Finite-Difference Modeling of Very-Low-Frequency Propagation in the Earth-Ionosphere Waveguide

Simulations of Very-low-frequency (VLF) transmitter signals are conducted using three models: the long-wave propagation capability, a finite-difference (FD) time-domain model, and an FD frequency-domain model. The FD models are corrected using Richardson extrapolation to minimize the numerical dispersion inherent in these models. Using identical ionosphere and ground parameters, the three models are shown to agree very well in their simulated VLF signal amplitude and phase, to within 1 dB of amplitude and a few degrees of phase, for a number of different simulation paths and transmitter frequencies. Furthermore, the three models are shown to produce comparable phase changes for the same ionosphere perturbations, again to within a few degrees. Finally, we show that the models reproduce the phase data of existing VLF transmitter–receiver pairs reasonably well, although the nighttime variation in the measured phase data is not captured by the simplified characterization of the ionosphere.