Phase Variation Model of VLF Timing Signal Based on Waveguide Mode Theory

Abstract

In integrated PNT systems, due to defects in satellite signals and long-wave signals, VLF signals can be an essential supplement. However, there is currently a lack of VLF timing systems in the world, and it is impossible to evaluate the impact of the propagation delay of these signals. Based on the theory of very-low-frequency propagation, this paper determines the waveguide mode propagation at ultra-long distances as the main research direction, establishes a signal phase change model, gives a theoretical formula for the phase velocity of VLF signals, and analyzes the main factors affecting the phase velocity of VLF signal propagation. Finally, combined with historical observation data, the phase change is predicted, compared, and analyzed. The results show that the theoretical calculation is consistent with the measured data. The average error of the delay prediction is 0.015 microseconds per 100 km, and the maximum error of the delay prediction is 0.152 microseconds per 100 km.

1. Introduction

PNT stands for positioning, navigation, and timing. People usually obtain location and time information through radio navigation. With the development of human civilization, human society’s demand for location and time information is also growing, and PNT systems are becoming much more important. Among them, the timing system is mainly based on satellites (GPS) and ground-based data (long and short waves). However, satellite navigation has a large coverage area and high accuracy, but the signal is fragile and easily interfered with, and it cannot enter water or the ground; short-wave timing has a long range and simple reception, but it has low accuracy and poor continuity, and it cannot enter water or the ground; long-wave timing has a stable phase and strong anti-interference ability, but the signal coverage range and depth of entry into water and the ground are limited. The propagation of long waves is mainly based on ground waves. The propagation path of ground waves is fixed, the random changes in the path are small, and the propagation delay estimation is easy. Therefore, it is easier to correct the propagation delay, and the timing accuracy is high, which can reach the microsecond level. The propagation of short waves is mainly based on sky waves. During the propagation process, the signal needs to pass through the troposphere and be refracted and reflected in the ionosphere. Compared with the sky waves of long-wave signals, the electron concentration required for the reflection of short-wave signals is higher, so the reflection height is higher. At the same time, electromagnetic waves propagate in the ionosphere, and the lower the frequency, the greater the attenuation. This makes the propagation distance of short-wave sky waves much greater than that of long-wave signals, but due to the complex and changeable ionospheric environment, the timing accuracy of short-wave signals is lower than that of long-wave ground waves, and it can reach the millisecond level.

Satellite-based radio navigation can achieve full coverage of the global surface and near-Earth space, with high positioning and timing accuracy, but the signal is very weak and is particularly susceptible to external interference; land-based radio navigation has strong signals, high reliability, and is not easily interfered with. Land-based radio navigation can be used as an important backup for satellite-based radio navigation systems. However, existing timing methods still have some defects, such as an inability to achieve ultra-long-distance timing, and due to the propagation characteristics of high-frequency radio signals, underwater radio timing cannot be performed. The frequency band of very-low-frequency radio signals is 3–30 kHz, corresponding to a wavelength of 100–10 km. The path attenuation of very-low-frequency waves is very low, at 2–3 dB per 1000 km [1], with almost no “attenuation”. Their transmission is very stable and reliable and can be used for long-distance communications; very-low-frequency waves can penetrate seawater to a depth of at least 10–40 m (30–130 feet) [2], so they can be used for underwater communications. Because of these characteristics of very-low-frequency signals, they can serve as an important supplement to existing ground-based timing systems.

Except for the five Alpha stations in Russia that use continuous-wave (CW) systems, the signals of other VLF stations are modulated in the MSK mode. Because of its good penetration of obstacles such as rocks and buildings and its good performance in long-distance transmission [3,4,5], it has a wide range of applications in military and detection functions. VLF grid station signals have a stable working mode. By studying their propagation path and the changing characteristics of the received signal, they can be used to infer relevant changes in the Earth’s ionosphere. With station signals and astronomical signals as the main observation objects, VLF receivers are widely used in solar activity monitoring, geospatial ground-based remote sensing, global marine communications, navigation systems, etc. VLF timing signals are generated by high-stability and high-precision time and frequency signals with good signal characteristics. They can not only provide timing services but also support research work such as geospatial environmental monitoring, solar activity monitoring, and communications [6,7,8,9,10], which can promote improvements in original innovation capabilities in related basic science and technology fields.

Currently, there are few navigation and timing systems that have a VLF. The United States previously built the Omega navigation system worldwide, but it was stopped for various reasons [11]. Only some stations still conduct low-frequency-related experiments. One of the research goals of the latest STOIC project in the United States is to use very-low-frequency signals to realize new ultra-long-range radio navigation and timing. The project has completed a navigation test, and the accuracy can reach 40 m after correction using the delay model. Russia previously had the only system in operation, the Alpha navigation system, which is now closed due to specific reasons.

The study of VLF radio wave propagation practice and theory began in the previous century. Although people have a good understanding of the propagation of this frequency range, based on our current knowledge, it is not possible to accurately predict the field, especially on complex paths.

Initially, several conceptual models explained the propagation of VLF radio waves. The three most popular propagation modes were: (1) ground wave; (2) sky wave, or ray theory; and (3) waveguide mode, or the theory of oscillation modes in a waveguide [12,13,14,15,16,17].

