## 3 Analysis of Accuracy and Performance
### Analysis of VLF Amplitude and Phase Characteristics
#### 3.2.2 An Analysis of the Amplitude Attenuation, Phase-Distribution, and Polarization  
Characteristics of VLF Electromagnetic Waves in the Horizontal Propagation Direction of the Ionosphere  
This paper provides two calculation examples to calculate the variation of the electric-field intensity with the distance when the horizontal dipole source is placed on the ground  
Figure 6: (**a**) A comparison of the daytime data of the phase between the algorithm proposed in this paper and the traditional algorithm; (**b**) a comparison of the nighttime data of the phase between the algorithm proposed in this paper and the traditional algorithm. and 75 km away from the ground. The following are the resulting graphs of the two simulation examples. Example 1: Horizontal dipole source; the source adopts a time-harmonic field and is placed on the ground. Observation points are set on the ground in sequence, and the horizontal distance from the source is 5-475 km. The curves of \(E_{r}\) and \(E_{\theta}\) varying with the distance obtained by the simulation are shown in Figure 7a. The polarization characteristics under this kind of calculation example are analyzed, and the variation curve of the polarization angle with distance is obtained through simulation, as shown in Figure 7b. Example 2: Horizontal dipole source. The source adopts a time-harmonic field and is placed 75 km away from the ground. Observation points are set in sequence at a height of 75 km from the ground. The horizontal distance from the source is 5-475 km. The curves of and varying with the distance are obtained through simulation, as shown in Figure 8a. The polarization characteristics under this kind of calculation example are analyzed, and the variation curve of the polarization angle with distance is obtained through simulation, as shown in Figure 8b. During the propagation of VLF electromagnetic waves, the variation of ionospheric electron density in the horizontal direction is relatively small, so the phase is also less affected by the ionosphere. The phase will be nonlinearly changed due to the influence of the ionosphere. When the frequency is 20 kHz in the horizontal direction during the propagation process, observation points are set every 5 km within the range of 1-200 km in the horizontal direction at the corresponding heights to record the data of the electromagnetic field with respect to time. Then, the phase at each position is obtained by using the phase-lag method, so that the phase-change curve of the field quantity Er in the horizontal direction can be drawn as shown in Figure 9. In fact, when the distance is not far, the horizontal variation of the ionosphere is relatively small, which means that the phase of electromagnetic waves propagating in the horizontal direction will not be affected. The phase-change characteristics of electromagnetic waves in the horizontal direction can also serve as a reference for comparison with other situations. Figure 7: (**a**) The variation curves of and with distance are obtained through simulation; (**b**) the curve of the polarization angle varying with distance. It can be observed that the phase of the electromagnetic wave propagating in the horizontal direction to 200 km at the ground height remains almost always linearly changing. This indicates that when the parameters of the ionosphere in the model do not change in the horizontal direction, the propagation of the electromagnetic wave in the horizontal direction will not be nonlinearly affected by the change in the ionosphere parameters, so its phase curve remains linearly changing. In fact, when the distance is not far, the change of the ionosphere in the horizontal direction is indeed relatively small. That is to say, the phase of the electromagnetic wave propagating in the horizontal direction will not be affected. The phase-change characteristics of the electromagnetic wave in the horizontal direction can also be used as a reference for comparison with other situations.