In the above analysis, we have focused on the statistical characteristics of the separation in JFs between O- and X-wave in the mid-latitude China region. The statistical mean, local time, season, geomagnetic activity, O-wave JF, and group path dependence are analyzed. The presented results are, in general, consistent with the previous theoretical and experimental results (Agy and Davies 1959; Kopka and Moller 1968; Davies 1990; Bennett et al. 1994; Lundborg et al. 1995). However, some new findings in the above statistical investigation are worth further discussions.

### Case of east–west propagation

The statistical investigations among the three paths show that the separation on the east–west path is minimal, but its relative change is much more prominent and more susceptible to local-time and seasonal variability. Figures 4 and 5 demonstrate that the separation on the east–west path regularly changes with local time. In general, the separation gradually increases after sunrise with one maximum near local noon, then slowly decreases before sunset, and grows up until reaching another maximum at midnight or pre-sunrise, and then decreases again with one minimum between 7:00 LT and 8:00 LT. According to the Appleton–Hartree formula with no collisions (Davies 1990), the refractive indices of the O- and X-modes are almost the same when a radio ray just enters the ionosphere. As the ray continues to pass through deep, the difference in the refractive index of the two modes becomes larger, and the propagation paths become more and more different. Therefore, we conjecture that the diurnal variation of the separation is related to the diurnal variation of hmF2 (the height of the peak electron density of F2-layer, see more details in Davies 1990). Then, we analyze the seasonal variation of hmF2 observed in Xi’an (one end of Link 1), as shown in Fig. 10. Comparing Fig. 10 with the top graph of Fig. 5, it can be found that the diurnal variation trend of the separation \({\mathrm{f}}_{\mathrm{X}}-{\mathrm{f}}_{\mathrm{O}}\) is highly similar to that of hmF2, especially in spring, autumn and winter. At night, when hmF2 goes higher, the separation becomes larger. After sunrise, when hmF2 decreases to a minimum, the separation is also minimized. At noon, when hmF2 rises to a maximum, the separation increases to a maximum too. This means that the separation varies quasi-linearly with hmF2. However, the linear relationship between separation and hmF2 in summer is no longer satisfied, in which case the hmF2 during daytime is comparable with that at night, while the separation at daytime is significantly larger than the nighttime separation.

The typical daytime oblique ionograms in spring and summer are reviewed to seek for possible explanations. Figure 11 presents a typical set of oblique-incidence F2-layer traces at 12:50 LT. The F2-layer trace circled by a red ellipse was taken on August 7, 2010 and the trace circled by a white ellipse was taken on March 7, 2010. Comparing these two sets of F2-layer traces, we find that the group path in summer is about 110 km higher than that in spring. It can be inferred that the difference between the equivalent reflection height of the F2-layer in spring and summer is about 100 km, which is evidently larger than the difference between spring’s hmF2 and summer’s hmF2. Further analysis of the summer oblique ionograms shows that the F2-layer trace uplifts mainly occur when a completely developed F1-layer or strong Es-layer exists. One possible interpretation is that the existence of the completely developed F1-layer or strong Es-layer increases the transmission length of F2 mode which further increases the difference in ray-path between O- and X-wave, thereby the frequency difference between O- and X-wave increases. In order to verify this idea, a 3D magnetoionic Hamiltonian ray-tracing toolbox PHaRLAP is adopted to simulate the changes in ray path when the F1-layer exists or not, which uses Jones and Stephenson (1975) 3D ray-tracing code for ray path calculation. The ionospheric model used in this verification is ‘IRI-2016’ with the ‘F1 model’ being set to be ‘Scotto-1997 no L’ or ‘None’ to simulate the condition with F1-layer or without F1-layer. Figure 12 presents the synthesized oblique ionograms with and without the F1-layer. It can be seen that when there is a F1-layer, the trace of the F2-layer is lifted by about 30 km, the JF of F2-layer O-wave is reduced by 0.15 MHz, and the separation gets larger by 0.025 MHz compared with the condition without F1-layer. Then, we set the critical frequency of the F1-layer to increase by 1.5 MHz, run the ray-tracing program, and find that the separation \({\mathrm{f}}_{\mathrm{X}}-{\mathrm{f}}_{\mathrm{O}}\) becomes larger by 0.05 MHz, which is close to the average value (0.062 MHz) of the difference in the separation between summer and other seasons during daytime. In summary, we believe that the separation is mainly affected by hmF2 above the reflection point, and the separation increases when there is an F1-layer or strong Es-layer.

