Radio Echoes from the Ionosphere

Why does a radio echo suggest a higher layer than the height where the wave actually turns? A pulse slows as it enters the plasma. Converting its round-trip delay into a height with the vacuum speed of light therefore places the apparent reflection above the actual turning point.

Send a radio pulse into an electron layer and follow its return or escape. Sweep the frequency, record the echo delays and compare the virtual height with the true turning height.

Model assumptions

This is an ideal cold, unmagnetized, collisionless plasma with one flat parabolic density layer and fixed ions. Paths and narrow-band group delays follow geometrical optics. Density is illustrative. Earth curvature, polarization splitting, absorption, antenna response, irregularities, multiple layers and hops, finite pulse bandwidth and full-wave cutoff scattering are omitted. The data are simulated forward predictions, with no measured amplitudes or density-profile inversion. They cannot forecast a real radio link.

Current ray prediction

Ground landing

343\,\mathrm{km}

True turning height

193\,\mathrm{km}

Echo travel time

2\,\mathrm{ms}

Vertical cutoff

8.98\,\mathrm{MHz}

Vertical sounding · simulated records

45 vertical test pulses; the current oblique ray is a separate prediction.

Virtual height from vertical echoes

Select a recorded point

Layer profile & echo timing

Prescribed density profile · ideal reference

The same 45 records · echo delay

Current ray · virtual height or slant range

299\,\mathrm{km}

Model, scale & sources

This is a forward simulation of one flat, horizontally uniform parabolic electron layer. The cold plasma has no magnetic field or collisions. Earth curvature, absorption, receiver noise, amplitudes and finite-bandwidth wave effects are omitted. It is not a reconstruction of a measured ionosphere.

All spatial axes use the same kilometre scale. Antennas, ray widths and gold pulse envelopes are enlarged; blue glow encodes density, not emitted light. Pulse positions follow model group travel time, shown slowly. The ground receiver is a predicted landing point.

The loaded 45-point run is an ideal simulated vertical acquisition with its own stored parameters. Steering or changing frequency preserves those records. Changing the layer clears them; a new sweep replaces the previous run. Current oblique delay and landing range come from the separate live ray solution.

Half the vertical echo delay converted at vacuum light speed gives virtual height. The corresponding oblique quantity is a virtual slant range, not a turning height. At the exact critical limit the ideal delay diverges; the visible path only approaches that layer.

h'=\frac{c\tau}{2}\quad(\theta_0=0),\qquad n^2(z_t)=\sin^2\theta_0

NOAA NCEI · Vertical Incidence Soundings
Read the institutional introduction on frequency sweeps, virtual height and critical frequency. No archived ionogram supplies this simulation’s records.
R. Fitzpatrick · Unmagnetized plasma waves
Read equations 9.27–9.35 on dispersion and group speed; the parabolic-layer solution is derived for this stated profile.
UMass Lowell · Digisonde Portable Sounder
Read the introduction distinguishing transmitter, receiving array and recorded signal. The enlarged crossed-dipole station is illustrative; polarization, coding and array processing are excluded.

One-sentence intuition: The group delay measures a virtual height; the turning point belongs to the density profile and the selected frequency.

Core mathematical model

The plasma changes the wave speed

\omega^2=c^2|\mathbf{k}|^2+\frac{N_e e^2}{\epsilon_0m_e},\qquad v_g=cn<c

This cold unmagnetized plasma law gives the group speed of a narrow-band pulse. The phase speed is different.

Horizontal symmetry fixes the ray

p=n\sin\theta=\sin\theta_0,\qquad n^2(z_t)=p^2

Angles are measured from vertical. Horizontal wave number stays fixed; the vertical component reaches zero at the turn.

Delay gives a virtual height

h'=\frac{c\tau}{2},\qquad \tau=\frac{2}{c}\int_0^{z_t}\frac{dz}{\sqrt{n^2(z)-p^2}}

The height definition applies to vertical soundings. For oblique paths half the light-speed delay is a virtual slant range.

An oblique wave can return at a higher frequency

f_m=\frac{1}{2\pi}\sqrt{\frac{N_m e^2}{\epsilon_0m_e}},\qquad f_{\mathrm{threshold}}=\frac{f_m}{\cos\theta_0}

This secant-law threshold assumes a flat, single layer. Its equality is an infinite-delay limit in the ideal parabolic ray model.

Common difficulties

Virtual height and a mirror

Typical misconceptionThe ionogram shows where a physical reflecting surface sits.

Better mental modelThe electron density changes continuously. A pulse bends and slows along its path; its travel time alone does not give the true turning height.

A cutoff and a working radio link

Typical misconceptionA returning path guarantees a usable signal at its landing point.

Better mental modelThis model calculates paths and delays. Absorption, antenna patterns, noise and a real receiver determine signal strength and practical availability.

Run the experiment

01
Start with a vertical echo
Select “Local echo” and press “Send one pulse”. Open “Layer profile & echo timing” to compare true turning height with virtual height.
What to observe: The pulse returns to the same sounder. The virtual height is larger because propagation slows inside the layer.
02
Record an ionogram
Press “New sweep”. Open “Electron layer”, increase peak density and repeat. Export “Raw CSV” to inspect each simulated delay.
What to observe: The return limit moves upward with the square root of peak density. The trace bends to long delays near the limit.
03
Change only the direction
Compare “Escape upward” with “Bend back”. Both use 9 MHz; only the launch direction changes.
What to observe: The vertical ray penetrates; the oblique ray turns and reaches a distant ground point. Its half-delay quantity is a virtual slant range.