A High-Precision Prediction Method of Atmospheric Absorption Attenuation on Over-the-Horizon Propagation Trajectories

Abstract

Abnormal refraction phenomena such as atmospheric ducts due to temperature inversions or rapid decreases in humidity often happen in the lower troposphere over the sea and coastal area, which can make low-elevation signals in the duct layer propagate beyond the horizon, and the ray trajectories extend horizontally over long distances. This paper uses ray tracing technology based on a second-order Taylor approximation to accurately predict the low-elevation ray trajectories within atmospheric ducts. The meteorologic parameters at the heights traversed by the rays are extracted to accumulate atmospheric absorption attenuation by line-by-line calculations, and a high-precision prediction method for atmospheric absorption attenuation in over-the-horizon propagation links is established; meanwhile, we also implement visualization of atmospheric absorption attenuation changes along the ray trajectories in atmospheric duct environments. By comparing the results of the atmospheric absorption attenuation models for horizontal terrestrial paths in the ITU-R P.676 recommendation and GJB Z87-1997 in atmospheric duct environments, we found that the high-precision model proposed in this paper can improve the prediction accuracy of atmospheric absorption attenuation by about 15% in surface ducts and 28% in elevated ducts, significantly improving the propagation performance of low-elevation signals under atmospheric ducts and other abnormal refraction conditions for electronic systems such as surveillance, detection, communication, and navigation.

1. Introduction

Microwaves, millimeter waves, visible and IR and even shorter wavelengths of radio waves experience absorption of electromagnetic energy by water vapor and oxygen molecules in the atmosphere, leading to energy level transitions and resulting in wave attenuation, known as atmospheric absorption attenuation. Atmospheric absorption attenuation is primarily influenced by meteorological factors such as water vapor density, temperature, and refractive index, as well as parameters of wireless systems like operating frequency, elevation, and signal propagation path length. Among the commonly used models for calculating atmospheric absorption attenuation in the troposphere, the ITU-R P.676 recommendations “Attenuation by atmospheric gases and related effects” [1] and the GJB Z87-1997’s “Handbook of refraction and attenuation of radiowave propagation for radar” [2] provide the prediction models for atmospheric absorption attenuation under two propagation conditions: slant paths (ground-to-air) and horizontal terrestrial paths (ground-to-ground). For slant path electromagnetic wave propagation in ground-to-air links, ITU-R recommendations offer a high-precision method involving line-by-line summation. This method precisely calculates the atmospheric specific attenuation at different heights along the propagation path by accumulating the resonant absorption spectra of oxygen and water vapor line-by-line [1,3,4,5,6]. It also considers the non-resonant Debye absorption spectrum of oxygen below 10 GHz, nitrogen attenuation above 100 GHz caused by atmospheric pressure, and additional continuous water vapor absorption discovered in experiments. Using atmospheric temperature, humidity, and pressure profiles, this method accounts for the curvature effect of radio waves due to atmospheric refraction and calculates the path length in each layer to obtain high-precision atmospheric absorption attenuation for ground-to-air links. For over-the-horizon propagation links caused by atmospheric ducts and other abnormal refraction phenomena, ITU-R 676 and GJB Z87-1997 approximate the radio wave propagation in the duct layer as straight-line propagation near the ground and use horizontal terrestrial paths models for calculations. This model assumes constant water vapor density near the Earth’s surface, estimates specific attenuation approximately, and uses straight-line distance to approach path length to obtain atmospheric absorption attenuation for over-the-horizon links [7,8,9,10].

In maritime and coastal regions, low-altitude atmospheric ducts occur frequently, and evaporation ducts over the sea almost permanently exist. In the actual atmosphere, multiple composite ducts are also very common [11,12,13,14,15]. Among these, evaporation ducts due to rapid decreases in water vapor near the sea surface typically occur at heights between 0 and 40 m. Surface ducts and elevated ducts are usually caused by temperature inversions and sharp decreases in humidity; surface ducts can extend to several hundred meters, and elevated ducts are up to several kilometers. When radio waves propagate in such abnormal refractive layers, especially in elevated ducts at heights of several kilometers, drastic changes in water vapor density and temperature inversions significantly impact the accuracy of traditional atmospheric absorption prediction models [16,17,18]. Additionally, the propagation trajectories of radio waves in ducts differ considerably from those at high elevation angles in ground-to-air links. Instead of approximate straight-line propagation, the waves reflect periodically in the duct, bouncing between the upper and lower boundaries of the duct. Radio waves propagating in the atmosphere remain with the highest water vapor and temperature in the low-altitude region, where oxygen and water vapor absorption is strongest, and the path length is significantly longer than in ground-to-air links.

