Research on a Simulation Model of a Skywave Over-the-Horizon Radar Sea Echo Spectrum

Abstract

The capability of a skywave over-the-horizon radar (SWR) to achieve the continuous observation of a wide range of ocean dynamics parameters via a single ionospheric reflection has been demonstrated by many scholars. In order to expand the method of SWR detection of ocean dynamics parameters, a simulation model of an SWR sea echo spectrum based on the Barrick sea surface scattering cross-section model (Barrick model) and 3D ray-tracing method, suitable for a narrow-beam, frequency-modulated continuous-wave radar system (FMCW), was established. Based on this model, we simulated ideal and contaminated SWR sea echo spectra, respectively with the 3D electron density data output by the International Reference Ionosphere (IRI) model. Then, we further analyzed the effects of the grazing incidence angle, scattering angle, scattering azimuth angle and fetch length on the sea surface scattering cross-sections, the retrieval precision of the sea surface wind direction, and the root-mean-square (RMS) wave height, using the simulation data calculated by the Barrick model. The results show that these angles and fetch length cause a small expansion and contraction of the scattering cross-section, and have no influence on the retrieval precision of the sea surface wind direction, but will affect the retrieval precision of the RMS wave height; the influence of the grazing incidence angle and scattering angle is ~2.5 times that of the scattering azimuth angle. The ideal SWR sea echo spectrum has small broadening, but the ionosphere phase contamination will cause serious broadening and shifting of the SWR sea echo spectrum, and the higher order nonlinear term has greater contamination.

1. Introduction

The continuous observation of large-area ocean dynamics parameters is of great significance to the study of air-sea interaction and climate change [1]. Neither satellites nor buoys can meet the requirements of high spatial resolution or continuous observation, due to their respective limitations [2]. Compared to satellites and buoys, high-frequency over-the-horizon radars—mainly including ground-wave radars, which propagate through diffraction on the sea surface, and skywave radars (SWRs), which rely on ionospheric reflection—are not limited by the curvature of the Earth, and can achieve high-temporal- and spatial-resolution detection of large-area ocean dynamics parameters.

Since Barrick proposed the sea surface first-order and second-order cross-section model of high-frequency radars [3,4], much in the way of research has been carried out on the ocean dynamics parameter retrieval of high-frequency radars over the past 50 years, such as on wave parameter retrieval algorithms based on the second-order scattering theory [5,6,7,8,9,10], sea surface wind direction retrieval algorithms based on the ratio of the positive and negative first-order Bragg peaks [1,11,12,13,14], and sea surface wind speed retrieval algorithms based on the position of second-order spectral peaks [15], although the retrieval theory of sea surface wind speed is immature. In the development of high-frequency radars, SWRs have the characteristics of high cost and serious ionospheric contamination. Therefore, most of the above studies on ocean dynamics parameter retrieval were carried out for ground-wave radar systems [16], such as CODAR [17], WERA [18], and OSMAR2000 [12].

Compared with ground-wave radars, SWRs have a larger range and longer distance detection capability; they can detect an area of ~3000 km from the ground using a single ionospheric reflection (although a detection range of more than 6000 km can be reached with multiple ionospheric reflections, since each reflection will cause serious signal attenuation, we consider only a single ionospheric reflection) [19]. However, the ionosphere is a double-edged sword; it will cause great contamination in the amplitude and phase of SWR sea echo signals, and will decrease the accuracy of the location, where radio waves illuminate the sea surface because of the characteristics of randomness, dispersion, and time-variation of the ionosphere. Therefore, the research on SWRs is mainly transferred to the development of echo data ionosphere contamination suppression algorithms and the modeling of high-frequency radio wave transmission paths in the ionosphere [20,21,22,23,24,25,26]. Although SWRs have many problems in the detection of ocean dynamics parameters, their unparalleled advantages—such as a large area, long distance, high spatial resolution, and real-time detection—still attract researchers to study them, and their ability to detect ocean dynamics parameters has been demonstrated by many researchers. For example, Maresca and Georges demonstrated the capability of SWRs to detect ocean wave parameters before and after a cold front passes [6]; the SMB algorithm (an empirical algorithm first developed by Sverdrup and Munk, and later modified by Bretschneider), based on the empirical relationship between ocean wave parameters and sea surface wind speed, in addition to the empirical algorithm based on the relationship between the −10 dB width of the first-order Bragg peak and sea surface wind speed, demonstrated the capability of SWRs to detect sea surface wind speed [1,27,28]. In addition, many scholars have demonstrated the capability of SWRs to detect sea surface wind based on the ratio of positive and negative first-order Bragg peaks [11,13].

The above retrieval algorithms are all empirical. The existence of ionospheric contamination causes it to be difficult to establish physical theories to retrieve ocean dynamics parameters. Therefore, most of the studies on SWR remote sensing of ocean dynamics parameters are based on processing the measured data. At present, some scholars have conducted simulation research on the sea echo spectrum of SWRs. Walsh et al., from Memorial University of Newfoundland, Canada, with the source field being that of a vertically polarized pulsed-dipole antenna, derived the first-order scattering cross-section equation for mixed-path ionosphere-ocean propagation [29]. On the basis of [29], Chen et al. derived the first-order and second-order scattering cross-section equations for the FMCW system, and they carried out simulations under different wind speeds, wind directions, and ionospheric conditions [30,31,32,33]. Maresca and Georges used the Barrick model to analyze the theoretical error caused by SWR data averaging, but they did not specify whether the effects of the grazing incidence angle, scattering angle, and scattering azimuth angle were considered [6]. Paladini studied the effects of different grazing incidence angles on the first-order cross-section of SWRs based on different ocean wave spectrums in the fetch-limited Mediterranean area, but he did not analyze the effects of the scattering angle and scattering azimuth angle; in addition, his paper did not involve the analysis of second-order cross-section and radar waveform parameters, nor did he analyze the effects of these angles on the retrieval of ocean dynamics parameters [34].

