A long-term over-the-horizon microwave propagation link was established over the Yellow Sea for a 54-day period. The measurements were carried out from September 20 to November 22, 2021, with interruption during October 1–10 when the system was off. The measurement location is presented in Figure 1A . We have specially developed X-band EM propagation systems that can operate all day long to automatically record data of propagation characteristics. The PL data and meteorological parameters such as wind speed (WS), air temperature (AT), relative humidity (RH), and air pressure (AP) were recorded. For the measured data used in the current study, the receiver (Rx) was deployed in Rizhao City (S1), Shandong Province, and the transmitter (Tx) was deployed in Yancheng City (S2), Jiangsu Province. The length of the propagation link was approximately 133 km. For the antenna heights in this measurement, the line-of-sight horizon was   D = 4.12 × ( h t + h r ) = 4.12 × ( 6 + 3 ) = 17.2 km in a standard atmosphere, where h _t is the Tx height, and h _r is the Rx height. The link length was approximately 7.7 times the line-of-sight length. As shown Figure 1A , the P1, P2, P3, and P4 are the reanalysis meteorological data grid points along the propagation link, which were used for simulation analysis.

Highly integrated and automated design enable the propagation system to operate for a long time in the marine environment. Photographs of the EM propagation systems are shown in Figures 1B, C . Figure 1D shows a schematic of this system. The EM propagation system is a combination of transceivers, and differentiates itself from other systems by setting different signal frequencies. The Tx includes a signal generator connected through an isolator to a power amplifier, and then to an omni antenna. The Rx includes a horn antenna connected through an isolator to a low noise amplifier, and then to a spectrum analyzer. The system configurations are shown in Table 1 .

The industrial computer inside is used for sending the configuration instructions to the signal generator and collecting the data from the spectrometer. The received signal level (RSL) was recorded at approximately 1-s intervals using the internal computer. The RSL was mainly influenced by three aspects. 1) The change of the duct environment during the measurement; 2) the fading caused by the multipath effect in the duct; 3) the fading caused by passing ships along the propagation path due to the low height of the transmission channel. In addition, the received signal also includes noise when the over-the-horizon propagation cannot be realized due to poor duct environments. The existing measurement data can’t well deduce the small-scale fading channel properties. In order to statistically analyze the connectable probability of this over-the-horizon link, and to provide reference for other electromagnetic systems, the PL (dB) was converted from the RSL (Yang et al., 2022b) using the system parameters shown in Table 1 and the following equation:

where P _t is the Tx power (42 dBm), G _t is the Tx antenna gain (7 dBi), G _r is the Rx antenna gain (20 dBi), P _r is the Rx amplifier gain (30 dB), SL is the system loss (10 dB), and RSL (dB) is the received signal level.

Reanalysis data were used to determine the variations of the meteorological parameters and to simulate the long-term inhomogeneous evaporation duct environments for the analysis of over-the-horizon EM propagation. This study used ECMWF ERA5 data (Hersbach et al., 2020, 5), which is the fifth generation of the ECMWF global climate and weather product with temporal coverage extending from 1979 to the present. The temporal resolution of the ERA5 dataset is 1 h and the horizontal resolution is 0.25° × 0.25°. ERA5 data have been widely used and verified suitable for studies on evaporation duct distributions (Sirkova, 2015; Huang et al., 2022; Wang et al., 2022a; Wang et al., 2022b). The parameters of AT at 2-m height, AP, WS at 10-m height, RH at 1000 hPa, and sea surface temperature (SST) were extracted to simulate the EDH distributions during the measurements. The data on P1, P2, P3, and P4 shown in Figure 1A were used for the EDHs simulation along the propagation link.

A long-range propagation mechanism is usually encountered at microwave frequencies owing to the trapping effect of ocean evaporation ducts. The propagation of EM radiation in the atmosphere depends upon the refractive index of air n, which varies in the troposphere due to the variation of pressure, temperature and water vapor content. The refraction index n is an electrical property of the propagation medium and is defined as the ratio between the speed of light in a vacuum, c ₀, and the speed of the wave through the medium, v. For radio waves the refractive index of the troposphere is given based on Debye theory (Debye, 1929):

where T (K) represents air temperature, P (hPa) represents the total atmospheric pressure, and e (hPa) represents the partial pressure of water vapor (Smith and Weintraub, 1953). The partial pressure of water vapor e may be derived from the specific humidity using the equation:

where q (kg/kg) represents the specific humidity, and ϵ is the ratio of the individual gas constant for dry air to that for water vapor.

