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

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.