In this paper, based on the Barrick model, we focus on a sea surface scattering cross-section model suitable for SWRs, which was used to analyze the effects of the grazing incidence angle, scattering angle, scattering azimuth angle, and fetch length on the first-order and second-order scattering cross-sections. The theoretical retrieval errors of the sea surface wind direction and RMS wave height caused by changes in these parameters were analyzed. Then, a simulation model of an SWR sea echo spectrum was established based on the Barrick model, 3D ray-tracing method, and frequency-modulated continuous-wave radar system (FMCW), which is suitable for narrow-beam SWRs. Finally, we simulated the SWR sea echo spectrum with and without ionosphere phase contamination using this model, based on assumed SWR parameters.

The rest of this paper is organized as follows: Section 2 introduces the theoretical models adopted in this paper, including an SWR ocean cross-section model, ocean wave spectrum models, and 3D ray-tracing method. Section 3 presents the data and simulation analysis, and a discussion of these is conducted in Section 4.

2. Materials and Methods

2.1. SWR Sea Surface Scattering Cross-Section Model

Figure 1 is a schematic diagram of SWR detection, where T is the transmitting station (SWRs can be regarded as monostatic systems), and S is the sea surface where the radio waves reach. The black, curved, and solid line is the incidence radio wave path (incidence path). The black, dotted line with an arrow represents the radio wave scattered back to the SWR receiving station. β is the elevation angle of the radio wave, α_i is the grazing incidence angle, α_s is the scattering angle, and ϕ_s is the scattering azimuth angle. SWRs transmit radio waves to the ionosphere in a range of elevation angles, and radio waves with a lower elevation angle often illuminate areas farther out to sea.

Barrick derived the first-order and second-order cross-section equations of the sea surface of high-frequency radars by using the boundary perturbation theory to solve Maxwell equations under the boundary condition that seawater is a perfect conductor [3]. The first-order cross-section equation is as follows:

(1)${\sigma }_1\left(\omega ,\theta \right)=2^4\pi k_0{}^4{(\cos {\varphi }_s-\cos {\alpha }_i\cos {\alpha }_s)}^2\sum _{m^{\prime }=\pm 1}S\left(-m^{\prime }{\stackrel{\rightharpoonup }{k}}^{\prime }\right)\delta \left(\omega -m^{\prime }{\omega }_B\right)$

where $\omega$ is the angular Doppler frequency, ${\omega }_B$ is the angular Bragg frequency, ${\omega }_B=\sqrt{g{k}_0{({\cos }^2 {\alpha }_i+{\cos }^2 {\alpha }_s-2 \cos {\alpha }_i \cos {\alpha }_s \cos {\varphi }_s)}^{\frac{1}{2}}},$ $k_0$ is the magnitude of the radar wave number vector ${\stackrel{\rightharpoonup }{k}}_0$, $S\left(-m^{\prime }{\stackrel{\rightharpoonup }{k}}^{\prime }\right)=S\left(k^{\prime },-m^{\prime }\theta \right)$ is the ocean wave spectrum, ${\stackrel{\rightharpoonup }{k}}^{\prime }$ is the ocean wave number vector that causes Bragg scattering and can be decomposed into $k_x^{\prime }=k_0\left(\cos {\alpha }_s \cos {\varphi }_s-\cos {\alpha }_i\right)$ and $k_y^{\prime }=k_0\left(\cos {\alpha }_s \sin {\varphi }_s\right)$, θ is the angle between the direction of the ocean wave and the direction of the radar beam, and $m^{\prime }=\pm 1$ represents the positive and negative Bragg frequencies, respectively, if ${\alpha }_i={\alpha }_s=0°,$ ${\varphi }_s=180°$. Equation (1) is simplified as the first-order cross-section of the sea surface of high-frequency ground-wave radars.

The second-order sea echo includes a double bounce from two first-order ocean waves as well as a single bounce from second-order ocean waves [35,36]; the second-order cross-section equation is as follows:

(2)$$\begin{gathered} {\sigma }_2\left(\omega ,\theta \right)=2^4\pi k_0{}^4{(\cos {\varphi }_s-\cos {\alpha }_i\cos {\alpha }_s)}^2 \\ \sum _{m,m^{\prime }=\pm 1}{{\displaystyle \int }}_{-\infty }^{+\infty }{{\displaystyle \int }}_{-\pi }^{\pi }{\left|\Gamma \right|}^{2}S\left(m\stackrel{\rightharpoonup }{k}\right)S\left(m^{\prime }{\stackrel{\rightharpoonup }{k}}^{\prime }\right)\delta \left(\omega -m\sqrt{gk}-m^{\prime }\sqrt{g{k}^{\prime }}\right)kdkd\theta \end{gathered}$$

where $\Gamma ={\Gamma }_H+{\Gamma }_{EM}$ is the sum of the electromagnetic coupling coefficient and hydrodynamic coupling coefficient, and $\stackrel{\rightharpoonup }{k}$ and ${\stackrel{\rightharpoonup }{k}}^{\prime }$ are the two ocean wave number vectors scattering radio waves, which satisfy the constraints of $k_x^{\prime }+k_x=k_0 \left(\cos {\alpha }_s \cos {\varphi }_s-\cos {\alpha }_i\right)$ and $k_y^{\prime }+k_y=k_0\left(\cos {\alpha }_s \sin {\varphi }_s\right)$. We can solve Equations (1) and (2) with an ocean wave spectrum S(·) to calculate the first-order and second-order cross-sections under different k₀, α_i, α_s, and ϕ_s conditions. If the ionosphere is very stable or changes slowly, the sea echo path is generally considered to be basically the same as the incidence path—that is, α_i = α_s and ${\varphi }_s=180°$. It should be noted here that the Barrick model used in this paper applies to the vertical polarization radio wave.