Ground waves are the sum of all field terms that appear on the Earth’s surface without an ionosphere. As early as 1907, Zennick theoretically discussed the problem of radio wave propagation on the Earth’s surface. Then, Sommerfeld and others made further analysis. Between 1936 and 1941 [18,19], Norton derived a calculation formula for field components for mathematical calculations. In 1957, Wait summarized the meaningful discussions and historical background of various aspects of ground waves by predecessors [20]. In 1962, Watt, Maxwell, and Mathews discussed the different electrical properties encountered on the Earth’s crust and gave typical values of the Earth’s conductivity under various conditions [21]. In the same year, Wait proposed the surface impedance concept, and Campbell and Norton also conducted related in-depth research later. After solving the research on ground wave propagation on uniform paths, Wait used analytical approximation methods in 1957 to study the problem of ground wave propagation on mixed paths proposed by Millington in 1949 [22], and the sea–land effect began to be discovered and explored.

Sky waves are usually combined with ground waves to calculate the total field at 2000–3000 km. Hollingworth studied the combined effect of VLF sky waves in 1926 [23]. Bremmer calculated the vertical electric field of sky waves in 1949. Norton used the geometric optics approximation method in 1959 to further correct sky waves. Norton studied the divergence and convergence effects of reflected waves of the same source in different directions due to the influence of the Earth’s curvature in 1959 [24]. Wait used the diffraction correction method in 1961 to solve the problem of the focus of the ray angle tangent to the Earth [25]. Johler proposed the theory of multiple wave jumps, but found it difficult to obtain the surface reflection coefficient of the specific reflection point. Wait and Walters proposed in 1964 that the equivalent reflection coefficient of the VLF band is significantly affected by the change in the conductivity height gradient. Its mutation at an altitude of 60–90 km causes the transition from the ion-free zone near the ground to the ion zone. The reflection of VLF sky waves mainly occurs in this altitude range. However, even after many experiments, the exact value of the reflection coefficient of the ionosphere could not be obtained through theoretical analysis and actual experiments. Later, Wait et al. conducted many theoretical and experimental studies and found that the total electric field is equivalent to the ground wave field within 400 km, the influence of the first-hop sky wave gradually becomes significant at 400–800 km, and the influence of the second-hop sky wave needs to be considered at 800–1500 km. The phenomenon farther away cannot be explained by the theory of ground-to-ground waves [26].

The waveguide mode propagation theory is a more reasonable explanation. The signal strength of VLF radio waves at ultra-long distances is much higher than that of the wave jump theory; that is, it is significantly lower than the signal attenuation rate of the wave strip theory. Its propagation mechanism is similar to the waveguide propagation of microwaves in metals; that is, it is conducted in the waveguide formed between the Earth and the ionosphere. Some scholars have developed mathematical expressions to describe and explain the VLF propagation field in the early days. Among them, Austin derived a formula consistent with the experimental results at a limited distance based on semi-experiments in 1911. Later, he made some corrections in 1926. Watson derived an expression for the electric field of the early propagation mode between two concentric spherical walls with preferential conductivity in 1919. Pierce derived an expression of great value for the non-attractive field based on theory and semi-experiments in 1952, making the Austin formula more consistent with experiments.

This paper aims to study the propagation characteristics of VLF(VLF) radio waves, building on previous research. It will explore the fundamental propagation mechanism of VLFVLF radio wave waveguide modes and establish a propagation model for these waves. The study will identify and analyze key factors that influence the phase change in VLFVLF radio wave propagation, examining these influences through mathematical relationships. Finally, the findings will be validated by combining them with historical measured data.

2. Materials and Methods

2.1. Subsection

The wavelength of VLF signals ranges from 10 to 100 km, while the height of the ionosphere is about 60 to 90 km above the Earth’s surface. When the height of the ionosphere is about 4 to 5 times the wavelength, its influence on the propagation of electromagnetic waves cannot be ignored. Unlike low-frequency signals, the frequency of VLF is lower, meaning their wavelength is longer. The wavelength of VLF signals in the lower frequency band even exceeds the effective reflection height of the ionosphere, which means that at long distances, the transmission method of VLF signals is different from the sky wave hopping method of low-frequency signals, but a method more like the waveguide propagation of microwaves in metals. There are two main ways for VLF signals to propagate: the ground wave transmission area within 400 km, the ground wave composite transmission area between 400 and 1500 km, and the waveguide mode propagation area beyond 1500 km. However, due to multimode interference, within 3000 km, there are multiple orders of modes simultaneously, and the signals interfere with each other greatly. At the same time, the attenuation rate of the signal at the same distance is positively correlated with its order. About 3000 km away, the multimode interference effect caused by the high-order mode signal has decayed to a negligible level. This area is called the single-mode transmission area. The prediction and correction of the propagation delay of VLF signals in the single-mode transmission area is the main research content of this paper.

There are two basic boundary conditions for waveguide propagation mode: (1) half the wavelength needs to be smaller than the inner diameter of the waveguide; and (2) the signal frequency needs to be greater than the speed of light/twice the waveguide width. The VLF propagation phase velocity is the parameter that directly affects the phase propagation delay, so it is necessary to study the numerical relationship between the phase velocity and other influencing factors such as frequency, ionospheric reflection characteristics, and ground reflection characteristics. When electromagnetic energy is guided between two reflecting boundaries, two important factors affect the electric and magnetic fields: (1) the shape and impedance characteristics of the boundary and (2) the modulus.

