The parabolic equation (PE) is one of the most commonly used mathematical tools for solving electromagnetic propagation problems in complex atmospheric refraction and irregular terrain scenarios.

The PE can be described as (Guo et al., 2021; Li et al., 2011): ∂ 2 u x , z ∂ z 2 + 2 i k ∂ u x , z ∂ x + k 2 m 2 − 1 u x , z = 0 , (1) where x and z represent the horizontal transmission distance of the sea surface and the altitude above the sea surface, respectively. u (x, z) is the electromagnetic attenuation function (it can be expressed as an electric field component in a horizontally polarized electromagnetic field or a magnetic field component in a vertically polarized electromagnetic field). k is the number of waves, and m is the atmospheric correction exponent of the refractive index.

It is impractical to directly solve the parabolic equation; thus, the split-step Fourier transform (SSFT) algorithm (Valtr and Pechac, 2007) was used to solve the discrete solution for reducing computational complexity. Assuming that the field distribution at the current position is known, the SSFT algorithm was used to solve the field distribution at a certain distance, and using recursive relations, we solved the field distributions at all positions.

Defining u (x ₀, z) as the field distribution at x ₀, the SSFT was used to obtain (Levadnyi et al., 2011): u x 0 + Δ x , z = exp i k 2 m 2 − 1 Δ x ⋅ F − 1 exp − i p 2 Δ x 2 k F u x 0 , z , (2) where p ≜ k sin θ, and θ is the wave velocity direction angle. F [⋅] and F ⁻¹ [⋅] represent the Fourier transform and inverse Fourier transform, respectively. Using the SSFT algorithm, u (x ₀ + 2Δx, z), u (x ₀ + 3Δx, z) etcetera can be worked out by iterating step size Δx. Then, the concerned field distribution at a position can be determined by changing the initial value u (x ₀, z), step size Δx, and times of iteration.

The performance of ducts is affected by the atmospheric refractive index under an evaporation duct environment over the sea. The formula for atmospheric refraction is as follows (Guo et al., 2014): N = 77.6 P T − 5.6 e T + 3.75 × 1 0 5 e T 2 , (3) where N and P represent atmospheric refractive index and atmospheric pressure, respectively. Moreover, e and T represent water vapor pressure and temperature, respectively.

Considering the influence of the curvature of the Earth, the corrected atmospheric refractive index is expressed as follows (Forster and Riechen, 2006): M = N + 0.15 z , (4) where M represents the corrected atmospheric refractive index. The relationship between M and m (the atmospheric correction exponent of the refractive index), can be described as (Smith and Weintraub, 1953): M = m − 1 × 10 6 . (5) Numerous prediction models are used for evaporation ducts. In recent years, the Paulus-Jeske (P–J) (Grimes and Hackett, 2014) model and Naval Postgraduate School (NPS) (Zhang et al., 2010) model have been widely used. This study used the NPS model to analyze the height and refractive index of the evaporation duct. In this model, the atmospheric temperature, humidity, wind speed, relative humidity, and other environmental parameters at a certain height were input, and then, the atmospheric refractive index profile of the evaporation duct is was obtained according to Eqs 3–5. The height of the evaporation duct was obtained by using the minimum corrected refractive index (Guo et al., 2014) as shown in Figure 2.

Evaporation ducts can be divided into three states, i.e., stable, neutral, and unstable states. Particularly, when the air-sea temperature difference (ASTD) is < 0 , it is an unstable state; when the ASTD is > 0 , it is a stable state; when the ASTD is = 0, it is a neutral state (Wang et al., 2020). Herein, we mainly studied the influence of relative humidity on electromagnetic propagation in stable and unstable states.

It was assumed that the input parameters of the NPS model are as follows: 1) The electromagnetic frequencies are 5, 10, and 15 GHz. 2) The transceiver antenna was 4 m high. 3) The antenna was in the horizontal polarization mode. 4) The wind speed at 10 m above sea level was 10 m/s 5) ASTD < 0 (the atmospheric and sea surface temperatures were 26 and 28°C, respectively), and ASTD > 0 (the atmospheric and sea surface temperature were 28 and 26°C, respectively). 6) The atmospheric pressure was 1,011.9 hPa. The path loss at the horizontal distance of 100 km at different electromagnetic frequencies and different relative humidity was obtained, as shown in Figure 3.