2.2. Ocean Wave Spectrum Models

The ocean wave spectrum S(k,θ) is an effective method with which to describe the state of the random sea surface, which can be regarded as the power spectrum density function of the ocean wave changing with direction and frequency. In general, S(k,θ) can be treated as the product of the nondirectional ocean wave spectrum f(k) and the directional factor G(Θ). The widely used f(k) includes the PM spectrum [37] and the JONSWAP spectrum [38]. The general form of the PM spectrum is as follows:

(3)$f_{\mathrm{PM}}\left(k\right)=0.0081k^{-4}exp(-0.74\left(\frac{g}{v^{2}k}{)}^2\right)$

where v is the sea surface wind speed at 19.5 m height. The PM spectrum is established based on fully developed ocean wave data, so the PM spectrum is more suitable for remote sensing ocean dynamics parameters of high-frequency radars under open-sea conditions. The general form of the JONSWAP spectrum is as follows:

(4)$$\begin{gathered} f_{\mathrm{JONSWAP}}\left(k\right)=\frac{0.0081}{2k^4}exp[\frac{-5}{4}\left(\frac{k_m}{k}{)}^2\right]\cdot {\gamma }^{exp[\frac{-1}{2{\sigma }^2{k}_m}\left(\sqrt{k}-\sqrt{k_m}{)}^2\right]}, \\ k_m=\frac{22g^{\frac{8}{3}}}{v{F}^{\frac{2}{3}}}, \\ \sigma =\left\{\begin{array}{c}0.07, k\le k_m\\ 0.09, k>k_m\end{array}\right., \\ \gamma =3.3 \end{gathered}$$

where k_m is the peak wave number and F is the fetch length, which is the measure of the length of water over which a given wind blows with regular intensity; σ is a parameter that controls the shape of the nondirectional ocean wave spectrum. The JONSWAP spectrum is more suitable for sea conditions with limited fetch, such as the Mediterranean [34].

G(Θ) is only a function of direction, which represents the distribution of ocean wave energy with direction. The model G(Θ) with a cosine form was used in this paper, as was proposed by Longuet-Higgins [39] and verified by Mitsuyasu [40]:

(5)$G_{COS}\left(\Theta \right)=A{\cos }^{2s\left(k\right)}\left(\Theta \right)$

where $\Theta =\left(\theta -{\theta }_w\right)/2$, θ and θ_w are the angles between the ocean wave direction, the sea surface wind direction, and the radar beam direction, respectively, $A=\frac{2^{2s-1}\Gamma \left(s+1\right)}{\pi \Gamma \left(2s+1\right)}$ is the normalization coefficient, s(k) is the directional spreading parameter—which controls the distribution of ocean wave energy in different directions—and the larger the value of s, the more concentrated the ocean wave energy spreading direction; that is, the ocean wave energy spreads in a narrower direction [41]. However, it is generally believed that s = 2 is reasonable [14], and $A=4/\left(3\pi \right)$.

2.3. Radio Wave Propagation Path Model

The 3D ray-tracing method is an effective method to quantitatively simulate the propagation path of high-frequency radio waves in the ionosphere, which mainly includes analytical [24,25,42,43] and numerical [26,44] ray tracing. The ray-tracing method used in this paper adopts the Runge–Kutta method to solve the Haselgrove ray square differential equation combining the Appleton–Hartree equation [25,44]. Ma Xin et al. used this method to successfully simulate the influence of rocket plume on the propagation path of high-frequency radio waves [45]. The equation is as follows:

(6)$\{\begin{array}{l}\frac{dr}{d{P}^{\prime }}=\frac{c}{\omega }{k}_r\\ \frac{d\theta }{d{P}^{\prime }}=\frac{c}{\omega r}{k}_{\theta }\\ \frac{d\theta }{d{P}^{\prime }}=\frac{c}{\omega r\sin \theta }{k}_{\phi }\\ \frac{d{k}_r}{d{P}^{\prime }}=-\frac{\omega }{2c}\frac{\partial X}{\partial \theta }+k_{\theta }^2\frac{c}{\omega r}+k_{\phi }^2\frac{c}{\omega r}\\ \frac{d{k}_{\theta }}{d{P}^{\prime }}=\frac{1}{r}\left(-\frac{\omega }{2c}\frac{\partial X}{\partial \theta }-k_r{k}_{\theta }\frac{c}{\omega }+k_{\phi }^2\frac{c}{\omega }\cot \theta \right)\\ \frac{d{k}_{\phi }}{d{P}^{\prime }}=\frac{1}{r\sin \theta }\left(-\frac{\omega }{2c}\frac{\partial X}{\partial \theta }-k_r{k}_{\theta }\frac{c}{\omega }\sin \theta +\frac{c}{\omega }{k}_{\theta }{k}_{\phi }\cos \theta \right)\end{array}$

where $P^{\prime }$ is the group path, and $r$, $\theta$, and $\phi$ are the components of the location of the radio wave in spherical coordinates. $k_r$, $k_{\theta }$ and $k_{\phi }$ are the magnitude of the components of the radio wave number vector in spherical coordinates; $c$ is the speed of light, $X=f_N^2/f^2$, $f_N$ is the plasma frequency, and $f$ is the radio wave frequency. $d{P}^{\prime }$ can be calculated as follows:

$d{P}^{\prime }=\frac{aX}{\frac{\partial X}{\partial r}+\frac{\partial X}{\partial \theta }+\frac{\partial X}{\partial \phi }}$

The step size of $d{P}^{\prime }$ varies with the electron density; $a$ is a constant with an empirical value of 0.01. The initial value of Equation (6) can be obtained from the latitude and longitude of the SWR transmitting station, the elevation angle $\beta$ (in Figure 1), and the azimuth angle ${\phi }_T$ (due north is 0°, and due east is 90°).