The boundary shape can choose a plane propagation model or a spherical propagation model according to the distance, and there are three main propagation modes in waveguide mode propagation, namely transverse electromagnetic mode (TEM), transverse electric mode (TE), and transverse magnetic mode (TM). Since TEM cannot exist in a rectangular or completely closed cylindrical waveguide, and the TE has no electric field in the propagation direction, the TM is the main part of the VLF waveguide propagation field. For propagation in the waveguide, after the energy enters the waveguide, it can be alternately reflected at the upper and lower boundaries. During resonance, the vertical section’s total phase change at any waveguide position is the same. The general form of mode resonance is that the phase shift after each boundary is reflected (reflected twice) is an integer multiple (n times) of 2π, where n is an integer, the mode number. For the planar waveguide model, as shown in Figure 1 below:

The downgoing wave and the upgoing wave can be regarded as rays propagating in free space. After being reflected by the two boundaries, the phase difference between before phase_1 and before phase_2 equals an integer multiple of 2π, which satisfies the resonance condition. It is derived that,

$$\mathrm{s}\mathrm{i}\mathrm{n} {\psi }_n=n\lambda /2h$$

(1)

In the above formula, n is the order of the mode, and h is the height between the ground and the ionosphere. This is the ideal case where the phase angle between the ground and the ionosphere reflection coefficient is 0. When considering the actual ionosphere, when the VLF is at a small incident angle, the phase of the ionization reflection coefficient is approximately equal to −π. At this time,

$$\mathrm{s}\mathrm{i}\mathrm{n} \psi \approx (n-{\displaystyle \frac{1}{2}})\lambda /2h$$

(2)

For example, at night, under first-order mode conditions (n = 1), f = 15 kHz, $\lambda =20 \mathrm{k}\mathrm{m}, h=90 \mathrm{k}\mathrm{m}$, $\mathrm{s}\mathrm{i}\mathrm{n} \psi \approx ({\displaystyle \frac{1}{2}})(20)/180=0.056$, so $\psi$ $\approx$ 3.2°. Phase velocity $v_p=f{\lambda }_g$, where ${\lambda }_g$ is the wavelength in the waveguide and ${\lambda }_o$ is the wavelength in free propagation.

$${\lambda }_g={\lambda }_o/\mathrm{c}\mathrm{o}\mathrm{s} \psi$$

(3)

Under the premise that $v_o$ is the speed of light, the phase velocity in the waveguide can be deduced as

$$v_p=v_o/\mathrm{c}\mathrm{o}\mathrm{s} \psi ={\displaystyle \frac{v_o}{{(1-{\mathrm{s}\mathrm{i}\mathrm{n}}^2 \psi )}^{\frac{1}{2}}}}\approx v_o(1+({\mathrm{s}\mathrm{i}\mathrm{n}}^2 \psi )/2]$$

(4)

The phase propagation speed of VLF in a waveguide is expressed as “phase velocity”. However, the phase velocity is not the speed at which information is transmitted in a waveguide. In a waveguide, the speed at which information is transmitted is called “group velocity” $v_g$:

$$v_g=v_o\mathrm{c}\mathrm{o}\mathrm{s} \psi =v_o{(1-{\mathrm{s}\mathrm{i}\mathrm{n}}^2 \psi )}^{{\displaystyle \frac{1}{2}}}\approx {\nu }_o[1-({\mathrm{s}\mathrm{i}\mathrm{n}}^2 \psi )/2]$$

(5)

Since the VLF timing requirement is mainly focused on the signal phase, it is sufficient to focus on the phase velocity. At the same time, the attenuation rate during signal propagation must also be paid attention to. The main attenuation in VLF waveguide mode propagation is divided into ground and ionospheric attenuation. Ground attenuation is mainly caused by limited ground conductivity. As shown in the figure above, the total vertical electric field equals the sum of $\mathrm{c}\mathrm{o}\mathrm{s} \psi$ and the plane wave electric field. Therefore, the wave impedance can be obtained:

$$E_z/H_x=\eta ={\eta }_o\mathrm{c}\mathrm{o}\mathrm{s} \psi ={\eta }_o{(1-{(f_c/f)}^2)}^{{\displaystyle \frac{1}{2}}}$$

(6)

where $\eta$ is the equivalent intrinsic impedance in the waveguide. The attenuation $\Delta {\alpha }_{{\sigma }_g}$ caused by the ground surface is

$$\Delta {\alpha }_{{\sigma }_g}\approx {\displaystyle \frac{8.686{\epsilon }_o^{\frac{1}{2}}{\pi }^{\frac{1}{2}}{f}^{\frac{1}{2}}}{{\sigma }^{\frac{1}{2}}h{[1-{(f_c/f)}^2]}^{\frac{1}{2}}}}$$

(7)

$$\Delta {\alpha }_{{\sigma }_g}\approx {\displaystyle \frac{4.6\times {10}^{-5}{f}^{\frac{1}{2}}}{{\sigma }^{\frac{1}{2}}h{[1-{(f_c/f)}^2]}^{\frac{1}{2}}}}$$

(8)

The signal attenuation caused by the ionosphere can be referred to as the ionosphere reflection coefficient because it is not always a good conductor. At the same time, attenuation can be obtained from the amplitude part of the mode resonance relationship:

$${\alpha }^{\prime }={\displaystyle \frac{R_g+R_i}{2d_1}}$$

(9)

where $2d_1$ is the effective distance between the refracted lines, and since the effective distance of the waveguide parameters and phase resonance conditions are known, the formula can be written as

$${\alpha }^{\prime }\approx {\displaystyle \frac{(R_g+R_i)\mathrm{s}\mathrm{i}\mathrm{n} \psi }{2h\mathrm{c}\mathrm{o}\mathrm{s} \psi }}$$

