= Ionospheric virtual height
{title2=$h'=c\tau_g/2$}
For a vertical outgoing and returning <radio wave>, define the virtual height $h'=c\tau_g/2$ 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:
$$
\omega_p^2(z)=a(z-z_0),\qquad a>0,\qquad
u_t=\frac{\omega^2}{a},\qquad z_t=z_0+u_t.
$$
Here the symbol $u_t$ denotes the penetration depth, and $a$ has units $\mathrm{s^{-2}\,m^{-1}}$. 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> is
$$
\tau_g=\frac2c\left[z_0+\int_0^{u_t}\frac{du}{\sqrt{1-u/u_t}}\right]
=\frac2c(z_0+2u_t),\qquad
h'=z_0+2u_t=2z_t-z_0>z_t.
$$
The 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> is
$$
\Psi(\omega)=\frac{2\omega}{c}\left[z_0+\int_0^{u_t}\sqrt{1-u/u_t}\,du\right]
=\frac{2\omega}{c}\left(z_0+\frac23\frac{\omega^2}{a}\right).
$$
Its <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 $h'$: <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 $f_p=\omega_p/(2\pi)$. Then
$$
z_t=70\,\mathrm{km}+30\,\mathrm{km}\left(\frac{f}{3\,\mathrm{MHz}}\right)^2,\qquad
h'=70\,\mathrm{km}+60\,\mathrm{km}\left(\frac{f}{3\,\mathrm{MHz}}\right)^2.
$$
At $f=3\,\mathrm{MHz}$, the real turning height is $100\,\mathrm{km}$ but the virtual height is $130\,\mathrm{km}$, corresponding to $\tau_g\simeq0.867\,\mathrm{ms}$ with $c\simeq3.00\times10^8\,\mathrm{m\,s^{-1}}$. 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.
\Image[/ionosphere-dispersive-virtual-height.png]
{title=<Cold-plasma radio dispersion> separates real turning height from <ionospheric virtual height>}
{description=Original linear-density-ramp model: the refractive index falls to zero at the 3 MHz turning point, while the group refractive index rises. The right panel shows the real turning height and the larger delay-derived virtual height as frequency changes. The 100 km and 130 km values are illustrative model outputs.}
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