Louis Fishman

Derivation and application of extended parabolic wave theories. I. The factorized Helmholtz equation

The reduced scalar Helmholtz equation for a transversely inhomogeneous half‐space supplemented with an outgoing radiation condition and an appropriate boundary condition on the initial‐value plane defines a direct acoustic propagation model. This elliptic formulation admits a factorization and is subsequently equivalent to a first‐order Weyl pseudodifferential equation which is recognized as an extended parabolic propagation model. Perturbation treatments of the appropriate Weyl composition equation result in a systematic development of approximate wave theories which extend the narrow‐angle, weak‐inhomogeneity, and weak‐gradient ordinary parabolic (Schrodinger) approximation. The analysis further provides for the formulation and exact solution of a multidimensional nonlinear inverse problem appropriate for ocean acoustic and seismic studies. The wave theories foreshadow computational algorithms, the inclusion of range‐dependent effects, and the extension to (1) the vector formulation appropriate for elas...

Factorization and Path Integration of the Helmholtz Equation: Numerical Algorithms

The propagator for the reduced scalar Helmholtz equation plays a significant role in both analytical and computational studies of acoustic direct wave propagation. Path (functional) integrals are taken to provide the principal representation of the propagator and are computed directly. The path integral is the primary tool in extending the classical Fourier methods, so appropriate for wave propagation in homogeneous media, to inhomogeneous media. For transversely inhomogeneous environments, the n‐dimensional Helmholtz equation can be exactly factored into separate forward and backward one‐way wave equations. A parabolic‐based (one‐way) phase space path integral construction provides the generalization of the Tappert/Hardin split‐step FFT algorithm to the full one‐way (factored Helmholtz) wave equation. These extended marching algorithms can readily accommodate density profiles and range updating, and further, in conjunction with imbedding methods, provide the basis for incorporating backscatter effects. In a complementary manner, for general range‐dependent environments, elliptic‐based (two‐way) path integral constructions lead to an approximate representation of the propagator (Feynman/Garrod) and a natural statistical (Monte Carlo) means of evaluation. Taken together, the path integrals provide the basis for a global analysis in addition to providing a unifying framework for dynamical approximations, resolution of the square root operator, and the concept of an underlying stochastic process. The one‐way marching algorithms are applied to ocean acoustic environments, seismological environments, and extreme model environments designed to establish their range of validity and manner of breakdown.

Derivation And Application Of Extended Parabolic Wave Theories

The parabolic approximation, applied to the propagation of time harmonic scalar waves, replaces the governing Helmholtz equation with a SchrOdinger equation. The primary computational advantage of this approximation is that it is first order in the range coordinate. The validity of the approximation limits variations in an inhomogeneous wave number field to be both small, over the total range of the experiment, and slow, as measured on a length scale determined by an averaged wavelength. Thus, the validity of the approximation can be severely strained in a number of applications; e.g., it is difficult to justify its application in the presence of regions of rapid, even discontinuous, changes near a surface that runs in the range direction. In this paper we consider the derivation of extended parabolic wave theories which retain the feature that they are first order in the range coordinate. Distinguishing the extended theories from the ordinary one, i.e. the SchrOdinger equation, is the manner in which the cross-range coordinates enter. In the ordinary theory, they appear via the two-dimensional Laplacian, a differential or local operator; in the extended theories they appear via non-local operators. Obtaining explicit forms for specific nonlocal cross-range operators requires the solution of factor-ordering problems which have their counterpart in the correspondence between quantized theories governing the evolution of dynamical systems and their related classical formulations. The operator construction can be related to the construction of both coordinate and phase space path integrals for the appropriate wave propagator and further provides for the formulation and exact solution of a multidimensional nonlinear inverse problem appropriate for ocean acoustic and seismic modelling.

Derivation and application of extended parabolic wave theories. II. Path integral representations

The n‐dimensional reduced scalar Helmholtz equation for a transversely inhomogeneous medium is naturally related to parabolic propagation models through (1) the n‐dimensional extended parabolic (Weyl pseudodifferential) equation and (2) an imbedding in an (n+1)‐dimensional parabolic (Schrodinger) equation. The first relationship provides the basis for the parabolic‐based Hamiltonian phase space path integral representation of the half‐space propagator. The second relationship provides the basis for the elliptic‐based path integral representations associated with Feynman and Fradkin, Feynman and Garrod, and Feynman and DeWitt‐Morette. Exact and approximate path integral constructions are derived for the homogeneous and transversely inhomogeneous cases corresponding to both narrow‐ and wide‐angle extended parabolic wave theories. The path integrals allow for a global perspective of the transition from elliptic to parabolic wave theory in addition to providing a unifying framework for dynamical approximation...

