A method to test HF ray tracing algorithm in the ionosphere by means of the virtual time delay

It is well known that a 3D ray tracing algorithm furnishes the ray’s coordinates, the three components of the wave vector and the calculated group time delay of the wave along the path. The latter quantity can be compared with the measured group time delay to check the performance of the algorithm. Simulating a perfect reflector at an altitude equal to the virtual height of reflection, the virtual time delay is assumed as a real time delay. For a monotonic electronic density profile we find a very small relative difference between the calculated and the virtual delay for both analytic and numerical 3D electronic density models.

💡 Research Summary

The paper presents a practical method for validating three‑dimensional (3‑D) high‑frequency (HF) ray‑tracing algorithms used to model ionospheric propagation. Ray‑tracing codes typically output the ray’s spatial coordinates, the three components of the wave‑vector, and the calculated group‑delay (t_calc) along the ray path. In operational settings the calculated delay can be compared with a measured group‑delay (t_real) to assess the fidelity of both the ray‑tracing algorithm and the underlying ionospheric model. However, direct measurements of t_real are rare because they require specialized sounding experiments.

To overcome this limitation the authors introduce a “virtual reflector” concept. They place an ideal, perfectly reflecting surface at the virtual reflection height h′_v that would be obtained from ionospheric sounding (or from an analytical model). The ray is then imagined to travel from the transmitter (point A) to the virtual reflector (point C) and back to the ground (point B) along a straight line in vacuum. The travel time of this geometric path, t_virt = |ACB|/c, is taken as a surrogate for the unavailable measured delay. The validity of this substitution rests on two classic ionospheric theorems:

1. Breit‑Tuve theorem – For a flat, horizontally stratified ionosphere the group‑delay of a wave propagating at the group velocity v_g = c/n_g (where n_g is the group refractive index) equals the time a light ray would need to travel the same virtual path at the speed of light c. Hence t_calc = t_virt.

2. Martyn’s theorem – Relates the oblique and vertical virtual heights through the secant law cos φ · f_ob = f_v (φ is the incidence angle). It guarantees that the same virtual height h′_v applies both to vertical and oblique propagation, allowing the same reflector height to be used for any launch angle.

The ray‑tracing implementation used in the study is based on the Hamiltonian formulation of Jones and Stephenson (1974). Six coupled differential equations for the spherical coordinates (r, θ, φ) and the corresponding wave‑vector components (k_r, k_θ, k_φ) are integrated numerically. The Hamiltonian H is defined as

H = ½