As shown in Figure 3, the path loss versus relative humidity curve is insensitive to the transmission frequency value. When the relative humidity is over 40%, with an increase in the relative humidity, the path loss remains stable at first and then gradually increases. By comparing Figures 3A,B, it can be seen that the maximum relative humidity under an unstable state is higher than that under the stable state, and the higher the frequency of the electromagnetic wave, the greater the tolerable maximum relative humidity. With an increase in relative humidity, the increase in the path loss rate in the stable state is greater than that in the unstable state, and the higher the frequency of the electromagnetic wave, the greater the path loss. Therefore, when the relative humidity is over 70%, higher transmission frequencies can be used to achieve a longer distance transmission, owing to the considerably smaller path loss. When the relative humidity is below 65%, a lower transmission frequency results in a lower loss value. However, under the condition of extremely high relative humidity, the path loss rapidly increases, and the height of the evaporation duct is significantly low, so the evaporation duct cannot be used to achieve reliable communication on the sea.

Wind speed does not directly affect the intensity of the evaporation duct, but it can accelerate the air-sea exchange rate and thus indirectly affect the intensity of the evaporation duct, subsequently affecting electromagnetic propagation.

It was assumed that the relative humidity at 10 m above sea level was 40%; ASTD < 0 (the atmospheric and sea surface temperatures were 13 and 14°C, respectively) or ASTD > 0 (the atmospheric and sea surface temperatures were 14 and 13°C, respectively), and other parameters were set as in Section 3.1. The path loss was obtained at a distance of 100 km in the horizontal direction under different transmit frequencies and different wind speeds, and the results are shown in Figure 4.

As shown in Figure 4, the influence of wind speed on electromagnetic wave path loss is significantly different under stable and unstable states. In the unstable state, when the wind speed is low, the path loss is considerably high. With an increase in wind speed, the path loss decreases rapidly. When the wind speed exceeds a certain value, the path loss remains almost stable. This is because in the unstable state, the height of the evaporation duct increases with increasing wind speed. In particular, when the wind speed is 20 m/s, the height of the duct is nearly 30 m, and the performance of the evaporation duct is satisfactory. Under the stable state, the path loss tends to increase with increasing wind speed because the height of the evaporation duct decreases with increasing wind speed, the strength of the evaporation duct decreases, and then, the path loss increases.

Comparing Figures 4A,B, it can be seen that in the unstable state, when the wind speed is high, the evaporation duct can be used to achieve long-distance communication on the sea; in the stable state, the electromagnetic wave of 10 GHz can be used to achieve wireless communication on the sea, because this frequency is insensitive to wind speed. Under this frequency, the communication is more stable, which is more suitable for long-distance wireless communication.

In this section, it is assumed that the environmental factors were maintained constant except for the atmospheric temperature and sea-surface temperature. The relative humidity was 40%. The wind speed was 10 m/s. The temperature setting is as shown in Table 1, and other parameter settings are the same as those in Section 3.1.

To analyze the influence of temperature on electromagnetic propagation under the unstable and stable states, it is assumed that the sea-surface temperature gradually varies from 5 to 35°C, the atmospheric temperature varies with the sea-surface temperature, and the ASTD remains constant. The influence of sea-surface temperature, atmospheric temperature, and ASTD on the electromagnetic propagation of 15 GHz electromagnetic waves is obtained, and the results are shown in Figure 5.

Comparing Figures 5A,B, it can be seen that under the condition that the ASTD remains unchanged when the sea-surface temperature or atmospheric temperature is higher or lower, the path loss fluctuates. When the sea surface temperature is between 20 and 27°C, the path loss is relatively stable and low.

Considering the influence of ASTD on electromagnetic propagation, the trend of the four curves under the unstable state is closer than that under the stable state. That is, the influence of ASTD on path loss under unstable states is moderately small. When the sea-surface temperature is below 20°C, the smaller the temperature difference, the lower the loss, and the smaller the fluctuation. When the sea-surface temperature is over 27°C, the smaller the temperature difference, the greater the path loss, and the greater the fluctuation. Under stable states, the minimum average loss can be obtained when the ASTD is 2°C. Only considering the influence of temperature on electromagnetic propagation, the evaporation duct can be selected to realize wireless communication over the sea under the sea-surface temperature of 15 ∼ 28°C.

The influence of relative humidity, wind speed, temperature, and ASTD on path loss is analyzed in this section. It can be seen that under the condition of evaporation duct, the environmental factors significantly affect the strength of evaporation ducts. According to the simulation results, the following conclusions can be drawn. Under the unstable state, the higher the wind speed, the lower the relative humidity, and the higher the temperature, the smaller the path loss. That is because this climate condition is more conducive to the evaporation of sea water and the formation of a stronger evaporation duct layer. Under the stable state, it is easier to form a stronger evaporation duct layer if the relative humidity is lower, the wind speed is lower, the temperature is higher, and the ASTD is higher, consequently reducing path loss. Observing the results presented in this section, it can be determined whether the evaporation duct can be used to achieve wireless communication on the sea according to the actual climate conditions.