The research for atmospheric gases mostly focuses on the components. Yu. M. Timofeev estimated the trends for TCAs of 13 climatically important gases for the first time in Russia, and trends of the main greenhouse gases, CO₂, CH₄, and N₂O, determined with high accuracy (~0.02%), are positive and equal 0.52, 0.42, and 0.28% per year, respectively [4]. Osazuwa G. Agbonlahor used nano-porous activated carbon for atmospheric passivation of the graphene channel, Extreme Gradient Boosting (XGBoost), K-nearest neighbors (KNN), and Naïve Bayes supervised ML models for atmospheric gas identification were developed, with model feature contributions studied using the Game theoretic approach, Shapley Additive explanations (SHAP) [5]. For the method of atmospheric gas attenuation in complicated environments, WU Ling-da used the method in CCIR, which provides a simplified approach to calculating attenuation by modeling atmospheric absorption attenuation as a linear function of height, thereby enabling the determination of attenuation values at various heights based on the attenuation measured at sea level [6,7].

The ray tracing technology for low-elevation angles in atmospheric ducts is used to precisely trace the propagation trajectories of over-the-horizon links, which can accurately reproduce the periodic propagation trajectories of radio waves in atmospheric ducts. During the tracing process, the method retrieves actual height, path length, and other trajectory parameters at different propagation distances. Then, it extracts temperature, pressure, water vapor density, and refractive index parameters at these heights [19,20,21,22]. Using a line-by-line summation approach, the model performs high-precision calculations of atmospheric absorption attenuation for low-elevation over-the-horizon links. Based on the ray trajectories, the model also visualizes the changes in atmospheric attenuation along the propagation paths. This approach not only accurately demonstrates the electromagnetic wave propagation trajectories but also precisely reflects the trends in atmospheric absorption attenuation along different elevation angle paths [23,24,25,26,27,28].

2. Ray Trajectory Prediction Techniques for Over-the-Horizon Links in Ducts

The algorithm principle of ray tracing technology is the geometric optical approximation, which can vividly explain the positioning process of the radar system and accurately determine the specific position of the target. Ray tracing can not only track the propagation trajectory of electric waves in normal atmospheric environments but also track the ray trajectory of radio waves in the abnormal propagation of atmospheric ducts.

Snell’s law of the spherical layered atmosphere is [8,9,22]:

$$n_0(r_e+h_0)\cos {\theta }_0=n(r_e+h)\cos \theta$$

(1)

Here ${\theta }_0$, $\theta$ and $n_0$, $n$ are the elevation angles and refractive index at first height layer $h_0$ and higher layer $h$, $r_e=6371 \mathrm{km}$, is the Earth’s radius. When $h_0<<r_e$, $n\approx 1$ and makes $m(h)=n(h)+{\displaystyle \frac{h}{r_e}}$, Formula (1) can be approximated by [8,9,22]:

$$m(h)\cos \theta =m(h_0)\cos {\theta }_0$$

(2)

Formula (2) is the form of Snell’s law under flat ground, and $m(h)$ is called the modified refractive index. In this case, it is more convenient to deal with the atmospheric duct problem. We can get the ray trajectory parameters by [8,9,22]:

$$\left\{\begin{array}{l}{\theta }_{i+1}=\sqrt{{\theta }_i^2+2\Delta m_i}\\ x_{i+1}=x_i+{\displaystyle \frac{{\theta }_{i+1}-{\theta }_i}{g_i}}\\ l_{i+1}=l_i+{\displaystyle \frac{(m_i-{\theta }_{i+1}^2/2)({\theta }_{i+1}-{\theta }_i)+({\theta }_{i+1}^3-{\theta }_i^3)/3}{g_i}}+d{x}_i\\ {\phi }_i=x_i/r_e\\ d_{i+1}=d_i+\sqrt{{(r_e+h_i)}^2+{(r_e+h_{i+1})}^2-2(r_e+h_i)(r_e+h_{i+1})\cos ({\phi }_{i+1}-{\phi }_i)}\end{array}\right.$$

(3)

In the Formula (3), ${\theta }_i$ and ${\theta }_{i+1}$ denote the elevation angles at heights $h_i$ and $h_{i+1}$, surface distances $x_i$ and $x_{i+1}$, respectively. The geometric optical path lengths (apparent distances) are denoted as $l_i$ and $l_{i+1}$, while the actual path lengths are $d_i$ and $d_{i+1}$. The corresponding geographic angles are ${\phi }_i$ and ${\phi }_{i+1}$.

During the computation, a fixed height step is maintained. The duct capture of a ray is determined by the sign of the radical term in the equation for ${\theta }_{i+1}$. A positive value indicates the ray remains uncaptured, while a negative value signifies capture at the current position with a directional reversal. The elevation angle is then reset to 0, and propagation resumes in the opposite direction. When the detected value is ≤0, the ray is considered to have hit the ground, at which point the elevation angle is reset to 0, and the ray continues upward after total reflection off the ground. Under uniform atmospheric conditions and a flat reflective surface, the captured ray undergoes periodic jumps within the duct [8,9]. The algorithm flow chart is shown in Figure 1.

