Uniform Asymptotic Expansion of the Generalized Bremmer Series

The Bremmer coupling series solution of the wave equation, in generally inhomogeneous media, requires the introduction of pseudo-differential operators. In this paper, in two dimensions, uniform asymptotic expansions of the Schwartz kernels of these operators are derived. Also, we derive a uniform asymptotic expansion of the one-way propagator appearing in the series. We focus on designing closed-form representations, valid in the highfrequency limit, taking into account critical scattering-angle phenomena. Our expansion is not limited by propagation angle. In principle, the uniform asymptotic expansion of a kernel follows by matching its asymptotic behaviors away and near its diagonal. The Bremmer series solver consists of three steps: directional decomposition into upand downgoing waves, one-way propagation, and interaction of the counter-propagating constituents. Each of these steps is here represented by a kernel for which a uniform asymptotic expansion is found. The associated algorithm provides a fundamental improvement of the parabolic-equation and phase-shift/phase-screen style methods applied in ocean acoustics, integrated optics and exploration seismology. Center for Wave Phenomena and Department of Mathematical and Computer Sciences, Colorado School of Mines, Golden, CO 80401-1887, USA ([email protected]). M.V. d.H. would like to thank Mobil for financial support of this research. Applied Mathematical Sciences, Ames Laboratory and Department of Mathematics, Iowa State University, Ames, IA 50011-3020, USA ([email protected]). The work of A.K. G. was supported by the Applied Mathematical Sciences subprogram of the Office of Energy Research of the United States Department of Energy under contract No. W-7405-Eng-82. 1 M.V. de Hoop, A.K. Gautesen Uniform asymptotic Bremmer series 2