Then we come to multi-hop: Known elevation can be based on geometric knowledge to get the relationship between the hopping angle and angle of incidence as shown in Figure 1, we take the ionosphere height h = 100 km and the Earth’s radius R = 6371 km.

We have:

$d=2R\left(\mathrm{arccos}\left(\frac{R\mathrm{cos}\partial}{h+R}\right)-\partial \right)$ (3)

2.3. Correction of Attenuation in Medium

1) Since radio waves propagate in the line of sight range, the radio wave propagation mode is mainly diffractive propagation, and the subject requires the study of reflection. Therefore, it is not necessary to discuss the diffraction correction of the sea surface within the viewing distance range.

2) The following picture suggest that when a point light emits signals in all directions, the energy of all the electric waves that can reach ③ only comes from the loss of electric wave propagation in the area ②, and the energy loss of the electric waves emitted in the range ① is not considered. For that case the coefficients need to be corrected with correction ratio which is mean of angle, see Figure 2.

Since the corresponding propagation distance in scattering is infinite, all

Figure 1. Earth and atmosphere.

Figure 2. Relation of angles.

signal losses are taken into account, so there is no need to correct the scattering losses.

3) The calm sea surface reflection coefficient by the law of refraction refraction. Because of the law of refraction refraction: ${n}_{1}{\theta}_{i}={n}_{2}{\theta}_{i}$ .

We calculate the reflection coefficient: with:

${Q}_{1}=L\times 4\text{\pi}{d}^{2}\times {R}_{1}$

${R}_{1}=\frac{1}{2}\left({r}_{1}^{2}+{r}_{2}^{2}\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}{r}_{1}=\frac{{\lambda}_{2}\mathrm{cos}{\theta}_{i}-{\lambda}_{1}\mathrm{cos}{\theta}_{t}}{{\lambda}_{2}\mathrm{cos}{\theta}_{i}+{\lambda}_{1}\mathrm{cos}{\theta}_{t}}$

${r}_{\text{2}}=\frac{{\lambda}_{\text{1}}\mathrm{cos}{\theta}_{i}-{\lambda}_{2}\mathrm{cos}{\theta}_{t}}{{\lambda}_{\text{1}}\mathrm{cos}{\theta}_{i}+{\lambda}_{2}\mathrm{cos}{\theta}_{t}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\lambda}_{i}=\sqrt{\frac{{\mu}_{i}}{{\epsilon}_{i}}}\text{\hspace{0.17em}}\left(i=1,2\right)$ (4)

The relationship between the reflection coefficient and the incident angle is shown in Figure 3.

Based on the empirical formula in the longley-Rice model, we can deduce that the attenuation of the first reflection at calm sea surface is:

4) Rough turbulent ocean surface reflection coefficient correction

① The establishment of 3D turbulent ocean wave model based on ocean wave spectrum.

Because there is sea slope in turbulent sea compared to the calm sea surface, a three-dimensional turbulent ocean wave model needs to be established to study it.

② Create a wave model

The wave is described by a stationary stochastic process with ergodicity. The wave is viewed as a superposition of waves and swells in a simple cosine wave of infinitely different amplitudes, of varying frequency and of an incipient phase. That is, for the composition of wave propagation direction relative to the wind direction angle. θ is the angle which wave spread by in x. We use the wave spectrum function in reference:

$S\left(\rho ,\theta \right)=\frac{8.1\times {10}^{-3}{g}^{2}}{{w}^{5}}\mathrm{exp}\left[-0.74{\left(\frac{g}{uw}\right)}^{4}\right]\frac{2}{\text{\pi}}{\mathrm{cos}}^{2}(\theta )$

Figure 3. The relationship between angle and reflection confficient.

$S\left(\rho ,\theta \right)=\frac{8.1\times {10}^{-3}{g}^{2}}{{w}^{5}}\mathrm{exp}\left[-0.74{\left(\frac{g}{uw}\right)}^{4}\right]\frac{\text{8}}{\text{3\pi}}{\mathrm{cos}}^{2}(\theta )$

Create a random wave model shown in Figure 4.

From this model, we get the slope distribution function of each discrete sea surface randomly according to the wave spectrum (Figure 5).

③ Set up the sea coordinate system

Set up the sea coordinate system as shown. Let the origin be located on the sea surface where the study object is located. The X and Y axes are located on the horizontal plane of the coordinate system. The positive direction of Y axis is the position of the detector and the positive direction of Z axis is upwar, see Figure 6.

We change the coordinates, maintaining the Z axis direction unchanged. The coordinate system is rotated clockwise:

$\{\begin{array}{l}{z}_{u}={S}_{x}\mathrm{cos}\gamma +{S}_{y}\mathrm{sin}\gamma \\ {z}_{v}=-{S}_{x}\mathrm{cos}\gamma +{S}_{y}\mathrm{sin}\gamma \end{array}$ (5)

Figure 4. Wave mode.

Figure 5. Spectrum.

Roughness correction factor $\xi $ :

$\xi \text{=}\frac{\text{\pi}P\left({z}_{x},{z}_{y}\right)}{\text{4}\mathrm{cos}{\omega}_{i}{\mathrm{cos}}^{4}\beta}$

Rough correction factor and the relationship between the incidence angle shown in Figure 7.

Because: The reflection coefficient on the turbulent sea is the reflection coefficient on calm sea surface with rough correction factor.