3. Atmospheric Absorption Attenuation Evaluation Model

3.1. Traditional Horizontal Path Atmospheric Absorption Simple Attenuation Model

For atmospheric absorption attenuation in waveguide propagation, the GJB Z87-1997 model uses horizontal path attenuation to approximate atmospheric absorption attenuation in ducts [2].

$$A_{gs}=({\gamma }_o+{\gamma }_w)d$$

(4)

In the formula, d is the ground distance from the signal emission to the target, while the ground attenuation rates of oxygen and water vapor are denoted as ${\gamma }_o$ and ${\gamma }_w$, respectively [2]:

$${\gamma }_o=\left\{\begin{array}{l}[7.19\times {10}^{-3}+{\displaystyle \frac{6.09}{f^2+0.227}}+{\displaystyle \frac{4.81}{{(f-57)}^2+1.5}}]f^2\times {10}^{-3}\hspace{1em}\hspace{1em}\hspace{1em}(f<57 \mathrm{GHz})\\ 14.9\hspace{1em}\hspace{1em}\hspace{1em}(57 \mathrm{GHz}\le f\le 63 \mathrm{GHz})\\ [3.79\times {10}^{-7}f+{\displaystyle \frac{0.265}{{(f-63)}^2+1.59}}+{\displaystyle \frac{0.028}{{(f-118)}^2+1.47}}]{(f+198)}^2\times {10}^{-3}\hspace{1em}\hspace{1em}\hspace{1em}(63 \mathrm{GHz}<f<350 \mathrm{GHz})\end{array}\right.$$

(5)

as shown in the following equation:

$${\gamma }_w=[0.05+0.0021\rho +{\displaystyle \frac{3.6}{{(f-22.2)}^2+8.5}}+{\displaystyle \frac{10.6}{{(f-183.3)}^2+9}}+{\displaystyle \frac{8.9}{{(f-325.4)}^2+26.3}}]f^2\rho \times {10}^{-4}\hspace{1em}(f<350 \mathrm{GHz})$$

(6)

From the above equation, it can be seen that the attenuation rate of oxygen γ_o in the horizontal path model is only related to frequency, while the attenuation rate of water vapor γ_w is only related to frequency and water vapor density ρ, which is taken at the height of the emission position.

3.2. High-Precision Implementation Method of Atmospheric Absorption Attenuation Based on Ray Trajectory of Beyond-Visual-Range Link

This paper is based on ray trajectory prediction technology of the atmospheric waveguide beyond the line-of-sight link, and uses the method of atmospheric attenuation rate of inclined path line-by-line accumulation to calculate the high precision of dry air and water vapor attenuation rate; the true path of ray in the process of line by line accumulation is obtained, and realizes the high-precision prediction of atmospheric absorption attenuation of ray trajectories of beyond the line-of-sight links in atmospheric ducts.

Atmospheric attenuation can be calculated by integrating the resonant absorption lines of oxygen and water vapor line by line, while considering the non-resonant Debye absorption of oxygen below 10 GHz, with the nitrogen attenuation mainly caused by atmospheric pressure above 100 GHz and the additional continuous water vapor absorption observed in experiments. The atmospheric attenuation rate can be precisely calculated using atmospheric temperature, humidity, and pressure parameters. The method for calculating the line-by-line integrated atmospheric attenuation rate is as follows:

$$\gamma ={\gamma }_o+{\gamma }_w=0.182 f{N}^{″}\left(f\right)\hspace{1em}\hspace{1em}\mathrm{dB}/\mathrm{km}$$

(7)

In the formula: ${\gamma }_o$ (dB/km) represents the attenuation rate of dry air (including nitrogen caused by oxygen and atmospheric pressure, as well as non-resonant Debye attenuation); ${\gamma }_w$ (dB/km) denotes the attenuation rate of water vapor; $f$ (GHz) is the frequency; and $N^{″}\left(f\right)$ is the imaginary part of the complex refractive index associated with frequency:

$$N^{″}\left(f\right)={\displaystyle \sum _i{S}_i{F}_i}+N_D^{″}\left(f\right)$$

(8)