In the troposphere, the refractive index varies between 1.000250 and 1.000400 n-units. Because it is so close to unity, the refractive index of the troposphere is represented by a quantity called radio refractivity N, which is given by:

Under ducting conditions, EM propagation is refracted toward the earth’s surface so that it becomes trapped in a layer. To determine these conditions, the quantity modified refractivity M which takes into account the curvature of the earth’s surface, is often used. The modified refractivity M is defined by:

Where r _e (m) is the earth’s radius and z (m) is the altitude. In the regions with a negative vertical slope of M, EM propagation is refracted toward the surface and may become trapped in a leaky atmospheric duct.

The evaporation duct exists primarily due to a significant negative vertical gradient in humidity near the sea surface. In the evaporation duct layer, the M decreases as the sea surface height increases. When a certain height is reached, the M reaches a minimum value, and then increases with height. The height corresponding to the minimum point is defined as the EDH.

In this study, an evaporation duct model was used to calculate the evaporation duct distributions using ERA5 reanalysis data covering the measurement period. Many evaporation duct models based on Monin–Obukhov Similarity Theory, including the Paulus–Jeske model (Paulus, 1985), Musson–Gauthier–Bruth model (Musson-Genon et al., 1992), Liu–Katsaros–Businger model (Babin and Dockery, 2002), Babin–Young–Carton model (Babin, 1996), and NAVSLaM (Frederickson et al., 2000, 2), have been developed to obtain M-profiles. NAVSLaM has been verified against radiosonde measurements (Anderson et al., 2004; Pozderac et al., 2018; Wang et al., 2018; Wang et al., 2019), and is considered ideal for estimating EDH (Babin and Dockery, 2002). This study used NAVSLaM to diagnose the evaporation duct distributions.

As equation (3) and (4) shows, the pressure, temperature and partial pressure of water vapor profiles are needed to calculate the M-profiles. The NAVSLaM model derive these profiles from the following equations:

where T(z) and q(z) are the temperature and specific humidity at a given altitude z above the sea surface. z _0θ and z _0q represent integration constants called the temperature and specific humidity roughness length, respectively. θ _* and q _* are the Monin-Obukhov temperature and specific humidity scaling parameters, respectively. The values of z _0θ , z _0q , θ _* and q _* are calculated using the TOGA COARE 3.0 bulk flux algorithm (Fairall et al., 2003). κ is von Karman’s constant. Γ _d is adiabatic lapse rate. L is the Obukhov length. ψ _h   denotes the temperature function. g , R , correspond to gravity acceleration, and gas constant, respectively. T _v   is the mean value of virtual temperature at height of z ₁ and z ₂ .

The processed ERA5 data were input into NAVSLaM, and then the distributions of evaporation ducts during the measurement period were obtained.

For free space microwave propagation, PL can be expressed by the free-space loss using the following equation:

where PL _FSL is the free-space PL (dB), f is the signal frequency (MHz), and d is the propagation distance (km).

For the marine environment, reflection from the sea surface cannot be ignored, and PL can be modeled using the 2-Ray model. The reflection coefficient for a vertically polarized wave approaches -1, and the 2-ray PL model can be simplified (Lee et al., 2014) as follows:

where PL _2−ray is the 2-Ray PL (dB), λ is the wavelength (m), D is the propagation distance (m), h _t is the Tx height (m), and h _r is the Rx height (m).

Long-range propagation of microwaves over the sea is greatly affected by the duct-trapping mechanism. The parabolic equation (PE) method has been widely used to predict EM propagation in the troposphere. The standard PE can be obtained from the Helmholtz equation:

where u is a scalar component of the electric field, z is the height, x is the range, k ₀ is the free-space wave number, and M is the modified refractivity. Parameter u is given by the Fourier split-step solution of PE as follows:

where F[▪]  and F ⁻¹[▪] are the Fourier transform and the Fourier inverse transform, respectively, p is the transform variable, and δx is the range increment. More detailed information on the Fourier split-step PE solution can be found in Goldhirsh and Dockery (1998). The PL expressed by the PE field function u (PL _PE ) can be calculated as follows:

Numerous models based on the PE method have been developed, e.g., the Terrain Parabolic Equation Model (Barrios, 1994), Parabolic Equation Toolbox (Ozgun et al., 2020), and Advanced Propagation Model (APM) (Barrios et al., 2002). The APM is applied in the Advanced Refractivity Effects Prediction System, which is widely used in assessment of EM instruments. Moreover, the APM has been widely used and tested in different duct environments (Barrios et al., 2002; Shi et al., 2015a; Shi et al., 2015c). In this study, we used the APM to simulate the effects of evaporation duct inhomogeneity on EM propagation.

