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This paper reports the first application of an optimized geodesic, three-dimensional (3-D) finite-difference time-domain (FDTD) grid to model impulsive, extremely low-frequency (ELF) electromagnetic wave propagation within the entire Earth-ionosphere cavity. This new model, which complements our previously reported efficient 3-D latitude-longitude grid, is comprised entirely of hexagonal cells except for a small, fixed number of pentagonal cells. Grid-cell areas and locations are optimized to yield a smoothly varying area difference between adjacent cells, thereby maximizing numerical convergence. Extending from 100 km below sea level to an altitude of 100 km, this technique can accommodate arbitrary horizontal as well as vertical geometrical and electrical inhomogeneities/anisotropies of the excitation, ionosphere, lithosphere, and oceans. We first verify the global model by comparing the FDTD-calculated daytime ELF propagation attenuation with data reported in the literature. Then as one example application of this grid, we illustrate a novel ELF radar for major oil deposits.