The presence of a turbulent band of ELF hiss throughout the plasmasphere has been modeled by following unducted whistler mode waves from a restricted region of cyclotron resonant growth in the outer plasmasphere. The most unstable waves originate near the equatorial plane within a modest cone of wave normal angles aligned along the ambient magnetic field. As the waves propagate to higher latitude, their wave normal direction becomes progressively oblique, and ray paths generally migrate outward until they reach the plasmapause. A major fraction of the incident waves internally reflect at the steep plasmapause density gradient and subsequently propagate inward to populate most of the plasmasphere. Only a minor part of the wave energy is able to escape from the plasmasphere, primarily at high latitude. A detailed study of the internally trapped ray paths shows that certain waves can return to the equatorial growth region with field-aligned propagation vectors and thus experience further amplification. this is particularly important for the maintenance of quiet time hiss when a single transit of the growth region is insufficient to amplify the background incoherent cyclotron noise to detectable levels. The model calculations demonstrate that the observed frequency spectrum and polarization of plasmaspheric hiss not only are controlled by the relative rates of cyclotron gowth and Landau damping but also are strongly influenced by the wave propagation characteristics.