2b. Prior modeling of east-west asymmetry of VLF propagation
The reflection of radio waves from the underside of the ionosphere became an active area of research during the 1950s [see the historical references given by, e.g., Barber and Crombie , 1959; Wait and Spies , 1960; Wait and Spies , 1964; Yabroff , 1957]. The problem was nontrivial due to the anisotropy of the dielectric, associated with the gyration of charged particles about the geomagnetic field. This was especially true for VLF waves, whose height of reflection occurs in the lowermost ionosphere, namely the D-layer. The strong electron-neutral collision rate in the D-layer further complicates models of VLF reflection. The models needed to address practical challenges, e.g.:
(a) What is the VLF reflectivity?
(b) How does it depend on solar zenith angle?
(c) How does the reflectivity depend on angle-of-incidence?
(d) How does the reflectivity depend on local propagation magnetic azimuth (reckoned clockwise from local magnetic North) and on local magnetic declination (”dip angle”)?
(e) How does the reflectivity depend on electron-neutral collision rate?
Starting late in the 1950’s, sharp-boundary treatments of the collisional, anisotropic VLF reflection process were set up analytically and solved numerically with newly available digital computers [Barber and Crombie , 1959; Wait and Spies , 1960;Wait and Spies , 1964; Yabroff , 1957]. The first numerical model of an arbitrarily-layered (rather than just a sharp boundary) D-layer [Piggott et al. , 1965; Pitteway , 1965] followed quickly, although its physical implications appear to have been only slowly appreciated. The Pitteway model for the continuously varying D-layer solved the Maxwell Equations for the altitude-dependent, anisotropic, and complex susceptibility tensor. All of the sharp-boundary models, as well as the Pitteway model, dealt with the elementary reflection of an incident plane wave.
Such plane-wave models are excellent for providing insights on ”process” questions, such as those cited in the previous paragraph. However, for long-range ”multi-hop” propagation, it is more efficient, though less heuristically instructive, to cast the problem in terms of waveguide modes in the spherical-shell Earth-ionosphere waveguide (EIWG). The modes are akin to cylindrical waves from a point source within a parallel-plane waveguide, except that the waveguide elements are (approximately) concentric spherical surfaces [see the illuminating tutorial by Cummer , 2000]. A waveguide model provides a point-to-point complete description of the VLF transmission along any given Great Circle path. This includes all portions of the path. The first portion consists of 3-dimensional expansion of the wavefield into a hemisphere. The next portion takes account of the first ionospheric reflection, which effectively is a transition to spherical-shell EIWG propagation. This transition needs many higher-order modes to describe the wavefield, because at such a short range (e.g, < 1000 km) a broad range of plane-wave ”angles of incidence” are at play [Cummer , 2000]. Ultimately, however, at longer range the waveguide modes simplify. For a vertical-dipole source near ground level, and a vertical-dipole receiver also near ground level, the modes simplify at large distances to the fundamental Transverse Magnetic (TM) mode. Thus the transmission characteristics vary from 3-dimensional expansion into a hemisphere, to a single 2-D, fundamental TM mode in the waveguide.
The waveguide approach was perfected in the Long Range Propagation Capability, or LWPC [Pappert and Ferguson , 1986] suite of computer codes developed by the United States Navy. The LWPC includes an atlas of Earth-surface conductivity. The user can select a D-layer model, usually exponential profiles of electron density and of electron-neutral collision rate. The LWPC contains ”everything” in one master code suite. LWPC uses just an approximation of the D-layer electron-density profile, but that is justified by the impossibility of knowing any better profile at any given instant.
One adverse side-effect of its end-to-end completeness is that the LWPC blurs (to the LWPC user) the role of local parameters, such as solar magnetic propagation azimuth and local magnetic dip angle. These vary along the path, but the LWPC’s end-to-end approach path-integrates over their local variations, and all the user sees is the result of the path integration. Thus, despite its completeness, premiere accuracy, and reliability, the LWPC is not pedagogically illuminating for exploring individual local processes in isolation.