2.4. SWR Sea Echo Simulation Method

The scattering cross-section calculated by the Barrick model is the theoretical Doppler spectrum, and it should be obtained from the time-series backscattered SWR signal by two Fourier transforms. However, the Barrick model only provides the amplitude, and does not contain the phase of the Doppler spectrum, so we cannot obtain the slow time-domain signal of SWRs through the inverse Fourier transform—that is, the time-domain sea echo signal cannot be obtained only through the Barrick model. However, the biggest difficulty in SWR remote sensing of ocean dynamics parameters is ionospheric contamination suppression, among which the phase contamination is the most difficult to suppress [20,46,47], because the amplitude contamination can often be suppressed by normalization and other methods after the phase contamination is suppressed.

In order to simulate the SWR ocean time-domain echo spectrum (RD spectrum), we combined the 3D ray-tracing method with the Barrick model to obtain the theoretical scattering cross-sections, which were brought into the FMCW system to obtain the RD spectrum. Figure 2 shows the simulation process. It should be noted here that the effect of the ionosphere on SWR sea echo simulation is reflected in the propagation path of the radio waves, which was calculated from the electron density data and ionospheric height by a 3D ray-tracing method.

2.4.1. SWR Parameter Settings

SWRs achieve high angular resolution through a large antenna array. According to the research of Georges and Maresca on SWR beamwidth settings, a reasonable second-order echo spectrum can be obtained with a beamwidth of less than 2°; to reduce contamination significantly, beamwidths as narrow as 0.5° are required [6,48]. Therefore, we only studied sea echo simulations with a narrow beamwidth. The SWR parameters are shown in Table 1. It should be noted here that we did not simulate large-scale antenna arrays; that is, the synthetic antenna pattern and transmitting/receiving antenna gain of large-scale antenna arrays were not simulated. Ionosphere and path attenuation, etc., were also not taken into account in the simulation. However, these multiplicative factors can be simulated according to the actual radar parameters and the remote sensing environment. In addition, the SWR parameters in Table 1 can theoretically be set to any value according to the actual application. For this paper, we only conducted simulations based on the parameters in Table 1.

2.4.2. Data

The 3D ray-tracing method requires ionosphere height and electron density data in order to calculate the path of high-frequency radio waves in the ionosphere. In this paper, the International Reference Ionosphere (IRI) model is used to provide electron density data near the transmitting station. The IRI model is an empirical model that combines observation data and models to provide globally distributed electron density data. We used electron density data with a latitude of 30°34′N, a longitude of 118–128°E, and an altitude of 60–400 km, where the resolution of the longitude and latitude was 0.1°, the resolution of the altitude was 1 km, and the data time was 11 June 2019 at 12:00. It should be noted here that although the ionosphere is a medium with irregular spatiotemporal changes, the simulations for the entire coherent integration time were based on the ionospheric constant conditions. This is because the measured large-scale electron density data with high temporal resolution cannot be obtained, and the IRI model cannot reflect the drastic changes in electron density within the 80 s coherent integration.

Figure 3 shows the distribution of electron density at five locations by height, and the distribution of electron density at 80, 90, and 100 km height with latitude and longitude. We can see from Figure 3a that the electron density maximum appears at ~300 km. The change in electron density with latitude and longitude is not obvious below 230 km, and the electron density decreases with latitude above 230 km, but remains stable with longitude. The reason for showing electron density data at heights of 80, 90, and 100 km is because the radio waves are reflected in the ionosphere at almost 100 km in our simulations. As seen from Figure 3b–d, electron density increases rapidly with height, and electron density changes more significantly with longitude than with latitude. Therefore, we predicted that the direction of the high-frequency radio waves would be slightly shifted to the north, which was confirmed in our simulations.

The sea surface wind direction (${\varphi }_w$) and wind speed ($WS$) data were calculated from 10 m wind speed components $u$ and $v$ of the newest version of the European Centre for Medium-Range Weather Forecasts (ECMWF) reanalysis data—ERA5. The data acquisition time was the same as the electron density data. What needs to be explained here is that the 10 m $WS$ needs to be converted into 19.5 m $WS$ according to the relationship between the $WS$ and height [49]. Since ERA5 comprises standard grid data with a resolution of 0.25°, ${\varphi }_w$ and $WS$ need to be interpolated to the illuminated sea areas, which can be calculated based on the SWR parameters and the electron density data. As shown in Figure 4, $T$ is the location of the transmitting station. The variation range of $WS$ is 2–8 m/s, and the larger $WS$ is concentrated at the middle and long distance from $T$. The variation range of ${\varphi }_w$ is mainly 180–280° (due north is 0°, and due east is 90°). Additionally, ${\varphi }_w$ gradually changes from north wind to northeast wind from the long distance to the short distance, and northeast wind accounts for most of the area.

3. Results

3.1. The Influence of Grazing Incidence Angle, Scattering Angle, Scattering Azimuth Angle, and Fetch Length on the First-Order and Second-Order Cross-Section

The first-order and second-order scattering cross-sections were calculated by incorporating Equations (3)–(5) into Equations (1) and (2). We analyzed the influence of the grazing incidence angle (${\alpha }_i$), the scattering angle (${\alpha }_s$), the scattering azimuth angle (${\varphi }_s$), and the fetch length ($F$) on the scattering cross-sections. What needs to be explained here is that we used a discrete $k$ algorithm to solve Equation (2). This algorithm, which is described in detail in [50], decomposes the ocean wave number space from two-dimensional to one-dimensional, and adds up the second-order spectrum from all one-dimensional data [50]. Assuming that the sea surface wind speed is 15 m/s, the sea surface wind direction is 30°, and the radio wave frequency is 15 MHz, Figure 5 shows the sea surface scattering cross-sections of SWRs under different conditions (the scattering conditions are indicated in each figure). According to Mao et al.’s research on the influence of the fetch length ($F$) on sea surface current retrieval with high-frequency radars [51], $F=30, 150, 400 \mathrm{km}$ corresponds to short, medium, and long fetch, respectively. Since the grazing incidence angle (${\alpha }_i$) and the scattering angle (${\alpha }_s$) have the same influence on the scattering cross-section, we only studied the influence of the scattering angle (${\alpha }_s$).