Another noteworthy phenomenon is that the separation on east–west path is slightly weakened under active geomagnetic conditions compared to quiet conditions. Based on Davis’s approximation formula for the cases of magnetic east–west propagation, it is known that the separation is proportional to the square of \({\mathrm{f}}_{\mathrm{H}}\), where \({\mathrm{f}}_{\mathrm{H}}=\frac{\mathrm{eF}}{2\mathrm{\pi m}}\) (F is the total intensity of the Earth’s geomagnetic field; e and m are the charge and mass of electron, respectively). In general, F at any point on the Earth’s surface is more than 30,000 nT (Davies 1990). During the main phase of the magnetic storm, F may drop by 40–500 nT (Clilverd et al. 1998). It may cause a 0.3–3% drop in the separation, which is consistent with the statistical one (about 0.01 MHz). However, we must keep in mind that the reduction of the separation during geomagnetic disturbance is lower than the frequency resolution. So we argue that the separation is less sensitive to magnetic disturbances.

### Case of quasi north–south propagation

Our statistics confirms that the separation on the quasi north–south propagation path is the most significant among the three links, which agrees with previous studies. This means that the magneto-ionic splitting on north–south path is more prominent than that on east–west path. Furthermore, the separation on north–south path does not vary significantly with the local time and season, and also has a weak correlation with O-wave JF and group path. In addition, the comparison between Figs. 5 and 10 shows that the diurnal variation trend of the separation on Link 3 is obviously different from that of hmF2. In the book "Ionospheric radio", Davies (1990) has pointed out that the angle between the wave normal and the magnetic field is small in the case of north–south or south–north propagation, so the propagation is quasi-longitudinal, and the separation in JFs between O- and X-wave is approximately expressed as \({f}_{X}-{f}_{O}\approx {f}_{H}cosI\). It shows that the separation is mainly determined by the gyrofrequency \({f}_{H}\) and the magnetic dip \(I\), and is hardly affected by the ionospheric activity. This may be the reason why the separation on Link 3 has a weak correlation with local time, season, O-wave JF and group, and hmF2. According to the analysis in the previous section, the change of the total intensity F is very small compared to the background field, which cannot cause significant changes in \({\mathrm{f}}_{\mathrm{H}}\) and \(\mathrm{I}\). So there is no obvious abnormality in the mean value of the separations during the period of geomagnetic disturbance.

### Simulation of the dependence on solar activity variations

The aforementioned analysis shows that the separation to east–west path is susceptible on ionospheric variability, and it cannot be interpreted well by Davies’s relationship. Davies (1990) pointed out that the left-handed O-wave and the right-handed X-wave introduced by magneto-ionic splitting traveled different paths in the ionosphere and it was not possible to integrate analytically to obtain the precise ray path, but it was practical to do so by means of point-to-point ray tracing. Therefore, we make use of 3D ray tracing to analyze the dependence of the separation on solar activity.

The diurnal variations of the separation are conducted on Link 1 covering three levels of solar activity (R = 10, 70, and 120, respectively, represent the low, medium, and high solar activity) and four months (January, April, July, and October, characterize four different seasons), as shown in Fig. 13.

It is observed that the double-peak variation of the separations with one maximum in local noon and another maximum in local midnight or pre-sunrise is favored for all solar activities in different seasons. In January and October, the separation at night is more prominent than that during the daytime, and the maximum at night occurs at low solar activity while the largest value during daytime is found at the high solar activity. In April, the separation during the low solar activity phase is higher at night and slightly lower in the daytime which is consistent with the observations over Link 1, while the separations during medium and high solar activities are larger in the daytime than that at night. In July, the separation during the day is more obvious when compared with that at night for all solar activities. According to IRI, the F1-layer critical frequency at the midpoint of link 1 is calculated, and the result shows the occurrence of the F1-layer is higher than that in January, April, and October. Combining the previous simulation, we infer that the increase in the occurrence of F1-layer in IRI model is the main factor responsible for the obviousness of the separation during the day in July.

### Simulation of the dependence on length and direction of propagation

In order to investigate sensitivity to the length and direction of propagation, calculations were performed for different path configurations with a central receiver at Xi’an. The results obtained by the 3D ray-tracing method discussed in this paper are shown in Fig. 14. The figure above shows the variability of the separation as a function of the direction of propagation for two different transmission lengths that are 1000 km and 2000 km, while the figure below shows the variability of the separation against the length of propagation for four different transmission azimuth angles of 45°, 90°, 145°, and 180°. As we can see, the separation varies approximately as the cosine function of the propagation direction with maximums at north–south direction and minimums at east–west direction in mid-latitude China region. Due to the different transmission directions, the separations on radio links with ground range greater than 500 km present obviously distinct patterns. For east–west propagation path, the separation decreases to a minimum near ground range of 2000 km and then increases very slowly with increasing ground range, while for north–south propagation path, it gradually increases with increasing ground range. The relationship of the separation with transmission length is partly inconsistent with the results simulated by Bennett et al. (1994) which show a tendency to decrease below 1000 km and then increase very slowly for a case with azimuth equal to 0 (of north–south propagation). Based on the use of an equivalent operating frequency, Bennett took into account the effect of the Earth’s magnetic field in an analytical ray-tracing program. The use of the equivalent operating frequency led to deviations in the ray path, which caused calculation errors, and resulted in the inconsistency with 3D ray tracing.