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 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.