It can be seen from Figure 5 that as the scattering angle (${\alpha }_s$) increases, the position of the Bragg peaks approaches a Doppler frequency of zero, and the power of the Bragg peaks has an increasing trend. This is because the ocean wave number that produces Bragg scattering is greater than the peak wave number of the ocean wave spectrum $k_m$, resulting in $f\left(k_{B,large}\right)<f\left(k_{B,small}\right)$. However, as the scattering azimuth angle (${\varphi }_s$) increases, the scattering cross-section shows the opposite characteristics. In addition, the influence on the scattering cross-section of the scattering azimuth angle (${\varphi }_s$) is smaller than that of the grazing incidence angle (${\alpha }_i$) and the scattering angle (${\alpha }_s$). Comparing Figure 5a,c,e or Figure 5b,d,f, it can be seen that as the fetch length ($F$) increases, the scattering cross-sections calculated from the JONSWAP spectrum gradually tend to that of the PM spectrum, and the power of the scattering cross-section gradually increases. This is due to two reasons: (1) during the process of transforming the short fetch into the long fetch, $f\left(k\right)$ increases over most of the range of $k$, which leads to an increase in the scattering cross-section’s power; (2) at the same time, $k_m$ becomes smaller, and the energy of $f\left(k\right)$ will be concentrated at the smaller $k$, which leads to a decrease in $f\left(k\right)$ power at some $k>k_m$ range.

Although it can be seen from Figure 5 that the grazing incidence angle (${\alpha }_i$), the scattering angle (${\alpha }_s$), and the scattering azimuth angle (${\varphi }_s$) have little influence on the scattering cross-section, the magnitude of their influence on the ocean dynamics parameter retrieval precision must be evaluated. Here, two empirical retrieval equations are used to evaluate the magnitude of this influence on the sea surface wind direction and RMS wave height retrieval precision. The sea surface wind direction retrieval equation adopted here was fitted by Harlan and Georges based on SWR sea echo data, ship data, buoy data, and NMC model data [11]:

(7)$\left|{\theta }_w\right|=3.75R+90°$

where $R$ is the ratio of the positive and negative Bragg peaks, and the unit is dB, while $\left|{\theta }_w\right|$ is the absolute value of the angle between the sea surface wind direction and the radar beam.

The RMS wave height retrieval equation adopted here was fitted by Maresca and Georges based on SWR sea echo data and buoy data [6]:

(8)$k_{0}h=0.8R_e{}^{0.6}$

where $R_e$ is the ratio of the energy of the second-order sidebands on both sides of the higher Bragg peak to the first-order energy.

We used the scattering cross-sections calculated from the PM spectrum in Figure 5a,b to calculate the retrieval values and relative errors of $\left|{\theta }_w\right|$ and $h$; the results are shown in Table 2. In order to draw stronger conclusions, we used the same method to calculate the retrieval results of $\left|{\theta }_w\right|$ and $h$ with different $WS$ (5 m/s, 15 m/s, and 25 m/s) and ${\varphi }_w$ (30°, 90°, and 150°); the results are shown in Table 2 and Table 3. It can be observed that the grazing incidence angle (${\alpha }_i$), the scattering angle (${\alpha }_s$), the scattering azimuth angle (${\varphi }_s$), and the $WS$ will not cause additional errors in the $\left|{\theta }_w\right|$ retrieval. This is due to the synchronous change in the positive and negative Bragg peaks, causing $R$ to remain unchanged. However, these angles, the $WS$, and the ${\varphi }_w$ will cause additional errors in $h$ retrieval, and the additional errors caused by these angles are greater at a lower $WS$. However, ${\varphi }_w$ had little effect on these errors, and these errors appear to be symmetrical with $\left|{\theta }_w\right|=90°$. In addition, the errors caused by the grazing incidence angle (${\alpha }_i$) and the scattering angle (${\alpha }_s$), are ~2.5 times that of the scattering azimuth angles (${\varphi }_s$). In practice, the elevation angle range of SWRs is generally small [52], the return path and the incidence path of radio waves are often not very different, and the ionosphere is not sensitive to the Bragg peaks [6]. Therefore, it is reasonable to assume ${\alpha }_s={\alpha }_i, {\varphi }_s=180°$ in one sweep duration when these angles cannot be accurately obtained and the ionosphere changes slowly.

We also studied the influence of the fetch length on the retrieval errors of $\left|{\theta }_w\right|$ and $h$ based on the scattering cross-sections calculated using the JONSWAP spectrum in Figure 5d,f; the effects of different $WS$ and ${\varphi }_w$ were also investigated. The results are shown in Table 2 and Table 3. The errors in estimating the fetch length do not affect the $\left|{\theta }_w\right|$ retrieval precision, but they have a great influence on the $h$ retrieval, especially at a lower $WS$. The effects of ${\varphi }_w$ are also small and symmetrical. Therefore, it is necessary to evaluate the fetch length before retrieving ocean dynamics parameters other than $\left|{\theta }_w\right|$. It should be noted here that the large difference between the retrieval values of $\left|{\theta }_w\right|$ and $h$ with the target values is because Equations (7) and (8) are established with measured data. Nevertheless, Equations (7) and (8) can be directly used for our study, because the results calculated by Equations (7) and (8) are the only theoretical errors caused by the grazing incidence angle (${\alpha }_i$), scattering angle (${\alpha }_s$), scattering azimuth angle (${\varphi }_s$), and fetch length ($F$). Therefore, the relative errors in Table 2 and Table 3 have great reference significance, and we need to pay attention to the corresponding parameters (${\alpha }_i, {\alpha }_s, {\varphi }_s, \mathrm{and} F$) when retrieving ocean dynamics parameters with measured data.

