Factorization and path integration of the Helmholtz equation: Numerical algorithms

Abstract

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 parabolicbased (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.

On Green's function retrieval by iterative substitution of the coupled Marchenko equations

“…The wavefields are decomposed into downgoing and upgoing constituents, where downgoing is indicated with superscript + and upgoing with superscript −. The decomposed wavefields are flux-normalized (Corones et al 1983;Fishman et al 1987) and can be related to the physical quantities of pressure and vertical particle velocity, using expressions of Wapenaar & Grimbergen (1996). We note that the coordinate system can be rotated and that reciprocity theorems for one-way wavefields have also been defined in curvilinear coordinate systems (Frijlink & Wapenaar 2010).…”

Green's function retrieval from reflection data, in absence of a receiver at the virtual source position

“…The downward and upward propagating, mutually coupled, constituents of the wave field are denoted by p þ and p À , respectively. In the space-frequency domain, the formal relation between one-way (i.e., down-going and up-going) and two-way (i.e., total) wave fields is given by [24][25][26][27][28][29][30][31] …”

Phase Space Methods and Path Integration: A Microscopic Approach to Direct and Inverse Wave Propagation.

“…-w/k. Consistent with taking the square root of the indefinite Helmholtz operator, the corresponding symbols, generally, have both real and imaginary parts characterized by oscillatory behavior (4,6], as illustrated in Figure 2. Nonuniform and uniform perturbation solutions corresponding to definite physical limits (frequency, propagation angle, field strength, field gradient) recover several known approximate wave theories (ordinary parabolic, rangerefraction parabolic, Grandvuillemin-extended parabolic, half-space Born, Thomson-Chapman, rational linear) and systematically lead to several new full-wave, wide-angle approximations [2][3][4]6].…”

“…As the operator symbol is not simply quadratic in p, the configuration space Feynman path integral formulation is not appropriate, necessitating the more general phase space construction [4,7]. This results in a parabolic-based (one-way) Hamiltonian phase space path integral representation of the propagator in the form [3,7] N-i N G (xx iO,x t ) -lim…”

“…The exact pseudo-differential evolution equation (2) and, in general, the wide-angle extended parabolic approximate equations derived from the ana'ysis of the composition equation [2][3][4]6] are singular integrodifferential wave equations. Solution representations for such pseudodifferential equations can be directly expressed in terms of infinitedimensional functional, or path, integrals (7,8], following from the Markov property of the propagator.…”