In the formula, $S_i$ is the intensity of the i-th absorption line, $F_i$ is the shape factor of the absorption line, and the above expression is the sum of all spectral lines. $N_D^{″}\left(f\right)$ The following is the contribution of the continuous absorption spectrum, including atmospheric pressure-induced nitrogen absorption and Debye absorption, to the imaginary part of the refractive index:

$$\begin{array}{ll}{S}_i& =\left\{\begin{array}{l}{a}_1\times {10}^{-7}P{(300/T)}^3\exp [a_2(1-300/T)]\hspace{1em}oxygen\\ b_1\times {10}^{-1}e{(300/T)}^{3.5}\exp [b_2(1-300/T)\hspace{1em}water vapor\end{array}\right.\\ & =\left\{\begin{array}{l}{a}_1\times {10}^{-7}(P_{tot}-e){(300/T)}^3\exp [a_2(1-300/T)]\begin{array}{c}\hspace{1em}oxygen\end{array}\\ b_1\times {10}^{-1}e{(300/T)}^{3.5}\exp [b_2(1-300/T)\begin{array}{c}\hspace{1em}water vapor\end{array}\end{array}\right.\end{array}$$

(9)

In the formula, $P$ represents the dry air pressure (hPa), $e$ is the water vapor pressure (hPa), while the temperature T is in K; the total atmospheric pressure is $P_{tot}=P+e$, while the temperature is in K. The water vapor pressure can be calculated using the water vapor density:

$$e={\displaystyle \frac{\rho T}{216.7}}$$

(10)

where $\rho$ is water vapor density in g/m³. The shape factor of the absorption spectrum is:

$$F_i={\displaystyle \frac{f}{f_i}[\frac{\Delta f-\delta (f_i-f)}{{(f_i-f)}^2+\Delta f^2}+\frac{\Delta f-\delta (f_i+f)}{{(f_i+f)}^2+\Delta f^2}]}$$

(11)

where $f_i$ is the frequency of the absorption line; $\Delta f$ is the width of the absorption line:

$$\Delta f=\left\{\begin{array}{l}\hspace{1em}{a}_3\times {10}^{-4}(p {\theta }^{(0.8-a_4)}+1.1 e \theta )\hspace{1em}oxygen\\ b_3\times {10}^{-4}(p{\theta }^{b_4}+b_{5}e{\theta }^{b_6})\hspace{1em}\hspace{1em}\hspace{1em}water vapor\end{array}\right.$$

(12)

Due to the Doppler broadening effect, the spectral line width requires further correction to:

$$\Delta f=\left\{\begin{array}{l}\hspace{1em}\sqrt{\Delta f^2+2.25\times {10}^{-6}}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}oxygen\\ 0.535\Delta f+\sqrt{0.217\Delta f^2+{\displaystyle \frac{2.1316\times {10}^{-12}{f}_i^2}{\theta }}}\hspace{1em}water vapor\end{array}\right.$$

(13)

In the formula:

$$\delta =\left\{\begin{array}{l}\hspace{1em}(a_5+a_6\theta )\times {10}^{-4}(p+e){\theta }^{0.8}\hspace{1em}oxygen\\ 0\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}water vapor\end{array}\right.$$

(14)

The spectroscopic data for oxygen attenuation $a_1$~$a_6$ and spectroscopic data for water vapor attenuation $b_1$~$b_6$ are defined in

Table 1. Spectroscopic data for oxygen attenuation.

