The above discussion merely illustrates a simple picture of mode and ray theory. More detailed treatments require a large computer program. In particular, it is difficult to solve the problem of the horizontal and vertical inhomogeneities of the waveguide. The effect of the Earth's curvature is, that near the antipode the field strength slightly increases. Due to the influence of the Earth' magnetic field, the medium becomes anisotropic so that the ionospheric reflection factor in reality is a matrix. This means that a vertically polarized incident wave after reflection at the ionospheric D-layer converses to a vertically and a horizontally polarized wave. Moreover, the geomagnetic field gives rise to a nonreciprocity of VLF waves. Waves propagating from east to west are more strongly attenuated than vice versa. There appears a phase slipping near the distance of the deep interference minimum of Eq. 3. During the times of sunrise and/or sunset, there is sometimes a phase gain or loss of 360° because of the irreversible behavior of the first sky wave.