The EM propagation systems and automatic weather stations were deployed in the coastal areas of the Yellow Sea. These systems acquired continuous data both of the PL of the 133-km over-the-horizon link at the frequency of 9.4 GHz, and of the atmosphere parameters of AP, RH, WS, and AT. Figure 2 shows the measured PL (788,806 points) during autumn 2021. It can be seen that the PL data fluctuated greatly with even 60 dB decrease from 205 dB to 145 dB, and exhibited strong diurnal variation. Owing to the limitation of the maximum power loss that can be measured with our measurement equipment, PL below the noise floor could not be monitored. As shown in Figure 2 , for the 133-km link (7.7 times the line-of-sight length), over-the-horizon propagation often occurred in different duct environments. This shows potential for application to long-range maritime broadband communication systems. Detailed statistical analysis of the PL is presented in the next section. A typical example of over-the-horizon propagation, as shown in the orange box in Figure 2 , will be analyzed in the section 3.5 to reveal the reasons for the drastic PL fluctuation of the channel.

Figure 3 shows the temporal evolutions of AP, RH, WS, and AT measured by the automatic weather stations installed at S1 and S2. It can be seen that the atmospheric parameters also fluctuated greatly with time. As shown in Figure 3A , the monitored AP was broadly similar at both locations. The RH and AT showed strong diurnal variation characteristics, as shown in Figures 3B, C , respectively. The RH varied greatly with values as low as 23% and as high as 98%. Generally, RH was low during the day and high at night, which might be attributable mainly to the light. The AT also exhibited strong diurnal variation with high temperatures during the day and low temperatures at night. The mean values of the measured AT were 23.3, 16.2, and 11.5 °C during September, October, and November, respectively, showing a general trend of decrease toward winter. As shown in Figure 3D , WS also varied greatly with values as low as 0 m/s and as high as 20 m/s, which would have marked influence on the evaporation of seawater. Overall, a large number of measured data were recorded, and both the PL and the meteorological parameters varied substantially.

The cumulative distribution function (CDF) is used in the statistical analysis of the measured PL. The CDF of the random variable  X has the following definition:

where P(X≤t) represents the probability of a random variable X which takes the value that is less than or equal to that of the t . The CDF is used to determine the likelihood that a random observation taken from the population will be less than or equal to a particular value.

Table 2 presents some basic statistical characteristics of the measured PL, and Figure 4 shows the empirical CDF of the measured PL during different months. With the maximum measurable power loss of 200 dB, our measurement system connected with a probability of 52.6% during the measurement period. The PL higher than 200dB was noise, and we only show the CDF of PL below 200dB. The PL distribution of three months in autumn is basically the same as shown in Figure 4 . The probabilities of PL of less than 200 dB were 85.9% in September. However, since the measurement in September started on September 20, 2021, and only lasted for about 10 days, the statistical results of this month may not be representative. The probabilities of PL of less than 200 dB were 48.2% in October, and 53.2% in November, respectively. The mean values of PL were 189 dB in September, 195 dB in October, and 192 dB in November, respectively. The probabilities of PL of less than 200, 190, 180, and 170 dB during the measurement period were 52.6%, 31.3%, 18.5%, and 7.3%, respectively.

For a communication system, the communication system capability A is defined as follows (Yang et al., 2021):

where P _t is the Tx power, G _t is the Tx antenna gain, G _r is the Rx antenna gain, L is receiver sensitivity, SL is the system loss, and M is the system margin.

Thus, for a communication system with system capacity of 190 dB, the available signal probability over 133 km in the Yellow Sea was approximately 31.3% during the measurement period. For a real-time communication system, this available signal probability obviously cannot meet the requirements; however, over-the-horizon communication capability can be improved by shortening the propagation distance or by increasing the communication system capability. For quasi-real-time communication systems, such as observation buoys, over-the-horizon communication could be achieved during the better communication environments when the EDH is high causing small PL and large throughput can be achieved.

The evaporation duct distributions during the measurement period were obtained using the ECMWF ERA5 reanalysis data. It can be seen from Figure 5A that EDH varied dramatically during the measurement period. However, there was little difference in EDH at different positions along the link, which reflects characteristics of horizontal inhomogeneity. The reason that PL distribution is basically the same in autumn is that the climatic conditions are basically the same, which leads to similar distribution of EDHs. Table 2 presents some statistical characteristics of the simulated EDH at P3 during the measurement. The mean EDHs were 11.5m in September, 11.1 m in October, and 9.9 m in November. Variations in EDH over the ocean come primarily from variations in four environmental parameters: SST, AT, RH, and near-surface WS. The regular monthly variation of environmental parameters leads to the monthly variation of the EDH to be general, except in some extreme weather conditions. The mean value of EDH at P3 during the measurement period was 10.6 m, which was relatively low. This led to a small probability of communication via the over-the-horizon link in autumn. Figure 5B shows the empirical CDF of EDH at P3 during the measurement period. The probabilities of EDH greater than 5, 10, and 15 m during the measurement period were 97.2%, 50.7%, and 11.9%, respectively.

