## 1. Introduction

[2] Since the early days of radio wave communications, high frequency (HF) has been extensively used for long-distance radio service, despite the difficulties due to the varying ionospheric structures. HF radio ray tracing in the ionosphere mainly involves refraction and reflection due to ionospheric structure on a large scale compared with the radio wavelength. The tracing results can be efficiently evaluated through the estimated parameters of ground range, reflection height, phase path, and group path. Ray tracing has applications in operation frequency and power management and prediction of HF broadcasting or communication systems as well as in experimental studies on over-the-horizon radar systems and direction finding systems. For these beyond line-of-sight systems their performance depends critically on accurately tracing the ray through a realistic ionospheric model. Some works of ray tracing have combined scattering and the time varying nature of the ionosphere and provided the additional parameters of field strength, absorption, and Doppler shift, but a detailed discussion is beyond the scope of this study.

[3] The theory of HF radio waves in the ionospheric plasma started early by different groups working on full wave theory [*Försterling*, 1942; *Rydbeck*, 1944; *Budden*, 1952] and geometrical optics [*Poeverlein*, 1948, 1949; *Hines*, 1951; *Haselgrove*, 1955]. Since then, two main types of ray tracing in the ionosphere emerged: numerical and analytic. *Bennett et al.* [2004] have discussed and reviewed the theoretical basis and a number of applications and techniques of ray tracing. Over all, the most widely distributed numerical ray-tracing program is probably that developed by Jones [*Jones*, 1966]. The Jones program provides for a number of models of ionospheric *N*_{e} distribution and the Earth's magnetic field. *Reilly* [1991] also developed a three-dimensional numerical ray-tracing program which is especially applied to point-to-point propagation and homing-in calculation using approximate partial derivatives of latitude and longitude with respect to launch angle. There are other numerical ray-tracing programs that have been adopted by different groups. However, they are not necessarily freely available. The analytic tracing mostly applies to spherically symmetrical ionospheres where the vertical structures are reconstructed by a choice of suitable functions. Thus the ray-tracing parameters can be efficiently and analytically calculated, and the analytic tracing methods, more properly, are closed-form methods. For a typically analytic method, *Norman and Cannon* [1997] introduced a segmented method for analytic ray tracing (SMART) on the fully analytic ionospheric model (FAIM) [*Anderson et al.*, 1989] and with the Earth's magnetic field. SMART is able to accurately trace the ray by segmenting the ionosphere horizontally as well as vertically and, hence, including horizontal *N*_{e} gradient effects are included. Analytic ray-tracing techniques are much faster than numerical approaches, but they are more restricted in the ionospheric models to which they can be applied.

[4] Here we have attempted to present another numerical and stepped ray-tracing method on a phenomenological ionospheric *N*_{e} model constructed from the FS3/COSMIC data. The primary propose of this construction is to provide subsidiary ray-tracing results through a simple and easily accessible ionospheric *N*_{e} model for applications in communication and wave propagation. The phenomenological ionospheric model provides temporal and synoptic variations in three-dimensional (latitude, longitude, and altitude) *N*_{e} and maintains the continuity in the first and second *N*_{e} derivatives for providing reliable radio propagation predictions and electrostatic field determinations. The concentrations of *N*_{e} at certain altitudes have characteristic maximum peaks, forming the ionospheric F2, F1, E, and D layers. Herein we report the results of ray-tracing parameters of ground range, reflection height, phase path, and group path at conditions with or without an Earth-centered magnetic dipole and horizontal *N*_{e} gradients.