Fast Algorithms for Long-Range Wave Propagation over Complex Terrain

Abstract

Parabolic Wave Equations are an area of extensive research in the description of wave propagation. The Split-Step Fourier (SSF) method solves the Parabolic Wave Equation spectrally and is a method of choice for long-range propagation through atmosphere. Split-Step Fourier methods, however, are unable to sparsely represent fields and require repeated forward and inverse Fourier Transforms. Furthermore, Radiation Boundary Conditions (RBCs) are cumbersome to implement due to the Periodic Boundary Conditions enforced by the spectral propagator. This thesis solves the one-way wave equation in 2D and 3D with Gabor Transforms, representing propagating fields as a sum of locally supported frame functions with spatial shifts and frequency modulations. Gabor Transforms easily exploit sparsity in the space-frequency representation of structured fields. By precomputing the propagation characteristics of each frame function, a Gabor transformed wavefront can be efficiently propagated from one spatial slice to the next. RBCs are trivially implemented by removing frame functions that propagate outside the computational domain (i.e. beyond certain height bounds) from consideration, a feat that is impossible using classical split-step Fourier methods. Phase screens, formerly requiring immense computational resources to be applied in the spatial domain, are implemented in the Gabor domain. The choice of Gabor frame is critically important for sparsification; the optimum window width must be selected to match the field characteristics of a particular scenario. Conventional Gabor frames have uniform window size for all spatial and frequency shifts. However, real-world problems rarely have uniform field complexity. This thesis will demonstrate the use of the jigsaw puzzle Gabor frame to optimally sparsify propagating fields while maintaining accuracy. The jigsaw puzzle frame is characterized by Gabor window functions that have different widths at different locations in space, rather than uniform window widths over all space. This framework naturally and efficiently accommodates the multi-scale nature of realistic propagation scenarios. This thesis introduces hybrid solvers in 2D and 3D to describe propagation over terrain. In 2D, this thesis outlines a hybrid SSF-FD solver that combines the efficiency of SSF and the modeling flexibility of FD-based schemes. The solver maintains the SSF approach of advancing free-space propagating fields using spectral propagators, but invokes a localized FD scheme to account for field interactions with terrain. From an operational perspective, the hybrid solver slices up the computational domain just like standard SSF- and FD-based solvers. In 3D, a similar hybridization is performed between a sparse 3D Gabor propagator for fields moving through upper atmosphere, and a SSF solver augmented with Impedance Boundary Conditions for fields about the terrain. These hybridizations substantially reduce the computational complexity of propagation problems by only using expensive field descriptions for propagation near terrain, and using fast propagators for all other space.