Similar to time division multiplexing (TDM) and frequency division multiplexing (FDM) technologies, code division multiplexing (CDM) technology divides the physical channel into several code channels with little correlation or orthogonal to each other in the code domain, and the users transmit different data symbols on different code channels for achieving the parallel transmission of data symbols and increasing the data rate. The m-sequence codebook is provided in Figure 8. As shown in Figure f8a, c _m = (c _m1, … , c _mn , … , c _mN ) is the spread spectrum code-sequence on the mth code channel. The value of m ranges from 1 to M. M represents the number of code channels. N represents the spreading code length of each code channel, that is, each spreading code sequence includes N chips. The nth chip on the mth code channel of the m-sequence is represented as c _mn (m = 1, … , M, n = 1, … , N).

In this study, m-sequence was used in CDM in the form of a single-code multiple stream, that is, only one m-sequence was required to be generated, and the spreading code sequences of all the code channels can be obtained by the cyclic shift of m-sequence. However, it is required to ensure that the number of multiplexed code channels and the code length satisfy M ≤ N. If M > N, more than two code channels will have the same spreading code sequence, thus violating the principle of code division multiplexing. Provided that the spreading code sequence of the mth code channel is c _m = (c _m1, … , c _mn , … , c _mN ), the spreading code sequence of the (m + l) th code channel, that is, c _m+l = (c _m(N−l+1), …, c _mN , c _m1, … , c _m(N−l)), can be obtained via cyclically shifting the mth spreading code, c _m , to the right by l bits, where l takes the value from (− m + 1, N − m). The codebook of the m-sequence is described below by considering N = 7 and M = 5 as an example. Assuming that the m-sequence is c ₁ = (+1, +1, −1, −1, −1, +1, −1) when N = 7, the codebook is described in Figure 8B.

As shown in Figure 8B, c ₂, c ₃, c ₄, and c ₅ are the spreading code sequences obtained by shifting c ₁ by 1, 2, 3, and 4 bits, respectively, to the right.

In the communication system, CDM is generally realized using the spread spectrum technology. Figure 9 shows the transceiver block diagram of a BPSK direct-sequence spread-spectrum signal.

As shown in Figure 9, b = (b ₁, … , b _m , … , b _M ) is a binary data symbol, distributed to different code channels for parallel transmission after serial-to-parallel conversion, that is, b ₁ ∼ b _M are respectively transmitted on the first code channel, and b ₂ is transmitted on the 1st ∼ Mth code channel. The spreading code sequence of each code channel is c ₁, c ₂, … , c _M , and the signal at the transmitter after spreading processing is expressed as: s = ∑ i = 1 M b i c i . (6)

After the channel effect, the received signal is expressed as: s ′ = h ∑ i = 1 M b i c i + n , (7) where h represents the channel gain. Assuming that the channel gain is the same for each chip symbol, a scalar can be uesd to represent the channel gain. n represents the noise; n = (n ₁, … , n _n , … , n _N ), where n _n represents the noise on the nth chip. For the convenience of research, it is assumed that the statistical characteristics of the noise superimposed on each chip are the same and are all represented by n.

The signal representation at the receiver is described below by taking the κth path as an example. The received signal can be expressed as: s ′ = h b κ c κ + h ∑ i = 1 , i ≠ κ M b i c i + n , (8) where the value of κ ranges from 1 to M. The κth receiver signal after the despreading is expressed as: b κ ′ = s ′ c κ T = h b κ c κ c κ T + h ∑ i = 1 , i ≠ κ M b i c i c κ T + n c κ T , (9) where b κ ′ comprises three parts. The first part can obtain the originally transmitted signal b _κ . The second part represents the inter-symbol interference (ISI) of other symbols on the κth signal. The third part is additive noise.

The autocorrelation function of an m-sequence with the period N is: r c = c i c i + τ T = c i c j T = N i = j − 1 i ≠ j (10) where τ is the number of bits of the cyclic shift and τ = j − i. From the balance of m-sequences (the number of 1 s is always one more than the number of 0 s in the maximum length sequence), and there is one more −1 than +1 in c _i .