fi (GHz)	a1	a2	a3	a4	a5	a6
50.474214	0.975	9.651	6.690	0.0	2.566	6.850
50.987745	2.529	8.653	7.170	0.0	2.246	6.800
51.503360	6.193	7.709	7.640	0.0	1.947	6.729
52.021429	14.320	6.819	8.110	0.0	1.667	6.640
52.542418	31.240	5.983	8.580	0.0	1.388	6.526
53.066934	64.290	5.201	9.060	0.0	1.349	6.206
53.595775	124.600	4.474	9.550	0.0	2.227	5.085
54.130025	227.300	3.800	9.960	0.0	3.170	3.750
54.671180	389.700	3.182	10.370	0.0	3.558	2.654
55.221384	627.100	2.618	10.890	0.0	2.560	2.952
55.783815	945.300	2.109	11.340	0.0	−1.172	6.135
56.264774	543.400	0.014	17.030	0.0	3.525	−0.978
56.363399	1331.800	1.654	11.890	0.0	−2.378	6.547
56.968211	1746.600	1.255	12.230	0.0	−3.545	6.451
57.612486	2120.100	0.910	12.620	0.0	−5.416	6.056
58.323877	2363.700	0.621	12.950	0.0	−1.932	0.436
58.446588	1442.100	0.083	14.910	0.0	6.768	−1.273
59.164204	2379.900	0.387	13.530	0.0	−6.561	2.309
59.590983	2090.700	0.207	14.080	0.0	6.957	−0.776
60.306056	2103.400	0.207	14.150	0.0	−6.395	0.699
60.434778	2438.000	0.386	13.390	0.0	6.342	−2.825
61.150562	2479.500	0.621	12.920	0.0	1.014	−0.584
61.800158	2275.900	0.910	12.630	0.0	5.014	−6.619
62.411220	1915.400	1.255	12.170	0.0	3.029	−6.759
62.486253	1503.000	0.083	15.130	0.0	−4.499	0.844
62.997984	1490.200	1.654	11.740	0.0	1.856	−6.675
63.568526	1078.000	2.108	11.340	0.0	0.658	−6.139
64.127775	728.700	2.617	10.880	0.0	−3.036	−2.895
64.678910	461.300	3.181	10.380	0.0	−3.968	−2.590
65.224078	274.000	3.800	9.960	0.0	−3.528	−3.680
65.764779	153.000	4.473	9.550	0.0	−2.548	−5.002
66.302096	80.400	5.200	9.060	0.0	−1.660	−6.091
66.836834	39.800	5.982	8.580	0.0	−1.680	−6.393
67.369601	18.560	6.818	8.110	0.0	−1.956	−6.475
67.900868	8.172	7.708	7.640	0.0	−2.216	−6.545
68.431006	3.397	8.652	7.170	0.0	−2.492	−6.600
68.960312	1.334	9.650	6.690	0.0	−2.773	−6.650
118.750334	940.300	0.010	16.640	0.0	−0.439	0.079
368.498246	67.400	0.048	16.400	0.0	0.0	0.0
424.763020	637.700	0.044	16.400	0.0	0.000	0.000
487.249273	237.400	0.049	16.000	0.0	0.000	0.000
715.392902	98.100	0.145	16.000	0.0	0.000	0.000
773.839490	572.300	0.141	16.200	0.0	0.000	0.000
834.145546	183.100	0.145	14.700	0.0	0.000	0.000

and

Table 2. Spectroscopic data for water vapor attenuation.

${\mathit{f}}_{\mathit{i}}$ (GHz)	${\mathit{b}}_1$	${\mathit{b}}_2$	${\mathit{b}}_3$	${\mathit{b}}_4$	${\mathit{b}}_5$	${\mathit{b}}_6$
22.235080	0.1130	2.143	28.11	0.69	4.800	1.00
67.803960	0.0012	8.735	28.58	0.69	4.930	0.82
119.995940	0.0008	8.356	29.48	0.70	4.780	0.79
183.310091	2.4200	0.668	30.50	0.64	5.300	0.85
321.225644	0.0483	6.181	23.03	0.67	4.690	0.54
325.152919	1.4990	1.540	27.83	0.68	4.850	0.74
336.222601	0.0011	9.829	26.93	0.69	4.740	0.61
380.197372	11.5200	1.048	28.73	0.54	5.380	0.89
390.134508	0.0046	7.350	21.52	0.63	4.810	0.55
437.346667	0.0650	5.050	18.45	0.60	4.230	0.48
439.150812	0.9218	3.596	21.00	0.63	4.290	0.52
443.018295	0.1976	5.050	18.60	0.60	4.230	0.50
448.001075	10.3200	1.405	26.32	0.66	4.840	0.67
470.888947	0.3297	3.599	21.52	0.66	4.570	0.65
474.689127	1.2620	2.381	23.55	0.65	4.650	0.64
488.491133	0.2520	2.853	26.02	0.69	5.040	0.72
503.568532	0.0390	6.733	16.12	0.61	3.980	0.43
504.482692	0.0130	6.733	16.12	0.61	4.010	0.45
547.676440	9.7010	0.114	26.00	0.70	4.500	1.00
552.020960	14.7700	0.114	26.00	0.70	4.500	1.00
556.936002	487.4000	0.159	32.10	0.69	4.110	1.00
620.700807	5.0120	2.200	24.38	0.71	4.680	0.68
645.866155	0.0713	8.580	18.00	0.60	4.000	0.50
658.005280	0.3022	7.820	32.10	0.69	4.140	1.00
752.033227	239.6000	0.396	30.60	0.68	4.090	0.84
841.053973	0.0140	8.180	15.90	0.33	5.760	0.45
859.962313	0.1472	7.989	30.60	0.68	4.090	0.84
899.306675	0.0605	7.917	29.85	0.68	4.530	0.90
902.616173	0.0426	8.432	28.65	0.70	5.100	0.95
899.306675	0.0605	7.917	29.85	0.68	4.530	0.90

:

The dry air continuous absorption caused by atmospheric pressure nitrogen absorption and Debye absorption is:

$$N_D^{″}(f)=fp {\theta }^2\left[{\displaystyle \frac{6.14\times {10}^{-5}}{z\left[1+{\left(\frac{f}{z}\right)}^2\right]}}+{\displaystyle \frac{1.4\times {10}^{-12} p {\theta }^{1.5}}{1+1.9\times {10}^{-5} f^{1.5}}}\right]$$