(10)

Therefore, the component due to ionization loss is

$${\alpha }_i^{\prime }\approx {\displaystyle \frac{R_i\mathrm{s}\mathrm{i}\mathrm{n} \psi }{2h{[1-{\mathrm{s}\mathrm{i}\mathrm{n}}^2 \psi ]}^{\frac{1}{2}}}}$$

(11)

This formula, combined with the resonance condition of the planar waveguide, can be written as

$${\alpha }_i^{\prime }\approx {\displaystyle \frac{R_i(2\pi n+{\varphi }_g+{\varphi }_i){\nu }_o/4\pi fh}{2^h{(1-{((2\pi n+{\varphi }_g+{\varphi }_i){\nu }_o/4\pi fh)}^2]}^{\frac{1}{2}}}}$$

(12)

At higher frequencies, $\psi \ll 1$, so,

$${\alpha }_i^{\prime }\approx R_i(2\pi n+{\varphi }_g+{\varphi }_i)v_o/8\pi f{h}^2$$

(13)

where ${\varphi }_g$ is the reflection mutation phase caused by ground parameters, and ${\varphi }_i$ is the mutation phase caused by ionospheric parameters. It can be seen that the reduction in the reflection coefficient phase ${\varphi }_i$ in the planar waveguide mode propagation model will reduce the attenuation rate. At the same time, under the premise that other parameters remain unchanged, ${\alpha }_i^{\prime }$ is inversely proportional to f and $h^2$.

As the propagation distance increases, and the effect of the earth’s curvature on VLF waveguide propagation becomes increasingly non-negligible, when the model is converted to a spherical model, there are three major changes compared to the plane model. First, the mode resonance relationship has changed, and the ray angle derived from the geometric relationship of the plane ground does not apply to the mode resonance of the spherical ground. Second, the energy diffusion speed and distance function in the spherical model is no longer linear but reaches a maximum value when it is about a quarter of a circle around the Earth. Third, the uplink and downlink waves associated with the reflected rays are no longer plane waves, and the boundary conditions of the spherical ground and the ionosphere must be considered. The spherical propagation model is shown in Figure 2 below:

As shown in Figure 2, the phase path_1 starts from P1 and reaches P1’ after being reflected from the ground and the ionosphere. The phase path_2 starts from P2 and reaches P2’ after being reflected from the ground and the ionosphere. The resonance condition is satisfied when the phase path_1 is an integer multiple of 2π longer than the phase path_2. That is,

$$\Delta l_g\beta +\Delta l_i\beta -{\varphi }_g-{\varphi }_i=2\pi n$$

(14)

Among them, $\Delta l_g\beta$ is the path difference before ground reflection, $\Delta l_i\beta$ is the path difference after ionospheric reflection, ${\varphi }_g$ is the phase angle of the ground reflection coefficient (phase shift generated by reflection), and ${\varphi }_i$ is the phase angle of the ionospheric reflection coefficient. According to the geometric relationship, we can know that

$$\Delta l_g=\Delta l_i={\displaystyle \frac{h(1-\frac{a}{a+h})}{{[1-\frac{2^a\mathrm{c}\mathrm{o}\mathrm{s} \theta }{a+h}+{(\frac{a}{a+h})}^2]}^{\frac{1}{2}}}}$$

(15)

Moreover, the ionosphere height h is much smaller than the Earth radius $a$, that is $h\ll a$, and the total difference between the two-phase paths can be derived as

$$2\Delta l\approx {\displaystyle \frac{\sqrt{2h(h/a)}}{[1-(1-h/a)\mathrm{c}\mathrm{o}\mathrm{s} \theta -h/a{]}^{\frac{1}{2}}}}\approx {\displaystyle \frac{\sqrt{2^{h(h/a)(1+h/a{)}^{\frac{1}{2}}}}}{(1-\mathrm{c}\mathrm{o}\mathrm{s} \theta {)}^{\frac{1}{2}}}}$$

(16)

Since $h\ll a$, that is, $h/a\ll 1$, we can get

$$\mathrm{c}\mathrm{o}\mathrm{s} \theta =(1-h/a)\{{\mathrm{c}\mathrm{o}\mathrm{s}}^2 \psi +\mathrm{s}\mathrm{i}\mathrm{n} {[(1+h/a{)}^2-{\mathrm{c}\mathrm{o}\mathrm{s}}^2 \psi ]}^{-{\displaystyle \frac{1}{2}}}\}$$

(17)

According to the approximation of $\mathrm{c}\mathrm{o}\mathrm{s} \psi \approx 1-{\psi }^2/2$ and $\mathrm{s}\mathrm{i}\mathrm{n} \psi \approx \psi$, we can conclude that

$$\mathrm{s}\mathrm{i}\mathrm{n} \psi \approx \psi \approx \Delta l/h\{1-(h/a)(h^2/2{\Delta }^{l^2})(1+h/a{)}^2\}(1-h/a)$$

(18)

This formula can also be written as

$$\mathrm{s}\mathrm{i}\mathrm{n} \psi \approx (1-h/a)(2\pi n+{\varphi }_g+{\varphi }_i)v_o/4\pi fh-{\displaystyle \frac{(1+h/a)2\pi f{h}^2}{a{v}_o(2\pi n+{\varphi }_g+{\varphi }_i)}}$$

(19)

When $a\to \infty$, the VLF spherical propagation model can be approximated as a plane propagation model, and the $\psi$ value of the plane propagation model can be obtained. Moreover since ${\varphi }_g$ and ${\varphi }_i$ depend on the incident angle, the cosine of the ionospheric reflection angle can be approximated as

$$\mathrm{c}\mathrm{o}\mathrm{s} \varphi \approx {\displaystyle \frac{ha+d^2/2}{a{[d^2(1+h/a)+h^2]}^{\frac{1}{2}}}}\approx {\displaystyle \frac{ha+d^2/2}{a{(d^2+h^2)}^{\frac{1}{2}}}}$$

(20)

Since $h/a\approx {10}^{-2}$ and $\mathrm{a}\approx 6400 km$, for long distances, the ionospheric reflection angle tends to a relatively constant value.

