problem in hand by connecting it to the gaussian two terminal source coding problem with covariance matrix distortion constraint our lower bound not only improves existing bounds in many cases but also allows us to prove sum rate tightness of the qb scheme when c is either relatively small or large and d is larger than some threshold furthermore our lower bound enables us to show that our new hybrid scheme performs within two b s from the optimal sum rate for all values of c and d

on various semiconvex relaxations of the squared distance function for the euclidean squared distance function f dist 2 k with k m n n we show that k is convex if and only if f equals either its rank one convex quasiconvex or polyconvex relaxations we also establish that if i k is compact and contractible or ii dim c k k n k is convex if and only if f equals one of the semiconvex relaxations when dist 2 p k is sufficiently large and for case i p m nxn for case ii p e k a k dimensional plane containing c k we also give some estimates of the difference between dist 2 p k and its semiconvex relaxations some possible extensions to more general p distance functions are also considered

on the generalized gaussian ceo problem this paper considers a distributed source coding dsc problem where l encoders observe noisy linear combinations of k correlated remote gaussian sources and separately transmit the compressed observations to the decoder to reconstruct the remote sources subject to a sum distortion constraint this dsc problem is referred to as the generalized gaussian ceo problem since it can be viewed as a generalization of the quadratic gaussian ceo problem where the number of remote source k 1 first we provide a new outer region obtained using the entropy power inequality and an equivalent argument in the sense of having the same rate distortion region and berger tung inner region among a certain class of generalized gaussian ceo problems we then give two sufficient conditions for our new outer region to match the inner region achieved by berger tung schemes where the second matching condition implies that in the low distortion regime the berger tung inner rate region is always tight while in the high distortion regime the same region is tight if a certain condition holds the sum rate part of the outer region is also studied and shown to meet the berger tung sum rate upper bound under a certain condition which is obtained using the karush kuhn tucker conditions of the underlying convex semidefinite optimization problem and is in general weaker than the aforesaid two for rate region tightness

the rate distortion function for the quadratic gaussian ceo problem a new multiterminal source coding problem called the ceo problem was presented and investigated by berger zhang and viswanathan recently viswanathan and berger have investigated an extension of the ceo problem to gaussian sources and call it the quadratic gaussian ceo problem they considered this problem from a statistical viewpoint deriving some interesting results in this paper we consider the quadratic gaussian ceo problem from a standpoint of multiterminal rate distortion theory we regard the ceo problem as a certain multiterminal remote source coding problem with infinitely many separate encoders whose observations are conditionally independent if the remote source is given this viewpoint leads us to a complete solution to the problem we determine the tradeoff between the total amount of rate and squared distortion deriving an explicit formula of the rate distortion function the derived function has the form of a sum of two nonnegative functions one is a classical rate distortion function for single gaussian source and the other is a new rate distortion function which dominates the performance of the system for a relatively small distortion it follows immediately from our result that the conjecture of viswanathan and berger on the asymptotic behavior of minimum squared distortion for large rates is true

rate distortion theory for gaussian multiterminal source coding systems with several side informations at the decoder in this paper we consider the separate coding problem for l 1 correlated gaussian memoryless sources we deal with the case where l sources work as partial side information at the decoder for the reconstruction of the remaining source the determination problem of the rate distortion region for this system is the so called many help one problem and it has been known as a highly challenging problem for almost 20 years in this paper we give a partial solution to this problem we determine the rate distortion region in the case where the l sources working as partial side information are conditionally independent if the remaining source we wish to reconstruct is given the additive white gaussian noise ceo problem is a special case of this we also discuss the relation of the result previous results of ours

vector gaussian multiterminal source coding we derive an outer bound of the rate region of the vector gaussian l terminal ceo problem by establishing a lower bound on each supporting hyperplane of the rate region to this end we prove a new extremal inequality by exploiting the connection between differential entropy and fisher information as well as some fundamental estimation theoretic inequalities it is shown that the outer bound matches the berger tung inner bound in the high resolution regime we then derive a lower bound on each supporting hyperplane of the rate region of the direct vector gaussian l terminal source coding problem by coupling it with the ceo problem through a limiting argument the tightness of this lower bound in the high resolution regime and the weak dependence regime is also proved