(15)

In the formula, the width coefficient in the Debye absorption spectrum is:

$$z=5.6\times {10}^{-4}p{\theta }^{0.8}$$

(16)

After obtaining the high-precision atmospheric attenuation rate, the atmospheric absorption attenuation calculated by line-by-line numerical accumulation is given by the following formula:

$$A_{gas}={\displaystyle \sum _{n=1}^k{a}_n{\gamma }_n}$$

(17)

In this formula, ${\gamma }_n$ represents the atmospheric attenuation rate calculated through the given equation, where n is the number of stratified layers. $k$ denotes the path length of the radio wave within the i-th layer. Here, instead of applying the ITU-R inclined path calculation method, we use the approach in Equation (3) to determine each layer’s path length. This enables us to quantify atmospheric absorption attenuation at every propagation step, ultimately accumulating these values line by line to obtain high-precision total attenuation for the entire beyond-the-horizon link.

4. Comparative Results Analysis and Visual Demonstration

4.1. Comparative Analysis of Atmospheric Absorption and Attenuation

Based on meteorological sounding data, this study employs both traditional horizontal path models and an enhanced high-precision model to predict atmospheric absorption attenuation in lines of sight beyond atmospheric ducts. The comparison of computational results between the two algorithms reveals the extent of the improvement achieved by the high-precision model. Figure 2 illustrates the temperature, water vapor density, and modified refractive index profiles measured in Qingdao on 8 June.

The modified refractive index profile reveals at least two distinct negative gradient layers along the vertical direction, indicating the stratification of composite atmospheric ducts. The surface duct layer is located at approximately 275 m above sea level, while the suspended duct layer occurs at around 2000 m. This anomalous refraction primarily results from the abrupt decrease in water vapor concentration and temperature inversion within the corresponding stratified layers, as illustrated in the figure.

The signal source is placed at 10 m and 1800 m above the ground, and the signal frequency is 10 GHz. The atmospheric absorption attenuation of the beyond-line-of-sight link is simulated using the high-precision ray trajectory model and the simplified horizontal path model.

Figure 3 demonstrates that atmospheric absorption attenuation in surface ducts is significantly higher than in elevated ducts. This occurs because surface ducts are positioned at lower altitudes, trapping electromagnetic waves within near-surface (or near-sea) layers where water vapor density is markedly higher than in suspended waveguides, resulting in more pronounced atmospheric absorption effects. Analysis of both models reveals that simplified models produce larger predictions than high-precision models, with the difference linearly increasing with distance. This stems from the simplified model’s approach of treating water vapor density as a constant value at the signal emission point and approximating path length as horizontal ground distance. In contrast, the high-precision model employs a periodic jump curve trajectory, with refined stratified processing of propagation steps. It also accumulates key atmospheric parameters (temperature, water vapor density, refractive index) along the path height. Consequently, as distance increases, the discrepancy between high-precision and simplified models becomes more pronounced. Below is the attenuation error distribution of the simplified model compared to the high-precision model for horizontal paths:

Figure 4 shows that the absolute attenuation error of the simplified model increases with the propagation distance, and can reach nearly 2 dB at 500 km. Especially for the elevated duct, the relative error is significantly larger than that of the surface duct because the elevated duct usually exists at an altitude of several kilometers where the temperature and water vapor density are relatively low; this results in the actual atmospheric absorption attenuation being significantly smaller than that in the lower atmosphere. However, the simplified model still uses traditional ground average parameters for statistical evaluation and calculation. Therefore, using the simplified model to calculate the absorption attenuation in elevated ducts leads to a larger error than that in the lower atmosphere, and the error on the propagation path in elevated ducts can reach nearly 40%.

Oxygen and water vapor in the troposphere are the primary absorbers of radio waves. The atmospheric absorption attenuation caused by these gases typically increases with frequency, accompanied by multiple resonant absorption frequencies. In the frequency band below 350 GHz, oxygen exhibits a series of absorption lines around 60 GHz, forming an oxygen absorption band, with an isolated absorption line at 118.74 GHz. Water vapor absorption lines are observed at 22.3 GHz, 183.3 GHz, and 323.8 GHz. Using two models, we calculated the frequency-dependent atmospheric absorption attenuation at 500 km altitude in two types of atmospheric duct stratification layers, as shown in Figure 5 and Figure 6:

The prediction results demonstrate that long-distance atmospheric absorption attenuation in duct stratification significantly exceeds that in normal atmospheric environments, particularly in surface duct stratification, where it can reach approximately 20 dB in the water vapor absorption window (22 GHz). The simplified model’s predictions surpass those of the high-precision model across most frequency bands, with the latter’s improvement becoming more pronounced as frequency increases. Specifically, the average error in surface duct stratification is 26.3 dB (relative error: 14.9%), while that in elevated duct stratification is 31.7 dB (relative error: 27.9%). The enhancement in atmospheric absorption attenuation prediction for elevated ducts is particularly notable.