For example, the ionosphere height is approximately 70 km during the day and about 90 km at night, and the corresponding $\mathrm{c}\mathrm{o}\mathrm{s} \varphi$ is approximately 0.15 and 0.16, respectively. Then, for the VLF electromagnetic wave with a frequency of 15 kHz, in the first-order mode during the day, when the ionosphere gradient is about 2–2.5, it can be concluded that ${\varphi }_i$ is about −2.7 radians, and ${\varphi }_g$ can be taken as −0.85 radians according to experimental data. It can be obtained that

$$\sin \psi \approx 0.1-0.054=0.046\approx 2.6°$$

(21)

According to the VLF spherical propagation model diagram, the wavelength at the intersection of the ray and the height $h^{\prime }$ is ${\lambda }_o/{\lambda }_{h^{\prime }}=\mathrm{c}\mathrm{o}\mathrm{s} (\psi +\theta /2)$, and the phase velocity can be deduced to be

$${\nu }_{h^{\prime }}=v_0^{\prime }/{[1-{\mathrm{s}\mathrm{i}\mathrm{n}}^2(\psi +\theta /2)]}^{{\displaystyle \frac{1}{2}}}$$

(22)

So, the phase velocity on the ground of the VLF spherical propagation model is

$$v_p=v_o(1-h^{\prime }/a){[1-{\mathrm{s}\mathrm{i}\mathrm{n}}^2(\psi +\theta /2)]}^{-{\displaystyle \frac{1}{2}}}$$

(23)

Since $\psi +\theta /2\ll 1$, the phase velocity can be approximated as

$$v_p\approx v_o\{1-h^{\prime }/a+{\displaystyle \frac{{\mathrm{s}\mathrm{i}\mathrm{n}}^2(\psi +\theta /2)}{2}}\}$$

(24)

Since $\psi \ll 1$, we can get:

$$\theta =(\pi /2-\varphi )-\psi$$

(25)

So, the phase velocity is approximately

$${\nu }_p\approx v_o\{1-h^{\prime }/a+{\displaystyle \frac{1}{8}}{[\psi +{({\psi }^2+2h/a)}^{{\displaystyle \frac{1}{2}}}]}^2\}$$

(26)

Assuming $\mathrm{s}\mathrm{i}\mathrm{n} \psi \approx \psi$, we have

$$\psi \approx (1-h/a)(2\pi n+{\varphi }_g+{\varphi }_i){\nu }_o/4\pi fh-{\displaystyle \frac{(1+h/a)2\pi f{h}^2}{a{\nu }_o(2\pi n+{\varphi }_g+{\varphi }_i)}}$$

(27)

Finally, when $h^{\prime }$ is the reference height, the phase velocity can be simplified to

$${\nu }_p\approx {\nu }_0{\{1-h^{\prime }/a+(1-h/a{)}^2[(2\pi n+{\varphi }_g+{\varphi }_i){\nu }_0/4\pi fh]}^2/2\}$$

(28)

Similarly, in the curved surface model, the attenuation rate also needs to be corrected because the effective reflection angle between the ionosphere and the ground changes.

$${\alpha }^{\prime }={\displaystyle \frac{R_g+R_i}{2d_1}}$$

(29)

From the above figure, we can see that $d_1=a\theta$, where $\theta$ is

$$\theta =(\pi /2-\varphi )-\psi \approx {({\psi }^2+2^{h/a})}^{{\displaystyle \frac{1}{2}}}-\psi$$

(30)

In the case of $\psi =x-y$, the above formula can be written as

$$\theta =2y=-{\displaystyle \frac{(1+h/a)4\pi f{h}^2}{a{\nu }_o(2\pi n+{\varphi }_g+{\varphi }_i)}}$$

(31)

The attenuation rate of the spherical ground model under the condition of $0<\psi \ll 1$ is

$${\alpha }^{\prime }\approx {\displaystyle \frac{(R_g+R_i)(2\pi n+{\varphi }_g+{\varphi }_i){\nu }_o}{8\pi (1+h/a)f{h}^2}}$$

(32)

In VLF radio wave propagation at lower frequencies ($f<10 kHz$), since $\psi$ becomes quite large, the attenuation rate formula of the plane ground propagation model can be directly used.

