Efficient and High-Accuracy Ray Tracing in Discretized Ionospheric Models
High-frequency (HF) ray tracing in complex ionospheric media generally involves a fundamental trade-off between path accuracy and computational efficiency, which directly affects practical applications such as over-the-horizon radar, ionospheric monitoring, and HF skywave communication systems. This paper presents RTM-GD, a ray-tracing framework that combines Hamiltonian ray integration with a continuously differentiable Galerkin--Difference (GD) interpolation strategy. Under discretized ionospheric grid conditions, the electron density is reconstructed as a $C^1$-continuous function within each grid cell, yielding an everywhere differentiable electron-density field for stable numerical integration and improved propagation-path accuracy. Numerical simulations and validations using measured HF oblique sounding data are conducted under diverse conditions, including different ionospheric states, low- and high-elevation angles, multiple operating frequencies, and both ordinary (O) and extraordinary (X) wave modes. Results show that RTM-GD consistently achieves sub-kilometer RMSE in both group-path and ground-distance metrics and sub-0.01-degree azimuth deviation relative to Richardson extrapolation, while reducing computational time by 98\%. Compared with Catmull--Rom interpolation, RTM-GD reduces the RMSEs of ray parameters by approximately one order of magnitude with less than 4\% additional computational cost. Measured-data validation based on ionogram synthesis further shows that the mean relative group-path error remains within 7\%, confirming reliable reproduction of practical HF oblique propagation characteristics. Overall, RTM-GD provides an accurate and computationally efficient framework for HF ray tracing in discretized ionospheric environments.