4.2. Visual Realization of Atmospheric Absorption Attenuation Along the Track of Over-the-Horizon Rays

To more clearly demonstrate the variation trend of electromagnetic wave absorption attenuation within atmospheric duct stratification, this paper combines ray tracing with atmospheric absorption attenuation calculations to visually represent how low-angle electromagnetic wave attenuation changes along over-the-horizon ray paths in an atmospheric duct environment. First, it presents the visualization results of atmospheric absorption attenuation at a signal frequency of 10 GHz, with the emission source positioned at different angles within the elevated duct layer.

In Figure 7, Figure 8 and Figure 9, the solid black lines represent duct top height, and the dashed black lines represent duct bottom height.

Figure 7 illustrates atmospheric absorption attenuation values using gradient colors from low to high. As shown, low-angle rays trapped by the duct form over-the-horizon propagation within it, with increasing atmospheric attenuation along their trajectories. High-angle rays penetrating the duct layer establish air-to-air links, while larger negative-angle rays reflect off the ground to form ground-to-air links. At long distances, absorption attenuation can exceed 5 dB.

Figure 8 visualizes atmospheric absorption attenuation along the ray trajectories of the emission source at different elevation angles within the surface duct layer.

In Figure 8, the low-angle rays propagate over the horizon in the surface duct, forming a ground-to-ground link. The absorption attenuation can exceed 10 dB over long distances, with a clear visual effect.

To compare the differences in atmospheric absorption attenuation among various types of atmospheric ducts, Figure 9 illustrates how atmospheric absorption attenuation varies with the ray trajectory when the emission source is launched at a 0° elevation angle at different altitudes.

The diagram depicts three duct layers, with surface ducts exhibiting maximum attenuation with distance. Elevated ducts at about 2000 m show moderate attenuation, while those at about 3200 m demonstrate minimal attenuation. Electromagnetic wave rays outside the duct layers exhibit varying degrees of curvature due to differing refractive index gradients at different altitudes. Significant curvature occurs when the gradient is pronounced, resulting in longer propagation paths and greater distances in the lower atmosphere. Conversely, smaller gradients lead to more linear ray trajectories, shorter propagation paths, and closer distances in the lower atmosphere. The atmospheric absorption attenuation along these paths remains relatively low, mostly within 2 dB.

5. Simulation and Accuracy Comparison Analysis of Path Propagation Loss

To further illustrate the accuracy of the high-precision implementation method, we use the original meteorological sounding data measured at sea level as the environment input data we use the American AREPS(3.6) software to calculate the propagation loss, and use this as the reference value to verify the accuracy of the new algorithm in this paper. The APM model of AREPS software is the only electromagnetic wave propagation model approved by N096/N61 that is applicable to the 100 MHz–57 GHz frequency band. This model combines geometric optics and the parabolic equation model, and can also include the absorption effects of oxygen and water vapor, forming an accurate hybrid model [29]. Figure 10 is the AREPS simulation of path propagation loss in the elevated duct provided by the software.

AREPS already has a large number of registered users (782 as of June 2002). Figure 11 show the users’ distribution of AREPS soft. Its representative users include the Coast Guard’s Multi-Sensor Performance Prediction (MSPP) tool, the Navy’s Integrated Tactical Environmental Support System, the North Atlantic Treaty Organization (NATO) Joint Environmental Support System, the Royal Australian Air Force’s Tactical Decision Aid, the 84th Radar Evaluation Air Force Squadron—Coastal Defense Radar Equipment, the Federal Aviation Administration, the GPS Joint Program Office, the Army Space Command, the National Science Foundation, the Antarctic Program—Microwave Satellite Communications, etc. [29].

In the process of comparative analysis, we also take the free space attenuation into account and combine it with the atmospheric absorption attenuation to form the propagation loss of the ray during propagation; we compare its accuracy with the output results of the AREPS software to verify the accuracy of the algorithm in this paper.

