# Estimating Electron Number Density via WSPR

WSPR is used by radio amateurs and radio scientists to measure radio propagation from VLF to VHF. On the order of one million WSPR spots occur daily worldwide. Most of these measurements are at HF (2-30 MHz) frequencies. Here is a discussion of one possible science application of WSPR in an NVIS configuration.

The units used are MKS.

## Background

The fundamental wave modes explored by Tonks and Langmuir [1] in the 1920s are responsible for important behaviors observed in plasmas such as those present in Earth’s ionosphere.
Key inflection points in behavior of externally excited waves traveling through a region of plasma are marked by the electron plasma frequency ω_{pe} or equivalently f_{pe}.
Where f ≪ f_{pe}, waves will not pass through the plasma–they will be reflected.
Where f ≫ f_{pe}, waves will pass through the plasma, with a detectable phase shift.
This phase shift is exploited to compute total electron content (TEC) via GPS receivers in the 1-1.5 GHz range (and beacons at other frequencies), a key quantity used in tomography of ionospheric number density reconstruction.

The electron plasma frequency is:

ω_{pe} = (N_{e} e^{2} / (m_{e} ε_{0}))^{1/2} [rad/s]

f_{pe} = ω_{pe}/(2π) = 1/(2π) (N_{e} e^{2} / (m_{e} ε_{0}))^{1/2} [Hz]

If we consider measurements from networks of radio amateurs using programs such as WSPR, which records SNR vs. time for numerous frequencies and disparate stations with a two-minute cadence, we may generate dynamic maps of electron density at the midpoint of each single-hop path. For this study, we initially focus on NVIS stations to keep within a single ionospheric hop. For world-wide communications, 10-20 or so hops may be involved, and in general the ionosphere and lithosphere will be heterogeneous, increasing the difficulty of the estimation problem.

## Estimation of electron number density in ionosphere F2 layer

An oft-cited approximation for the maximum number density in the F2 layer (NmF2) is given as:

f_{c} = 9 * (N_{max})^{1/2} [Hz]

or

N_{max} = f_{c}^{2} / 81 [m^{-3}]

This is an approximation for:

N_{e} = f_{pe}^{2} * (4π^{2} m_{e} ε_{0}) / e^{2}

assuming you know the critical frequency from a frequency swept (chirp) measurement from a vertical ionosonde.

For a locally stratified and homogeneous ionosphere near the midpoint of an NVIS path, we might estimate electron density N_{e} at an NVIS path midpoint, with assumptions on the height of the refracting layer.
Oblique incidence chirp ionospheric sounders have been known since 1964 and vertical incidence chirp ionospheric sounder since 1971 to be a highly power-efficient means of ionospheric characterization [2].
In the 1996, the benefits of adding time synchronization to remotely located oblique ionosondes was realized, giving absolute time delay, allowing virtual height to be estimated more accurately than with relative-only timing available from unsynchronized chirp transmitter and obliquely-located receivers.
Chirp sounders in modern times have seen further benefit from addition of broadband phase modulation to the stepped-frequency chirp.

## TEC and f_{0}F_{2} related statistically

TEC and f_{0}F_{2} have been related in a coarse sense as confirmed via observation [3] to be during nighttime hours:

TEC = 1.24 x 10^{-6} τ (f_{0}F_{2})^{2}

where τ is slab thickness in meters, taken to be 230 x 10^{3} in [3].

## References

[1] Tonks, L. and Langmuir, I. (1929). Oscillations in ionized gases. DOI: 10.1103/PhysRev.33.195. Physical Review 33:195-210.

[2] Barry, G. (1971). A Low-Power Vertical-Incidence Ionosonde. DOI: 10.1109/TGE.1971.271471. IEEE Transactions on Geoscience Electronics 9(2):86-89.

[3] Spalla P. and Ciraolo, L. (1994). TEC and f_{0}F_{2} comparison. Annali Di Geofisica, 37(5):929-938.