3.2. Simulation Results of SWR Sea Echo

3.2.1. Ideal SWR Sea Echo Simulation Results

Based on where the radio waves are reflected in the ionosphere, we calculated the reflection virtual heights ($h$), which were used to estimate the group path of each radio wave ($P_G$). Since it is assumed that the ionosphere remains stable throughout the coherent integration process, different $P_G$ values correspond to different time delays, so the range bins can be calculated by the distance resolution equation $c/\left(2B\right)$. Figure 6a–c show the $P_G$, the grazing incidence angles (${\alpha }_i$), and the reflection heights in the illuminated sea areas. We can see that, in general, a sea area with a larger $P_G$ corresponds to a farther ground distance, and a larger grazing incidence angle (${\alpha }_i$) corresponds to a smaller $P_G$ and to a higher reflection height. However, there are exceptions, which cause sea echoes from farther sea areas to be displayed on closer range bins. Figure 6d shows part of the propagation path of radio waves.

According to the Barrick scattering theory, only one or two ocean waves that satisfy the constraints can generate the sea echo spectrum, namely:

(9)$\omega -m\sqrt{gk}-m^{\prime }\sqrt{g{k}^{\prime }}=0$

where $\omega =2\pi \frac{2v_r}{\lambda }$, $v_r$ is the radial velocity. Taking ground-wave radars as an example: $\lambda =4\pi /\left|\stackrel{\rightharpoonup }{k}+\stackrel{\rightharpoonup }{k^{\prime }}\right|$, and $m\sqrt{gk}=m{\omega }_k=m{v}_{pk}k$, $m^{\prime }\sqrt{g{k}^{\prime }}=m^{\prime }{\omega }_{k^{\prime }}=m^{\prime }{v}_{p{k}^{\prime }}{k}^{\prime }$. $v_{pk}$ and $v_{p{k}^{\prime }}$ are the ocean wave phase velocities of the wave numbers $k$ and $k^{\prime }$, respectively. Therefore, Equation (9) can be transformed into:

(10)$v_r=\frac{m{v}_{pk}k+m^{\prime }{v}_{p{k}^{\prime }}{k}^{\prime }}{\left|\stackrel{\rightharpoonup }{k}+\stackrel{\rightharpoonup }{k^{\prime }}\right|}$

From Equation (10), we can see that the Doppler frequency in the sea echo is generated by the phase velocities of the two ocean waves (the Bragg peak corresponds to one ocean wave whose wavelength is half that of the radio wave) that satisfy the constraints, and the amplitude is determined by the nondirectional ocean wave spectrum, $f\left(k\right)$. Therefore, we assume that these ocean wave combinations generating a Doppler frequency can be analogized to one ocean wave with a specific scattering cross-section and direction.

Figure 7 shows the SWR RD spectra simulated by the sea echo signals returning along the incidence paths based on the PM spectrum and JONSWAP spectrum. The fetch length ($F$) required in the JONSWAP spectrum is estimated from Figure 4, at $F\approx 200 \mathrm{km}$. Due to the large range of azimuth angles (${\phi }_T$), only the RD spectra at two azimuth angles (90°, 100°) are shown here. This is because the $WS$ is small (mostly < 4 m/s) in the range of 80–90°, and the first-order and second-order cross-sections calculated by the Barrick model are very small. It can be observed that the power of the sea echo spectra calculated by the PM spectrum is stronger than that of the JONSWAP spectrum. There is a broadening of the Bragg peak in the Doppler spectrum, which is caused by the finite-time sampling and FFT. The power of the positive Bragg peak is stronger than that of the negative Bragg peak, due to the sea surface wind direction being towards the radar. The positive Bragg peak power corresponding to ${\phi }_T=90°$ is stronger than that of ${\phi }_T=100°$; this is because the sea surface wind direction in this area is mostly concentrated in 220–240°, which is closer to 180° with ${\phi }_T=90°$ than with ${\phi }_T=100°$. It should be noted here that because the distance resolution is 5 km, there are situations in which different elevation angles ($\beta$) correspond to the same range bin in the simulation (as shown in Figure 6). Similarly, there are also situations where some ocean areas cannot be illuminated. This results in a broadening of the Doppler spectrum in addition to no sea echoes at some range bins. $P_G$ calculated by the 3D ray-tracing method is not continuous, which means that there are no sea echoes in some range bins. Therefore, the lower power of the horizontal stripes displayed in Figure 7 is the result of interpolation after smoothing.

3.2.2. Contaminated SWR Sea Echo Simulation

The instability of the ionosphere allows sea echoes from the same sea area to return along different paths, so real-time ionosphere data have a positive effect on SWR signal path positioning, in addition to multipath and multimode contamination assessment. In general, the phase contamination caused by the ionosphere to the sea echo can be regarded as a polynomial function of the contamination phase, changing with time [53,54]. Here, we assume that the ionospheric phase contamination satisfies Equation (11) within one coherent integration period:

(11)$f_p=at+b{t}^2+c{t}^3$

where $a, b, \mathrm{and} c$ are coefficients, and $t$ is the coherent integration time series. Table 4 shows the four different combinations of coefficients $a$, $b$, and $c$. The ionosphere contamination is simulated by adding the contamination phase in Figure 7b.

Figure 8a–d show the four ionospheric contamination results corresponding to Table 4. Compared with Figure 7b, the ionosphere contamination has a serious broadening effect on the SWR RD spectrum, and the positive and negative Bragg peaks are no longer symmetrical around a Doppler frequency of zero. For the assumed polynomial ionospheric contamination function, the increase in the first-order and second-order coefficients (a and b) has little effect on the broadening and shifting. Comparing Figure 8a,c,d, it can be observed that the nonlinear term causes the first-order spectrum to deform. However, the linear term has little effect on the shape of the first-order spectrum, which can be seen by comparing Figure 8a,b. The increase in the cubic term coefficient (c) seriously broadens the RD spectrum, and seriously increases the occurrence of multiple first-order peaks, indicating that the high-order nonlinear contamination from the ionosphere to the SWR sea echo spectrum is the main factor that affects the ocean dynamics parameters’ retrieval precision. Compared with the weak Bragg peak, the strong Bragg peak and its nearby second-order sidebands have stronger anti-interference, which is why relevant scholars only use the data on the side of the strong Bragg peak for parameter retrieval.