Factorization, path integral representations, and the construction of direct and inverse wave propagation theories

The development of both direct and inverse wave propagation theories at the level of the reduced scalar Helmholtz equation is addressed. The principal constructive tool is the factorization analysis of the Helmholtz equation which derives from the physically suggestive picture of decoupled forward and backward wave propagation appropriate for transversely inhomogeneous environments. The factorization analysis, most significantly, provides for a more “microscopic” understanding of the Helmholtz wave propagation process. In so doing, the analysis underscores the intimate connection between the direct and inverse theories, provides an inherently multidimensional framework, and further serves as a focal point for a wide range of physical (the concept of an underlying stochastic process, the notion of strong and weak-coupling regimes, free motion on curved spaces) and mathematical (pseudo-differential operator theory, path integral methods, imbedding techniques) perspectives. The operator and path integral analysis provides for computational algorithms while foreshadowing extensions to more complex environments and other physical theories.

Phase space methods and path integration: the analysis and computation of scalar wave equations

Abstract The scalar Helmholtz equation plays a significant role in studies of electromagnetic, seismic, and acoustic direct wave propagation. Phase space, or ‘microscopic’, methods and path (functional) integral representations provide the appropriate framework to extend homogeneous Fourier methods to inhomogeneous environments. The two complementary approaches to this analysis and computation of the n -dimensional Helmholtz propagator are reviewed. For the factorization/(one-way) path integration/invariant imbedding approach, the exact solution of the Helmholtz composition equation for the Weyl square root operator symbol is presented in the quadratic case. The filtered, one-way, phase space marching algorithm is examined in detail and compared numerically with wide-angle, one-way, partial differential wave equations formally derived from approximation theory. For the second approach, which directly constructs approximate two-way path functionals, the feasibility of a Monte Carlo (statistical) evaluation of the Feynman/Garrod propagator is discussed.

Propagation modelling based on wave field factorization and invariant imbedding

The proposed propagation models address the propagation of CW acoustic fields through a transition region separating two different, range‐independent oceans. Wave field factorization refers to the conceptual decoupling of the radiation fields in the two range‐independent oceans, into positive and negative propagating components. Invariant imbedding refers to the reformulation of the propagation problem from a two‐point boundary value problem to an initial value problem. In the reformulation the extent of the transition region is an independent variable. The formulation is expressed in terms of reflection and transmission operators, which apply to the transition region. The proposed models are generalizations of existent models in reflection seismology. These latter models allow for no cross‐range variability of wave speed in either the two, now homogeneous, oceans nor in the transition region. The seismology models do accommodate P‐S coupling, which is neglected in the acoustic application. The incorporat...

Range Dependent Propagation Codes Based on Wave Field Factorization and Invariant Imbedding

In the absence of range variation the Helmholtz equation can be exactly factored into two one-way equations. These factored equations provide the bases for a number of computationally efficient codes, the most familiar of which is the ordinary parabolic wave theory. In this paper we discuss the reintroduction of range dependence to enable analysis of media with arbitrary variation. Two calculations will be presented. One is the reflection and transmission of an arbitrary field at an interface separating two transversely inhomogeneous media. The second is the reflection and transmission of an arbitrary field by a transition region separating two transversely inhomogeneous media. The second problem is addressed by an invariant imbedding procedure and taken together the solutions to the two problems represent an exact and complete generalization of Kennett’s method, in the context of scalar waves.

Direct and inverse wave propagation theories and the factorized Helmholtz equation

The reduced scalar Helmholtz equation for a transversely inhomogeneous half‐space supplemented with an outgoing radiation condition and an appropriate boundary condition on the initial‐value plane defines a direct acoustic propagation model. This elliptic formulation admits a factorization and is subsequently equivalent to a first‐order Weyl pseudo‐differential equation which is recognized as an extended parabolic propagation model. Perturbation treatments of the appropriate Weyl composition equation result in a systematic development of approximate wave theories while exact inversions for several nontrivial profiles provide for an analysis of strong refractive and diffractive effects. The analysis, in a natural manner, provides for the formulation and exact solution of a multidimensional nonlinear inverse problem appropriate for ocean acoustic, seismic, and optical studies. Backscatter effects in both the direct and inverse problems can be included through weak‐backscatter and spatial‐dimensional (imbedd...

Exact reflection and transmission operator symbols in two‐dimensional, invariant imbedding‐based, propagation modeling

Recognizing that typical ocean propagation problems are essentially scattering problems in terms of a transition region and transversely inhomogeneous asymptotic half‐spaces, wave‐field splitting, invariant imbedding, and phase space methods have reformulated the problem in terms of an operator scattering matrix characteristic of the transition region. This approach solves the elliptic (fixed‐frequency) scattering problem by well‐posed marching (one‐way) methods, and is centered on the reflection and transmission operator symbol equations. For several nontrivial, two‐dimensional, refractive index profiles, these equations (and the wave‐field equations) are solved exactly and analyzed. These results provide benchmark cases to test both the scattering operator and subsequent wave‐field calculations, providing a severe test of the stiff nonlinear solvers employed in the numerical implementation of the method. The exactly solved models incorporate symmetric and asymmetric wells, trapped modes, and sharp‐gradient features. Numerical results will be presented. [Work supported by NSF, AFOSR, ONR, ASEE, JEWC, and the S. N. Bose Centre for Basic Sciences.]