The solving process of the received signal-to-interference plus noise ratio (SINR) of the κth channel signal is given below. Assuming that the system uses a favorable power control, and the total power of the transmitter is S, the transmit power allocated to each symbol is identical (P _t = S/M), then, each code channel has the same signal-to-noise ratio (SNR) at the transmitter. Assuming that the SNR at the transmitter of the mth code channel is SNR _m , SNR _m = P _t /(N ₀ B) (m = 1, … , M). N ₀ represents the noise power spectral density and B represents the signal bandwidth. Sea obstacles are sparse, as we primarily consider large-scale fading, and the relationship between the received power and the transmitted power is expressed as: P r = P t × 10 - PL / 10 . (11) where PL represents the large-scale fading in the evaporation duct channel condition, and the expression is provided below. Finally, the SINR of the κth path signal is obtained according to (9): SINR κ = N P r M − 1 × P r + N 0 B (12)

Example: Assuming that N = 7 and M = 5, the m-sequence codebook is shown in Figure 8. For simplicity, it is assumed that the data symbols on each code channel are b ₁ = a, b ₂ = b, b ₃ = c, b ₄ = d, b ₅ = e, and the value is +1 or −1. The transmit signal is expressed as: s = + a − b + c − d − e + a + b − c + d − e − a + b + c − d + e − a − b + c + d − e − a − b − c + d + e + a − b − c − d + e − a + b − c − d − e T (13)

The receiver signal is expressed as: s ′ = h s + n = + a − b + c − d − e × h + n + a + b − c + d − e × h + n − a + b + c − d + e × h + n − a − b + c + d − e × h + n − a − b − c + d + e × h + n + a − b − c − d + e × h + n − a + b − c − d − e × h + n T (14)

Taking the first channel as an example, the received signal after despreading is expressed as: b 1 ′ = a ′ = s ′ c 1 T = 7 h a + h − b − c − d − e + − n (15) where the first part is the original symbol, the second part is the ISI, and the third part is noise. According to this example, the signal representation form at the receiver can be expressed simply, which is convenient for understanding.

It was assumed that the atmospheric environment is suitable, the electromagnetic wave frequency was 15 GHz, and other environmental parameters were set as in Section 4. The path loss is written as: PL = 4.419 × ln d + 112.74 dB . (16) where d is the transmission distance. The formula can approach the be used to obtain curve with R ² = 0.9759 (the closer R ² is to 1, the closer the two curves are). The fitted curve and the approximate curve are shown in Figure 10. The fitted curve is given by Eq. 16, and the approximate curve is drawn by the actual path loss. If the value where the path loss changes sharply is ignored, it can be determined that the error between the fitted curve and the approximate curve is minuscule. Therefore, we can use the fitted curve instead of the approximate curve as the path loss model under the evaporation duct environment.

The channel gain under the evaporation duct condition can be expressed as Eq. 16. Then, the link capacity of the CDM system under this channel condition is expressed as: C = ∑ κ = 1 M B × log 1 + SINR κ . (17)

In this study, it is assumed that the number of code channels and the code length are identical, M = N = 2046, and the system bandwidth B = 50 MHz. Then, the influence of the atmospheric environment, transmission distance, antenna height, and electromagnetic wave frequency on the system link capacity was studied, and the influence of code length on the link capacity was analyzed. The simulation curve is shown in Figure 11.

Figure 11A shows that the link capacity is approximately identical in both states when the relative humidity is between 30% and 55%, and the link capacity in the unstable state is greater than that in the stable state when the relative humidity is lower than 30% or higher than 55%. As shown in Figure 11B, the link capacity is significantly affected by the wind speed in the unstable state. In particular when the wind speed is high or low, the link capacity does not dramatically decrease. As shown in Figure 11C, in both states, the impact of sea surface temperature on the link capacity is approximately the same, except that the temperature is 11°C. Moreover, there is a stable and high link capacity at a temperature of 15 ∼ 27°C. Figure 11D shows the influence of transmission distance on the link capacity when the SNR at the transmitter is 30 dB. Without the CDM technology, the link capacity of the system is considerably low, as the transmission rate of a single code channel is 1/N times the transmission rate of the N parallel code channels. Particularly, the transmission rate at 100 km is only a few bps; thus, it is impossible to achieve maritime communication. When the code division multiplexing is adopted, the link capacity of the system is improved. In addition, the greater the number ofcode channels, the lower the impact of distance on link capacity. Therefore, increasing the number of code channels can help overcome the limitation of low link capacity caused by long transmission distances. As shown in Figure 11E, higher link capacity can be obtained using the evaporation duct channel model rather than the free space model, and with an increase in frequency, the link capacity shows a wave-like decline over the observation zone. Let the height of the receiver antenna be 4 m, and the heights of the transmit antenna be 4, 8, 12, and 16 m. A high link capacity can be obtained only when the antennas at the transceiver are highly matched. As shown in Figure 11F, in this part, we give a comparative simulation analysis of the performance of CDM and FDM. We consider that the filters has the ideal bandpass characteristics and no interference between sub-band channels. Then we analyze the link capacity performance of both with the same number of multiplexed band channels and code channels. Since FDM technology must have a guard interval (GI) between adjacent sub-band channels to ensure that there is no interference between sub-band channels, when GI = 0.1, the guard interval is 5 MHz, the link capacity of FDM is similar to CDM, but with the increase of GI, CDM has superior performance.