the time domain solution of the wide angle parabolic equation including the effects of sediment dispersion the wide angle time domain parabolic equation tdpe which is the inverse fourier transform of the wide angle parabolic equation pe is derived a numerical solution for the model is described and a benchmark calculation is presented the narrow angle tdpe is also considered and its error is analyzed and compared with the error of the narrow angle pe the tdpe is compared with the progressive wave equation which is shown to be restricted to narrow angle propagation for practical purposes in the sediment attenuation is assumed to depend linearly on frequency and the corresponding causal dispersion law is assumed the model is used to show that the effect of sediment dispersion on pulse propagation in the ocean can be significant

higher order parabolic approximations of the reduced wave equation abstract asymptotic solutions of order k n are developed for the reduced wave equation here k is a dimensionless wave number and n is the arbitrary order of the approximation these approximations are an extension of geometric acoustics theory and provide corrections to that theory in the form of multiplicative functions which satisfy parabolic partial differential equations these corrections account for the diffraction effects caused by variation of the field normal to the ray path and the interaction of these transverse variations with the variation of the field along the ray the theory is applied to the example of radiation from a piston and it is demonstrated that the higher order approximations are more accurate for decreasing values of k

a parabolic approximation method with application to global wave propagation motivated by the difficulty in using the splitting matrix method to obtain parabolic approximations to complicated wave equations we have developed an alternative method it is three dimensional does not a priori assume a preferred direction or path of propagation in the horizontal determines spreading factors and results in equations that are energy conserving it is an extension of previous work by several authors relating parabolic equations to the horizontal ray acoustics approximation unlike previous work it applies the horizontal ray acoustics approximation to the propagator rather than to the green s function or the homogenous field the propagator is related to the green s function by an integral over the famous fifth parameter of fock and feynman methods for evaluating this integral are equivalent to narrow angle approximations and their wide angle improvements when this new method is applied to simple problems it gives the standard results in this paper it is described by applying it to a problem of current interest the development of a parabolic approximation for modeling global underwater and atmospheric acoustic propagation the oceanic or atmospheric waveguide is on an earth that is modeled as an arbitrary convex solid of revolution the method results in a parabolic equation that is energy conserving and has a spreading factor that describes field intensification for antipodal propagation significantly it does not have the singularities in its range sliced version possessed by many parabolic equations developed for global propagation we then discuss two extensions of the method first to propagation along refracted geodesics and second to a description involving discrete local normal modes

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

the ceo problem multiterminal source coding we consider a new problem in multiterminal source coding motivated by the following decentralized communication estimation task a firm s chief executive officer ceo is interested in the data sequence x t sub t 1 sup spl infin which cannot be observed directly perhaps because it represents tactical decisions by a competing firm the ceo deploys a team of l agents who observe independently corrupted versions of x t sub t 1 sup spl infin because x t is only one among many pressing matters to which the ceo must attend the combined data rate at which the agents may communicate information about their observations to the ceo is limited to say r bits per second if the agents were permitted to confer and pool their data then in the limit as l spl rarr spl infin they usually would be able to smooth out their independent observation noises entirely then they could use their r bits per second to provide the ceo with a representation of x t with fidelity d r where d spl middot is the distortion rate function of x t in particular with such data pooling d can be made arbitrarily small if r exceeds the entropy rate h of x t suppose however that the agents are not permitted to convene agent i having to send data based solely on his own noisy observations y sub i t we show that then there does not exist a finite value of r for which even infinitely many agents can make d arbitrarily small furthermore in this isolated agents case we determine the asymptotic behavior of the minimal error frequency in the limit as l and then r tend to infinity