The reflection coefficient on the turbulent sea is:

${R}_{2}=\frac{\text{\pi}P\left({z}_{x},{z}_{y}\right){R}_{1}}{4\mathrm{cos}{\omega}_{i}{\mathrm{cos}}^{4}\beta}$ (6)

Figure 6. Relation of angle.

Figure 7. The relationship between angle of incidence and correction factor.

3. Model Calculation and Result Analysis

3.1. Optimize the Angle and Find the Maximum Number of Hops

As for S/N ratio explanation [5] , the main parameters of the world profile of atmospheric radio noise reported by CCIR-3 22 are the effective noise figure F_{a}. It is defined as the ratio of the external radio noise power received by the non-directional short vertical antenna to the noise power generated by the heat source at temperature T_{0} in the unit bandwidth that is the title of the signal to noise ratio [6] :

${F}_{a}=\frac{{P}_{n}}{k{T}_{0}B}$ (7)

P_{n} is the noise power received by the non-directional short vertical antenna.

K is boltzmann constant; B is the effective noise bandwidth of the receiver.

W is initial signal power.

As for ${F}_{a}=10\mathrm{lg}\frac{{P}_{n}}{W}$ , the critical condition is when the noise power is equal to the signal power，so according to the problem (what is the maximum number of hops the signal can take before its strength falls below a usable signal-to-noise ratio (SNR) threshold of 10 dB):

$W{\left(L\times {R}_{\text{1}}\right)}^{n}>{P}_{n}$

With W = 100W, we can get the remaining power after the reflection with

${W}_{1}=W\left(L\times {R}_{1}\right),\text{\hspace{0.17em}}{W}_{\text{2}}=W{\left(L\times {R}_{1}\right)}^{\text{2}},\text{\hspace{0.17em}}{W}_{\text{3}}=W{\left(L\times {R}_{1}\right)}^{\text{3}},\cdots ,\text{\hspace{0.17em}}{W}_{n}=W{\left(L\times {R}_{1}\right)}^{n}$

In the case of the incident angle is determined, the maximum number of hops is uniquely determined, we make an interval traversal of the angle and get angle of incidence and the maximum number of hops in Figure 8.

And maximum number is 14.

The size of the incident angle can affect the radio wave propagation time in the air and thus the reflection intensity, so we can get the relationship between the first reflection intensity and the incident angle in Figure 9.

3.2. Intensities Comparison of the First Reflections in Calm and Turbulent

The angle is 57˚ from the figure and we can get the first reflection strenghth [8] :

${P}_{1}=W{Q}_{1}=7,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{P}_{2}=W{Q}_{2}=3,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{P}_{1}>{P}_{2}$

4. Conclusion

By using original Longley-Rice formula, the minimum loss under the maximum transmission distance and the practicality of radio propagation in the navigation of ships are discussed. The multiple correction parameters is found, and the loss of radio propagation is calculated, which can be used to estimated loss and save

Figure 8. The relationship between angle of incidence and max-hop.

Figure 9. The relationship between angle of incidence and first strength.

energy. The model accurately describes the various types of losses in radio transmission acceptance, with great significance to the communication in the route. In addition to that, our paper is not only used to the situation above, but widely applies to kinds of fields such as marine radar communication, mountain exploration, etc. In more details, the correction factor we propose can also apply to many problems which have a traditional formula without matching situation.

Cite this paper

Chen, Y. , Han, L. , Huang, J. and Gui, Y. (2018) A Mathematical Model of Multi-Hop HF Radio Propagation.*Applied Mathematics*, **9**, 779-788. doi: 10.4236/am.2018.96054.

Chen, Y. , Han, L. , Huang, J. and Gui, Y. (2018) A Mathematical Model of Multi-Hop HF Radio Propagation.

References

[1] Li, J.-S. (2014) Study on AIS Sea Wave Propagation Model. Dalian Maritime University, Dalian, 32-45.

[2] Yi, Q.D. (2015) Research on Electromagnetic Wave Propagation model in Sea Area. Hainan University, Hainan.

[3] Fang, H. (2015) Research on Propagation Characteristics and Channel Modeling of Maritime Radio Waves. Hainan University, Hainan, 67-98.

[4] Yu, W.Z., Chi, X. and Ren, J. (2014) Transmission Model of Maritime Mobile Channel Based on ITM. Journal of Electronics, 40, 106-111.

[5] Jin, F. (2015) Radio Tracking Based on the Law of Radio Propagation in the Hilly Areas. Nanjing University of Posts and Telecommunications, Nanjing, 42-51.

[6] Xu C., Qiu, C.-C. and Li, L. (2015) Status and Analysis of Optimum Frequency Selection for Maritime HF Communications. Communications Technology, 1, 17-23.

[7] Zhao, Y.C. (2009) Research and Implementation of Radio Wave Propagation Prediction and Interference Analysis. National University of Defense Technology, Changsha.

[8] Peng, F.F. and Zhou, X.J. (2017) Prediction of Maritime Shortwave Telecommunication Links. Ship Electronic Engineering, 31, 125-127.

[9] Li, Z. (2018) Analysis of Maritime Radio Communication Time Based on Longley-Rice Model. Science and Technology Innovation and Application, 13, 16-17.

[10] Liu, J. (2018) On the Development Trend of Anti-Interference Technology for Ultra-Shortwave Radio Communication. Shandong Industrial Technology, 2, 117-118.

[11] Zhang, X.G. (2018) “Internet + Radio” to Achieve Efficient Radio Supervision. Science and Technology Economics Guide, 26, 192.