Ionospheric virtual height ($h^{\prime }=c{\tau }_g/2$)

For a vertical outgoing and returning radio wave, define the virtual height $h^{\prime }=c{\tau }_g/2$ from its round-trip group delay ${\tau }_g$, after removing instrument delays. It is the height a fictitious reflector would have if both legs traveled at $c$. It differs from both the real turning height $z_t$ and a height fitted to an oblique constant-path interference fringe model.

An original solvable profile makes the difference explicit. Approximate the region below $z_0$ by $n=1$, and prescribe a linear rise of squared plasma frequency above it:Here the symbol $u_t$ denotes the penetration depth, and $a$ has units ${\mathrm{s}}^{-2}{\mathrm{m}}^{-1}$. In the propagating part of the ramp, $n=\sqrt{1-(z-z_0)/u_t}$. In the leading geometrical-ray delay model, with a frequency-independent turning connection wave phase, the round-trip group delay isThe endpoint singularity is integrable: substitute $s=\sqrt{1-u/u_t}$ to obtain $2u_t$. This approximation does not resolve the finite-wavelength structure at the turning region. It illustrates the effect of slow group velocity before the return, rather than adding extra geometric altitude.

As an independent phase-based derivation, the propagation contribution to the round-trip wave phase isIts derivative $d\Psi /d\omega =2(z_0+2{\omega }^2/a)/c$ agrees with the delay integral. Using the phase integral itself as a distance at speed $c$ would give $z_0+2u_t/3$ instead of $h^{\prime }$: wave phase, group delay and real turning height are three different quantities.

Choose $z_0=70\mathrm{km}$ and stipulate $f_p(z)=3\mathrm{MHz}\sqrt{(z-z_0)/(30\mathrm{km})}$, where $f_p={\omega }_p/(2\pi )$. ThenAt $f=3\mathrm{MHz}$, the real turning height is $100\mathrm{km}$ but the virtual height is $130\mathrm{km}$, corresponding to ${\tau }_g\simeq 0.867\mathrm{ms}$ with $c\simeq 3.00\times 10^8\mathrm{m}{\mathrm{s}}^{-1}$. These stipulated parameters describe neither a measured profile nor a separate physical reflector at $130\mathrm{km}$. Because the ramp has no density maximum, it also has no finite upper penetration threshold; a real bounded E layer or F layer requires a richer profile.