There are two primary mathematical formulations associated with the VLF (3-30 kHz) and LF (30-300 kHz) bands corresponding to propagation in these bands, one involving a waveguide approach and the second characterized by wave-hop theory. As would be expected, the former approach is usually employed at VLF, while the latter approach is most utilized at LF.
In the waveguide formulation we find that the cavity between the earth and the ionosphere gives rise to the usual features one would expect, including cutoff frequencies and characteristic reflections from segments of the guide for which sharp electrical discontinuities arise. It is noteworthy that either Transverse-Electric (TE) or transverse-Magnetic (TM) modes may be excited, depending upon the model selected for the antenna current element. TE and TM waves both have their own personalities in terms of mode structure, excitation, height-gain factors, and attenuation rates. Nevertheless, they are not totally uncoupled because of the presence of the earth's magnetic field.
It is felt that neither the waveguide nor the wave-hop formulations are unassailable under all conditions, and should be viewed as complementary methods in a thoughtful analysis of VLF/LF coverage. There are a number of models that allow for the prediction field strength in these bands. Components of the U.S. Navy have long been active in this field (e.g., [Morfitt et al., 1982] [Houser et al., 1981] [Kelly et al., 1984]), although current research interest has waned along with the decline in use of communication and navigation systems at VLF/LF. The ITU-R has sanctioned two methods for theoretical calculation of field strength [ITU-R Rec. P.684, 1997], based upon wave-hop and waveguide methodologies.
From a propagation viewpoint, the ionosphere has a far greater impact on signals at VLF/LF than at ELF. Accordingly, the space weather impact is also greater. In the absence of significant space weather events, even VLF and LF bands are remarkably stable to benign ionospheric variabilities. Still, performance degradation in the VLF/LF bands, aside from radio noise competition, can result from a number of space weather factors, including solar flares and related phenomena. Sudden Phase Anomalies (SPA) occur at the same time as daytime solar x-ray events that produce an increase in the normal D layer ionization level. These anomalies, if uncompensated, may yield navigation errors of the order of 10 nautical miles or more. Other events of interest include magnetic storms and Polar Cap Absorption (PCA), the latter being caused by solar protons that gain access to the lower ionosphere within the polar cap. These events give rise to both phase and amplitude distortion of signals in the band. Although solar flare-related effects are of interest, the major problem areas in VLF/LF predictions may be related to other factors, including ground conductivity uncertainties and noise variability. There is considerable interest in the high-latitude region, owing to the combination of ionospheric effects, which dominate the arctic environment, and the existence of low conductivity zones characteristic of permafrost, sea ice, and the ice cap itself. Practical VLF circuits from America to Europe cross the Greenland ice cap, and these circuits are especially vulnerable to attenuation arising from ice cap traverse. Moreover, these paths, which are already degraded, are very sensitive indicators of solar proton events. Table 4-3 is a summary of ionospheric effects at VLF. Figure 4-3 shows the phase and amplitude variation during a PCA for an 18.6 kHz path between Washington, D.C and Sweden.
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