We replace the straight-line distance with the actual path length of the ray and use the free-space attenuation model to calculate the free-space attenuation [29]:

$${\mathrm{A}}_{free}=32.45+20\lg f+20\lg d$$

(18)

In Formula (17), $f$ is the frequency in MHz; $d$ is the actual path length of the ray in km, which is obtained through step-by-step calculation using Formula (3) to replace the original straight-line distance. We synthesize and accumulate the atmospheric absorption attenuation $A_{gas}$ and free-space attenuation ${\mathrm{A}}_{free}$ to obtain the total path propagation attenuation of the ray:

$${\mathrm{A}}_{\mathrm{tot}}={\mathrm{A}}_{gas}+A_{free}$$

(19)

Let the frequency be 4 GHz; we still select the radiosonde meteorological data of Qingdao area on 8 June in Figure 2 as the environmental input, with the antenna height of 800 m. The ray path attenuation at 0° elevation angle is calculated using the high-precision model, simplified model, and AREPS model in this section. The calculation results are shown in Figure 12:

Figure 12 shows the total path attenuation calculated by different models. It can be seen from Figure 12 that the high-precision model in this paper has a significantly higher degree of agreement with the calculation results of the American AREPS software than the simplified model; especially, the total attenuation values at close distances almost completely overlap. Taking the calculation results of the AREPS software as the reference values, we calculated the errors between the high-precision model and the simplified model versus the reference values, as shown in Figure 13:

It can be seen from Figure 13 that the calculation accuracy of the new high-precision model proposed in this paper is significantly better than that of the simplified model, and with the results of AREPS software as the reference value, the path attenuation errors of the high-precision model and the simplified model are 0.2 dB and 1.4 dB, respectively. Since ray optics cannot calculate the multipath fading and scattering effects, there is a large error compared with the AREPS calculation results of nearly 80 km. However, it is still near the statistical mean of the fading fluctuation amplitude and is closer to the median of the fading depth compared with the simplified model. The improvement includes not only the contribution from the gradual refinement of the calculation of meteorological parameters in the atmospheric absorption attenuation model, but also the contribution from the high-precision calculation of path length in both the absorption model and the free-space attenuation model.

However, since this method is based on the calculation of propagation attenuation along the ray trajectory path, it mainly focuses on the fast propagation link prediction under atmospheric refraction and absorption effects. Moreover, the calculation process of each ray is relatively independent, so it cannot predict the electromagnetic wave interference effect on large areas such as the APM model; therefore, the high-precision model proposed in this paper is only high precision relative to the simplified model in terms of current international standards; its calculation accuracy is comparable to the APM model in non-interference regions, and still has limitations in interference regions.

Figure 14 displays total path attenuation with ray trajectories, in which the elevation angle ranges are from −1° to 1°.

Considering free-space attenuation, the total path attenuation can be visible with trajectories; we can not only use it as a standalone tool but also in a wider range of applications, so that the method in this paper can be used as a component within a broader path loss estimation framework.

6. Conclusions

This study employed ray tracing technology based on Taylor’s second-order approximation to accurately predict low-angle ray trajectories within atmospheric ducts. During line-by-line calculations, atmospheric environmental parameters at ray passage altitudes are extracted, enabling refined accumulations of atmospheric absorption attenuation. A high-precision prediction model for atmospheric absorption attenuation in over-the-horizon propagation links is established, with visualization of attenuation variations along ray trajectories in atmospheric duct environments. Comparative analysis with current ITU-R Recommendation P.676 and the horizontal-path atmospheric absorption attenuation model from the GJB Handbook of Radar Wave Propagation, Refraction and Attenuation for Atmospheric duct Environments demonstrates that the proposed model improves prediction accuracy by approximately 15% for surface duct stratification and 28% for elevated duct stratification. To further illustrate the accuracy of the high-precision implementation method, we use the original meteorological sounding data measured at sea as the environment input data and use the American AREPS software to calculate the propagation loss, and use this as the reference value to verify the accuracy of the new algorithm in this paper. The path attenuation errors of the high-precision model and the simplified model are 0.2 dB and 1.4 dB, respectively. The improvement includes not only the contribution from the gradual refinement of the calculation of meteorological parameters in the atmospheric absorption attenuation model, but also the contribution from the high-precision calculation of path length in both the absorption model and the free-space attenuation model.

For the microwave over-the-horizon propagation effect in environments such as atmospheric ducts, traditional ray tracing algorithms are usually used to predict parameters such as ray trajectory, time delay, and grazing angle, as well as to conduct qualitative analysis of coverage. This study proposes a deterministic calculation method for propagation attenuation using ray trajectory as the carrier. It accurately calculates the ray trajectory parameters through the known distribution of atmospheric environmental parameters and uses the high-precision atmospheric absorption attenuation model and free space attenuation model to quantitatively calculate the propagation attenuation along the ray path.

The method proposed in this paper makes up for defects such as the low accuracy of statistical models and the slow calculation speed of the APM deterministic model, but there are still some limitations such as: it cannot predict the electromagnetic wave interference effect, and is just high precision relative to the simplified model in terms of the current ITU recommendation and GJB. Its calculation accuracy is comparable to the APM model in non-interference regions but weaker in interference regions. Therefore, our method’s advantages mainly lie in its more intuitive and clear visual demonstration, as well as its fast and relatively accurate propagation link calculation.