4. Discussion

The Barrick model was used for the SWR sea echo spectrum simulation in this paper. We first used the empirical retrieval algorithm of the sea surface wind direction and RMS wave height to analyze the influence of the grazing incidence angle (${\alpha }_i$), the scattering angle (${\alpha }_s$), the scattering azimuth angle (${\varphi }_s$), and the fetch length ($F$) on the retrieval results under different sea surface wind speed ($WS$) and direction (${\varphi }_w$) conditions. The results show that these parameters have no effect on the retrieval precision of the sea surface wind direction, but that the bias of 30° in these angles has a certain influence on the RMS wave height retrieval precision. In addition, the influence of the grazing incidence angle (${\alpha }_i$) and the scattering angle (${\alpha }_s$) is ~2.5 times that of the scattering azimuth angle (${\varphi }_s$). However, we consider ${\alpha }_s={\alpha }_i, {\varphi }_s=180°$ to be a reasonable approximation for one sweep duration when these angles cannot be obtained accurately and the ionospheric phase contamination cannot be removed. The fetch length ($F$) seriously affects the retrieval precision of the RMS wave height, and a fetch estimation error of 250 km can cause the relative error of the RMS wave height to reach more than 39%; therefore, it is necessary to evaluate the fetch length ($F$) before retrieving parameters other than the sea surface wind direction.

However, individual SWR cannot resolve ambiguous values of the sea surface wind direction. Combining ground-based receiving stations can solve this problem in the sea areas close to the coastline, and networking two SWRs can theoretically solve this problem in long-distance sea areas, but the cost is high. In this simulation experiment, it was found that the radio waves reflected by the ionosphere will reach the same sea area with different azimuth angles, which is the purpose of dual-station detection. Therefore, if the radio wave paths can be accurately located in practice, it is possible to eliminate the ambiguity of the sea surface wind direction using a monostatic SWR system in some sea areas, which may be a matter worth investigating.

We proposed a simulation model of the narrow-beam SWR sea echo spectrum by combining the Barrick model, the 3D ray-tracing method, and the FMCW system. In order to verify the capability of this model, we simulated the SWR sea echoes, which are based on the ocean environment that exists in nature, using the 3D electron density data output by the IRI model, including the presence (Figure 8) and absence (Figure 7) of the ionospheric phase contamination. The results show that the phase contamination of the ionosphere on the SWR sea echo spectrum will cause serious broadening, shifting, and first-order spectrum deformation, and that the influence of the high-order term is more obvious.

This model can be used as the basis for related research on SWRs, such as ocean dynamics parameter retrieval, ionospheric decontamination, and electron density retrieval; it has great flexibility, and can be adjusted according to the features of the actual SWR system, such as bandwidth, frequency, sweep duration, coherent integration time, etc. In addition, multiplicative factors such as transmitting power, antenna gain, and path attenuation can be added to this model in order to simulate a more realistic situation. In any case, solving the contamination from the ionosphere to the SWR signal is the biggest challenge, and the model proposed in this paper seems to find a direction for solving this problem. However, the effect of the spatiotemporal variability of the electron density was not considered. Even if the electron density data with high temporal and spatial resolution can be obtained, only the calculation of the propagation path will have a huge computational burden. Therefore, the dynamic ionospheric phase contamination of SWR sea echoes will be more complicated.

In summary, the question of how the ionosphere can be used perfectly is a difficult problem in SWR research. In the future, we will use the model proposed in this paper to carry out research in ionospheric phase contamination modeling.

Author Contributions

M.F. conducted all programming, calculations, and comparisons; he is also responsible for the figures and general text. H.F., W.A. and X.W. took part in the forward modeling, literature review, and analysis. X.Y., L.Z., C.G., Q.Z. and X.L. took part in all discussions and analysis at all stages of the research. All authors have read and agreed to the published version of the manuscript.

Funding

This research was funded by the National Natural Science Foundation of China, grant number 61771352.

Data Availability Statement

The data presented in this study are available upon request from the corresponding author.

Acknowledgments

Many thanks to the ECMWF for providing the ERA5 reanalysis data.

Conflicts of Interest

The authors declare no conflict of interest.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Figures and Tables

Figure 1. Schematic diagram of SWR detection: The black straight line with an arrow between T and S is the azimuth of the radar beam, which is generally considered to be 0°. The scattering azimuth corresponding to backscattering is 180°.

Figure 2. Diagram of the process to obtain the SWR ocean time-domain echo spectrum: [Forumla omitted. See PDF.] is the elevation angle, [Forumla omitted. See PDF.] is the azimuth angle, [Forumla omitted. See PDF.] is the radio wave frequency, [Forumla omitted. See PDF.] is the transmitting station, and [Forumla omitted. See PDF.] and [Forumla omitted. See PDF.] are the first-order and second-order cross-sections, respectively.

Figure 3. Distribution of electron density with height, latitude, and longitude. (a) Distribution of electron density at five locations by height. (b–d) show the distribution of electron density at 80, 90, and 100 km height with latitude and longitude, respectively.

Figure 4. Sea surface wind speed (a) and direction (b) in the illuminated sea area.

Figure 5. SWR scattering cross-sections under different grazing incidence angle, scattering angle, scattering azimuth angle, and fetch length conditions. The scattering conditions are indicated in each figure.