2.2. Determination of Parameters Affecting Phase Velocity

In waveguide mode propagation, the intensity of the resonant wave is much higher than other waves of the same source due to the superposition effect, so the research object of waveguide mode propagation is the resonant wave. The conditions for the formation of the resonant wave are related to the effective reflection height, wavelength, and mode number of the ionosphere. Figure 3 below is a schematic diagram of the resonant ray geometry:

If $\theta$ is the resonant angle that satisfies the resonance condition, the cosine of the n-order mode resonant angle is

$$C_n=(n-{\displaystyle \frac{1}{2}})\lambda /(2h)$$

(33)

where $c$ is the speed of light in vacuum, $h$ is the effective reflection height of the ionosphere, $a$ is the radius of the Earth, and $\lambda$ is the wavelength. Under ideal conditions, the speed of light in air is considered to be the same as that in a vacuum. At the same time, in waveguide mode propagation, under resonant conditions, the phase on the cross section perpendicular to the waveguide propagation direction at any position in the waveguide is the same. The propagation speed of the wavefront is defined as the phase velocity, which is also the quantity that most directly affects the phase change. Due to the mechanism of waveguide mode propagation, the phase velocity should be expressed as

$$v=c/S_n$$

(34)

where $S_n$ is $Sin\theta$ in the n-th order mode, so $v$ can be written as

$$v=c{(1-C_n^2)}^{-1/2}$$

(35)

When propagating over long distances, taking into account the effect of the Earth’s curvature, the phase velocity expression will be written as

$$v^{\prime }=v{\displaystyle \frac{a}{a+h}}=c{(1-C_n^2)}^{-1/2}{\displaystyle \frac{a}{a+h}}$$

(36)

Of course, the phase velocity expression described in the above formula is under ideal conditions. The three parameters that are ignored are (1) the effect of temperature and humidity changes in the air on the phase velocity, (2) the effect of the Earth’s conductivity on the phase velocity at the ground reflection point, and (3) the effect of the ionosphere state on the phase velocity at the reflection point. These three points will be used as research content to improve the accuracy of VLF propagation delay further.

According to the phase velocity expression, it can be concluded that the important factors affecting the phase velocity are mainly the cosine value of the resonance angle, the order of the waveguide mode, and the effective reflection height of the ionosphere. The effective reflection height and wavelength of the ionosphere determine the resonance angle. Under long-distance conditions, since the attenuation rate of the high-order mode is much greater than that of the first-order mode, the number of observable modes is only the first-order mode, and the phase velocity formula can be written as

$$v^{\prime }=c{(1-{\displaystyle \frac{{\lambda }^2}{16h^2}})}^{-1/2}{\displaystyle \frac{a}{a+h}}$$

(37)

It can be seen from the above formula that, under ideal conditions, the primary factors affecting the phase velocity are wavelength and the ionosphere’s equivalent height. The ionosphere equivalent reflection height varies significantly during the day and at night, resulting in a substantial difference in the phase velocity of the VLF radio wave signal during these periods, which in turn affects the propagation delay. Therefore, this paper focuses on the impact of ionosphere changes on propagation delay as its primary research objective.

2.3. Analysis of the Impact of Ionospheric Changes

The ionosphere itself is more active than the lower wall of waveguide propagation, the ground. The state of the ionosphere is constantly changing. Regarding the impact of ionosphere changes on propagation delays, this paper mainly analyzes the changes in the ionosphere itself and the changes in the ionosphere caused by the alternation of day and night.

2.3.1. Changes in the Ionosphere

As the “upper wall” in the propagation of VLF waveguide mode, the ionosphere reflects VLF electromagnetic waves. Its parameters, such as electron density, electron collision frequency, and effective conductivity, will affect the ionosphere’s reflection of VLF signals. Although humans have continued to study the characteristics of the ionosphere, a universal and highly accurate ionosphere model remains elusive. However, as a medium that reflects VLF electromagnetic waves, all influencing factors can be unified into the parameter “equivalent reflection height of the ionosphere” (hereinafter referred to as “ionosphere height”). Following the advancement of ionosphere research, the parameters influencing the ionosphere will be further refined to enhance the accuracy of the overall model.

The reflection of VLF radio waves by the ionosphere reduces the phase velocity of the waveguide mode, causing phase lag and absorption attenuation, which results in field strength attenuation. Changes in the ionosphere reflection height will directly affect the wavelength of the VLF radio wave in the waveguide and thus affect the phase velocity. During the propagation of VLF signals, due to the rotation of the Earth, the propagation path changes from day to night, and the ionosphere height changes accordingly, so the phase velocity also changes. Figure 4 below is a simulation of different frequencies changing with the ionosphere height:

In general, at the same frequency, the higher the ionosphere altitude, the lower the phase velocity of the VLF signal and the greater the signal propagation delay.

2.3.2. Day and Night Alternation

The impact of day and night changes on the propagation of VLF electromagnetic waves is primarily due to the sunlight irradiating the ionosphere, causing stable charge ionization, which in turn leads to drastic changes in ionospheric parameters, such as electron concentration, thereby affecting the propagation of VLF electromagnetic waves. After unifying the influencing factors of the ionosphere, such as ionosphere height, it is evident that the stable charge ionization caused by sunlight irradiating the ionosphere is not an instantaneous change. Therefore, the impact of day and night changes mainly considers the transition period between day and night.

During the entire day, the ionosphere height is low, transmission attenuation increases, phase velocity increases, and transmission remains relatively stable. At night, the ionosphere height is high, resulting in reduced transmission attenuation and phase velocity, as well as a reduced influence from ground characteristics. Consequently, stability worsens, mode interference effects are enhanced, and there are apparent directional and latitude effects. During the transition from day to night, the dividing line between day and night belongs to the medium mutation, and the ionosphere height mutation distance is approximately 20 km. This causes the waveguide mode conversion effect, resulting in sudden mode interference increases and a significant reduction in transmission stability.

The following figures show the phase change curves of the VLF radio waves with a frequency of 18 kHz received from the NBA station in Nairobi, 1385 km away from the transmitting source, and Bathurst, 6880 km away, within one day.