The effect of evaporation ducts on the over-the-horizon propagation link are analyzed in this section. The M-profiles calculated by NAVSLaM using the ERA5 data were input into the APM to simulate PL during the measurement period. Data for sites P4, P3, P2, and P1, shown in Figure 1A , were used as inputs into the models. For clarity, we present the results of November as an example to study the influence of evaporation ducts on the over-the-horizon link. Figure 6A shows the range-dependent M-profiles along the propagation link at 08:00 on November 10, 2021. The black curves are the M-profiles at different propagation distances. The red circles are the positions of minimum values of the M, and the corresponding height is the EDHs. Figure 6B shows the PL computed using APM and ray traces computed using the ray-optics approach (Zhou et al., 2018) based on the range-dependent modified refractivity profile. The simulation parameters are shown in Table 1 . For clarity, individual rays are traced and shown in black. It is shown that the results obtained using these two approaches is consistent with each other. The rays are strongly refracted towards the earth, followed by reflection off the earth surface only to be refracted back down. In other words, the rays are trapped, resulting in low PL beyond the horizon relative to the normal atmosphere. Figure 6C is extracted from the PL computed using APM in Figure 6B by fixing the Rx height to 3m, which shows the PL vs range from 0-133km. The PL is about 129 dB at 20 km. The PL at 120 km is about 142 dB, which is only 13-dB higher than that at 20 km. The PL exponent (Rouphael, 2009) is about 1.709.

The measured PL and the simulation results obtained from different PL models are presented in Figure 7 for comparison. For the 133-km link, the PL of the free space was 154 dB obtained by equation (8), while that of the 2-Ray model was 181 dB obtained by equation (9). The free space loss model and 2-Ray model do not consider the effect of evaporation duct. The APM can reflect the trapping effect of the evaporation duct on the EM waves, and it could be used to predict the trend of the measured PL. Correlation analysis showed the correlation coefficient between the simulated PL and the measured PL is 0.57. However, sometimes the prediction error was still large. For example, the prediction errors were large during 8-12 November. Although the model estimated that over-the-horizon signal can be detected at that time, it underestimated the PL. The possible reasons for the large prediction error are: 1) input environmental data deviation. Since the ECMWF data are reanalyzed data, there may be some errors in it, resulting in large prediction errors at some points; 2) NAVSLaM and the APM model prediction errors in some environments, such as extremely high and extremely low WS, atmospheric stable conditions (air-sea temperature difference (ASTD)>0);3) lack of modeling propagation environments such as rainfall, waves, tides, and local random weather process.

Figure 8 shows the correlation between the absolute error (AE) of the predicted PL and air–sea parameters when the measured PL was<200 dB during the entire measurement period. The absolute error AE of PL is defined as: AE _PL =|PL _measured−PL _predicted| . As shown in Figure 8D , the AE of the predicted PL was larger when the predicted EDH was either too low or too high. The AE of the predicted PL was smaller when the EDH was approximately 10 m. This means that the simulation model overestimated PL when EDH was low but underestimated PL when EDH was high, which led to many extreme values in the results. The correlations between the AE of the predicted PL and both the ASTD and the WS also indicated that EDH would be large when the absolute value of the ASTD was large and the WS was high, and also EDH would be low when the absolute value of the ASTD was small and the WS was low, both leading to greater prediction error. In summary, for an over-the-horizon link, the actual structure of an evaporation duct is horizontal inhomogeneous, caused by the difference between large-scale air–sea conditions and the locally random weather process. These complex environments cannot be well modeled. Therefore, the physics prediction models show some limitations in practice, as well as problems such as poor stability and large prediction errors in general, especially for an over-the-horizon propagation link.

In this section, an example analysis explores some reasons for the decrease of PL of the over-the-horizon EM propagation link in the Yellow Sea in autumn. As shown in Figure 7 , over-the-horizon propagation occurred during November 8–12 with low PL. Figure 9 shows the evolutions of air–sea parameters during November 6–12. As shown in Figure 9F , EDH increased from 7 to 15 m during November 6–8. The main reasons for the increase in EDH were reduction in RH, increase in WS, and increase in ASTD, which promoted evaporation of seawater. The reason for this series of changes in the meteorological parameters was that the wind changed from a sea breeze to a land breeze, as shown in Figure 9C . Figure 10 shows the changes of WS and wind direction (WD) during this period. The land breeze introduced dry and cold air to the link, which promoted evaporation of seawater and reduced the PL. Analysis of the over-the-horizon propagation examples during the measurement period revealed that factors such as increase in WS and reduction in RH were the main reasons for the increase in EDH and reduction in PL.