Uniform high-frequency approximations of the square root Helmholtz operator symbol

Abstract

The exact reformulation of the elliptic (two-way) Helmholtz equation in terms of one-way wave equations in a manner which is well-posed for marching is presented. This reformulation has application for large-scale, deterministic, direct wave propagation modelling and the corresponding layer-stripping inverse algorithms, in addition to providing a foundation for stochastic modelling calculations. The one-way wave equations are constructed in terms of appropriate square root Helmholtz and Dirichlet-to-Neumann operators which are explicitly represented in terms of their operator symbols in the Weyl pseudodifferential operator calculus. Moreover, the fundamental wave equation solutions are expressed as path integrals, directly in terms of the operator symbols, which immediately result in marching (one-way) computational algorithms. The analysis and subsequent computational algorithms crucially depend upon the construction of operator symbol approximations which are uniform over phase space, necessitating the extension of the usual pseudodifferential operator asymptotic theory. The development of uniform, phase space, operator symbol approximations is initiated with the derivation of a uniform, high-frequency approximation of the square root Helmholtz operator symbol in the two-dimensional case. A preliminary numerical evaluation of this uniform, high-frequency operator symbol approximation is presented, establishing comparisons with (1) exact operator symbol constructions, (2) families of rational approximation operator symbols, and (3) a uniform, low-frequency operator symbol approximation. These results are then applied to address several points pertinent to multidimensional, wave propagation modelling, computation, and inversion.