As shown in Figure 5, during the transition period between day and night, the phase change rate changes significantly, due to changes in the ionosphere height. The relative delay at night is greater than during the day, and the transition at sunrise is steeper than that at sunset. This indicates that ① the phase velocity during the day is higher than that at night, meaning that the increase in ionosphere height leads to a decrease in phase velocity and an increase in phase delay. ② In the early morning, the sun’s rays first irradiate the ionosphere and quickly produce ionization. After the sun’s rays stop irradiating at night, the ionization recombination process is relatively slow, and this conclusion can also be observed in the previous figure.

2.4. Classification Analysis of Factors Affecting Propagation Delay

The influencing factors related to the ionosphere can be uniformly quantified by the effective reflection height of the ionosphere, which allows for the quantification of changes in the phase velocity. The ground-related influencing factors cannot be directly expressed in numerical relationships due to the high randomness and complexity of the propagation path; however, they can be quantified and corrected later by collecting experimental data and differential stations. The changes in ground temperature, humidity, and weather are random factors that can only be corrected by subsequent differential methods to improve the accuracy of phase correction. The influence of the geomagnetic field on the phase propagation delay is relative, and its effect is also a sinusoidally related change, calculated based on the magnetic azimuth and geomagnetic latitude of the path. This part of the delay will be corrected together when calculating long distances and one-time delays. The impact of seasonal changes on the phase propagation delay is mainly summarized from the law, collecting a large amount of historical data, and modeling and predicting the temperature, humidity, sunrise time changes, sunset time changes, rainfall probability, climate change, etc., generated by the season. Quantitative analysis is performed on the variable parameters that can meet the control variable conditions to further correct the accuracy of the delay correction model. The impact of day–night changes on phase propagation delay is mainly because sunlight can cause the ionization of free electrons in the ionosphere, thereby affecting the change in electron concentration. From a quantitative perspective, it is equivalent to the effective reflection height of the ionosphere. Another impact of day–night changes on phase propagation delay is the two transition periods between day and night in the morning and evening. This period is when the phase propagation delay changes most violently, is the most unstable, and is the most complex of the day. However, the overall trend of the transition period is predictable, and the amplitude of the change can also be predicted by formula calculation.

According to the type of influence of the change in influencing parameters on the propagation delay of VLF radio waves, it can be divided into three categories: the delay that changes with the path and is relatively stable after the path stabilizes is the primary delay influencing parameter; in the propagation delay change in continuous signals, the delay influencing parameter that causes the signal-like periodic and substantial regular changes with a period of 24 h is the secondary delay influencing parameter; and the delay influencing parameter that has no regularity in the propagation delay and has a strong randomness and unpredictability in its changes is the random delay influencing parameter. The specific parameter classification is shown in

Table 1. Classification of parameters affecting propagation delay.

Name	Type	Order of Magnitude	Correction
Propagation distance	Systematic delay	Millisecond	Calculate deductions
Path magnetic bearing	Systematic delay	Hundreds of microseconds	Calculate deductions
Ionosphere equivalent altitude	Secondary delay	Ten microseconds	Model prediction deduction
Day and night changes	Secondary delay	Ten microseconds	Model prediction deduction
Earth conductivity	Random delay	Hundreds of nanoseconds	Differential correction
Seasonal changes	Random delay	Microsecond	Differential correction

:

According to previous research conclusions and theoretical analysis, the main factors affecting the phase delay changes in VLF radio signal propagation can be divided into two categories: quantifiable factors and non-quantifiable factors. The influence of the ionosphere, geomagnetic field, and diurnal changes can be quantified, while ground-related and seasonal changes cannot be quantified at present and represent a key research direction for future studies.

3. Results

Since the Russian Alpha navigation system has been shut down for unknown reasons in the past two years, this paper uses historical measured data. The data source is the article “Observation and Analysis of VLF Signal Phase and Field Strength during the Solar Eclipse on 22 July 2009”, which was published in the Journal of Time and Frequency in June 2011. The data are several sets of data from the Main, East, and West Stations to Chongqing, Qingdao, and Guilin, respectively, during the five-day measurement from 20 July 2009 to 25 July 2009. This paper selects six sets of data from these three stations in Chongqing and Guilin on 20 July 2009 [27].

3.1. Alpha Test Results—Chongqing

The receiving stations are the East Sub-station, West Sub-station, and Main Station of the Alpha Navigation System. The receiving point is Chongqing, China. The data time is 20 July 2009, the duration is 1 day, and the receiving coordinates are (29.5630° N, 106.5516° E). The phase change in the received signal is shown in the following Figure 6, Figure 7 and Figure 8:

As can be seen from Figure 6, Figure 7 and Figure 8 and

Table 2. Changes in receiving phase in Chongqing.

Station	Great Circle Distance	Estimated Daytime Delay	Estimated Time Delay at Night	Expected Delay Difference Between Day and Night	Measure Maximum Delay Difference
Main	3395.008 km	11,378.4 μs	11,407.4935 μs	29.0935 μs	27.3880 μs
East	3392.466 km	11,385.5 μs	11,409.5852 μs	24.0852 μs	29.2461 μs
West	6095.267 km	20,428.4 μs	20,480.6334 μs	52.2334 μs	47.2574 μs

, the propagation delay of the VLF signal during the day exhibits a highly periodic pattern, with the primary difference in its variations being the distinction between day and night. The primary reason for the phase difference between day and night is that the equivalent reflection height of the ionosphere varies significantly during the day and night, directly leading to a noticeable difference in phase velocity between the VLF signal of the same frequency during the day and at night. The propagation delays during the day and night are relatively stable. At night, because the ionosphere is more active than during the day, the propagation delay fluctuations are larger than those during the day. However, they are still significantly smaller than the propagation delay difference between day and night. The central station and the East Sub-station are relatively close to the big circle of Chongqing (3395.008 km and 3392.466 km, respectively), and the maximum value of the measured delay difference on the day is also relatively close; the West Station is far away from the big circle of Chongqing (6095.267 km), and the maximum value of the measured delay difference on the day is much larger than that of the Main Station and the East Station.

