Supplemantory materials for Application of the generalized force approach

1. Figure S1 shows the plasma frequency profile of the two-layer analytical model of the ionosphere, see Eq. (9) of the main text.2. Figure S2 illustrates the results of the low ray search using the direct optimization method where the initial arrangement of the vertices is chosen to be in the vicinity of the high ray of the F2 layer. Two-layer ionospheric model is defined by Eq. (9) of the main text. Operating frequency is 12 MHz. Circles indicate the positions of the vertices included in the calculation. The initial path is shown with empty circles and dashed line. The initial arrangement of the vertices along the previously found high ray of the F2 layer is perturbed by small, random displacements. The perturbations are not visible in the figure. The final, converged path representing the low ray reflected from the F2 layer is shown with blue filled circles and blue solid line. Intermediate path configuration where the minimal eigenvalue of the Hessian changes the sign is shown with grey filled circles and grey dashed line. Blue dashed lines indicate position of the E layer and the F2 layer.3. Figure S3 shows the ray configuration for the unperturbed ionosphere defined by Eq. (9) of the main text and operating frequency of 10 MHz.4. Movie movie.mov demonstrates the visualization of the point-to-point ray tracing using the generalized force approach. The disturbed two-layer ionospheric model is defined by Eq. (16) of the main text. The operation frequency is 10 MHz. The following information is presented in the upper panel of the frame at each iteration (from left to right): ray number and its type, number of movable vertices (P), value of the minimal eigenvalue of the Hessian (lambda_min), optical path length (S), maximal force on the movable vertices (F_max).