Efficient and High-Accuracy Ray Tracing in Discretized Ionospheric Models

High-Sfrequency (HF) ray tracing in the complex ionospheric medium generally faces a fundamental trade-off between path accuracy and computational efficiency. This paper presents a high-fidelity Ray Tracing Method (RTM) synergistically amalgamated with a continuously differentiable Galerkin-Difference (GD) interpolation strategy for three-dimensional electron density reconstruction. The RTM-GD can ensure analytically smooth gradients, and thus significantly enhance gradient continuity, numerical stability, and computational efficiency in Hamiltonian-based ray tracing. To systematically evaluate the applicability and performance of RTM-GD, we propose a four-stage experimental design. First, we conduct a grid-resolution sensitivity experiment to evaluate the convergence behavior and directional consistency of the interpolation method under varying spatial scales. Second, we perform an elevation-angle scanning experiment ranging from 3 to 65 degrees within a mid-latitude ionospheric environment. The results indicate that RTM-GD improves path accuracy by over an order of magnitude compared to Catmull-Rom interpolation, while achieving a 14.6-fold increase in computational efficiency relative to the Richardson extrapolation method. Third, we further conduct simulation experiments for high-elevation F2-mode propagation near the critical incident angle. RTM-GD achieves further error reduction compared to the second scheme, confirming its numerical stability and robustness. Finally, we compare synthetic oblique ionograms generated by RTM-GD with observed HF propagation characteristics. The results demonstrate that the model can effectively capture the typical dual-mode propagation behavior of the F2 layer. In summary, RTM-GD delivers accurate and efficent ray tracing in discretized ionospheric models, meeting the demands of HF propagation applications.