3.2. Alpha Test Results—Guilin

The receiving stations are the East Sub-station, West Sub-station, and Main Station of the Alpha Navigation System. The receiving point is Guilin, China. The data time is 20 July 2009, the duration is 1 day, and the receiving coordinates are (25.274° N, 110.2979° E). The phase change in the received signal is shown in the following Figure 9, Figure 10 and Figure 11:

As can be seen from Figure 9, Figure 10 and Figure 11 and

Table 3. Changes in receiving phase in Guilin.

Station	Great Circle Distance	Estimated Daytime Delay	Estimated Time Delay at Night	Expected Delay Difference Between Day and Night	Measure Maximum Delay Difference
Main	3982.786 km	13,348.4 μs	13,382.5305 μs	34.1305 μs	33.6587 μs
East	3562.756 km	11,933.0 μs	11,969.9636 μs	36.9636 μs	37.6818 μs
West	6680.961 km	22,391.3 μs	22,448.5525 μs	57.2525 μs	61.4393 μs

, the propagation delay of the VLF signal during this day exhibits a highly periodic pattern, with the primary difference in its changes being the transition between day and night. The primary reason for the phase difference between day and night is the substantial variation in the equivalent reflection height of the ionosphere between day and night, which directly results in a significant difference in phase velocity between VLF signals of the same frequency during daytime and nighttime. The propagation delays during the day and night are relatively stable. At night, because the ionosphere is more active than during the day, the propagation delay fluctuations are larger than those during the day. However, they are still significantly smaller than the propagation delay difference between day and night. The central station and the East Sub-station are relatively close to the Great Circle of Guilin (3982.786 km and 3562.756 km, respectively), and the maximum value of the measured delay difference on the day is also relatively close; the West Station is far away from the Great Circle of Chongqing (6680.961 km), and the maximum value of the measured delay difference on the day is much larger than that of the Main Station and the East Station.

3.3. Experimental Results Analysis

The above two groups of experimental results reflect the duplicate content, which well proves that the theory has a certain universality. First of all, the six groups of data share the most obvious common feature: the propagation delay is positively correlated with the propagation path distance. At the same time, the propagation delay during the day is significantly lower than that at night; that is, the phase velocity of the signal propagation during the day is significantly greater than that at night. The most direct reason is that sunlight irradiating the ionosphere causes the effective reflection height of the ionosphere to decrease due to the ionization phenomenon. This decrease in the effective reflection height increases the phase velocity, thereby reducing the propagation delay. At the same time, it can be found from the figure that the transition time from night to day is significantly shorter than that from day to night. This is because the ionization phenomenon caused by sunlight irradiating the ionosphere occurs quickly, and its recovery is relatively slow after losing sunlight. The difference in distance from the transmitter also leads to different day and night phase differences. The day–night phase difference is positively correlated with the propagation path distance. The average error between the predicted day and night propagation delay difference and the measured maximum delay difference in one day is 0.015 microseconds per 100 km, and the maximum error in delay prediction is 0.152 microseconds per 100 km. The phase fluctuations during night and day are related to real-time changing parameters, such as weather, temperature, and the Earth’s conductivity. At the same time, the phase fluctuations during the night are greater than those during the day because the ionosphere is more active than during the day.

In summary, the six experimental results have well demonstrated the correctness of the formula derived in this paper and the rationality of the proposed theory.

4. Discussion

The following conclusions and prospects are drawn from the above experimental data and theoretical analysis:

• The theoretical and experimental data on VLF waveguide mode propagation over long distances in the “Earth-ionosphere” waveguide mode are consistent, indicating that waveguide mode propagation enables VLF to achieve ultra-long-distance PNT applications.
• According to the phase velocity theory prediction formula, it can be known that the most important influencing factors of phase velocity are the ionospheric equivalent reflection height and the signal wavelength, and the actual test results are consistent with the predictions.
• The most important factor affecting the propagation delay is the propagation distance (milliseconds), followed by the change in the effective reflection height of the ionosphere during the day and night (tens of microseconds). The change in propagation delay is periodic (24 h) and remains constant as the propagation path is fixed. This part of the propagation delay can be predicted through modeling and calculation, and is the main correction object for the next step of propagation delay correction. The propagation delay of the VLF signal before correction is tens of microseconds, and the expected target after correction is microseconds.
• The random factors that affect the propagation delay include ionosphere properties, ground conductivity, weather, season, temperature, and humidity, among others. The fluctuation of this part of the propagation delay is highly random, and we can utilize machine learning and other methods to conduct a large number of actual measurement experiments, allowing for more in-depth research.
• Whether comparing three sets of data from different transmitting locations at the exact receiving location, or comparing two sets of data from different receiving locations at the same transmitting location, the start and end times of the day–night transition are different. This suggests that the day–night impact here should be related to the day–night conditions along the entire path, which can serve as the basis for subsequent research.