Figure 5. SWR scattering cross-sections under different grazing incidence angle, scattering angle, scattering azimuth angle, and fetch length conditions. The scattering conditions are indicated in each figure.

Figure 6. Parameters related to the reflection of radio waves by the ionosphere. (a) Group path ([Forumla omitted. See PDF.]). (b) Grazing incidence angles ([Forumla omitted. See PDF.]). (c) Reflection heights. (d) Radio waves’ paths.

Figure 6. Parameters related to the reflection of radio waves by the ionosphere. (a) Group path ([Forumla omitted. See PDF.]). (b) Grazing incidence angles ([Forumla omitted. See PDF.]). (c) Reflection heights. (d) Radio waves’ paths.

Figure 7. SWR RD spectra simulated by the sea echo signals returning along the incidence path based on the PM spectrum and JONSWAP spectrum. (a) PM spectrum, [Forumla omitted. See PDF.] = 90°. (b) PM spectrum, [Forumla omitted. See PDF.] = 100°. (c) JONSWAP spectrum, [Forumla omitted. See PDF.] = 90°. (d) JONSWAP spectrum, [Forumla omitted. See PDF.] = 100°.

Figure 8. Simulation results for the presence of ionospheric phase contamination in Figure 7b. The ionospheric phase contamination was calculated using Equation (11) and the coefficients ([Forumla omitted. See PDF.]) shown in Table 4. (a) a = 0.005, b = 0.005 and c = 0.005. (b) a = 0.01, b = 0.005 and c = 0.005. (c) a = 0.005, b = 0.01 and c = 0.005. (d) a = 0.005, b = 0.005 and c = 0.01.

Table 1

SWR parameters.

Parameters	Values
Longitude and latitude of T	32°N, 118.5°E (Nanjing, Jiangsu Province)
Radio wave f	10 MHz
Evaluation angle β	15–25°
Azimuth angle φT	80–100°
Azimuth resolution	0.5°
Bandwidth B	30 KHz
Sweep duration TS	0.2 s
Coherent integration sweeps N	400
Waveform ST	$exp[j\left(2\pi ft+\pi \frac{B}{T_s}{t}^2\right)]$

Table 2

$\left|{\theta }_w\right|$ and $h$ retrieval values as well as relative errors with ${\varphi }_w=30°$.

WS	Retrieved Parameter	${\mathit{\alpha }}_{\mathit{s}}=0°$ ${\mathit{\alpha }}_{\mathit{i}}=0°$ ${\mathit{\varphi }}_{\mathit{s}}=180°$	${\mathit{\alpha }}_{\mathit{s}}=30°$ ${\mathit{\alpha }}_{\mathit{i}}=0°$ ${\mathit{\varphi }}_{\mathit{s}}=180°$	Relative Error	${\mathit{\alpha }}_{\mathit{s}}=30°$ ${\mathit{\alpha }}_{\mathit{i}}=30°$ ${\mathit{\varphi }}_{\mathit{s}}=180°$	${\mathit{\alpha }}_{\mathit{s}}=30°$ ${\mathit{\alpha }}_{\mathit{i}}=30°$ ${\mathit{\varphi }}_{\mathit{s}}=150°$	Relative Errors	$\mathit{F}=150 \mathbf{km}$ ${\mathit{\alpha }}_{\mathit{s}}=30°$ ${\mathit{\alpha }}_{\mathit{i}}=30°$ ${\mathit{\varphi }}_{\mathit{s}}=180°$	$\mathit{F}=400 \mathbf{km}$ ${\mathit{\alpha }}_{\mathit{s}}=30°$ ${\mathit{\alpha }}_{\mathit{i}}=30°$ ${\mathit{\varphi }}_{\mathit{s}}=180°$	Relative Errors
15 m/s	$\left|{\theta }_w\right|$	4.207°	4.207°	0%	4.207°	4.207°	0%	4.207°	4.207°	0%
	$h$	0.402 m	0.346 m	16.1%	0.287 m	0.270 m	6.3%	0.028 m	0.046 m	39.1%
5 m/s	$\left|{\theta }_w\right|$	4.207°	4.207°	0%	4.207°	4.207°	0%	4.207°	4.207°	0%
	$h$	0.016 m	0.013 m	23.8%	0.010 m	0.009 m	11.1%	4.18 × 10−4 m	0.005 m	91.6%
25 m/s	$\left|{\theta }_w\right|$	4.207°	4.207°	0%	4.207°	4.207°	0%	4.207°	4.207°	0%
	$h$	3.645 m	3.076 m	18.5%	2.504 m	2.271 m	10.3%	0.040 m	0.096 m	58.3%

Table 3

$\left|{\theta }_w\right|$ and $h$ retrieval values as well as relative errors with $WS=15 \mathrm{m}/\mathrm{s}$.

${\mathit{\varphi }}_{\mathit{w}}$	Retrieved Parameter	${\mathit{\alpha }}_{\mathit{s}}=0°$ ${\mathit{\alpha }}_{\mathit{i}}=0°$ ${\mathit{\varphi }}_{\mathit{s}}=180°$	${\mathit{\alpha }}_{\mathit{s}}=30°$ ${\mathit{\alpha }}_{\mathit{i}}=0°$ ${\mathit{\varphi }}_{\mathit{s}}=180°$	Relative Error	${\mathit{\alpha }}_{\mathit{s}}=30°$ ${\mathit{\alpha }}_{\mathit{i}}=30°$ ${\mathit{\varphi }}_{\mathit{s}}=180°$	${\mathit{\alpha }}_{\mathit{s}}=30°$ ${\mathit{\alpha }}_{\mathit{i}}=30°$ ${\mathit{\varphi }}_{\mathit{s}}=150°$	Relative Errors	$\mathit{F}=150 \mathbf{km}$ ${\mathit{\alpha }}_{\mathit{s}}=30°$ ${\mathit{\alpha }}_{\mathit{i}}=30°$ ${\mathit{\varphi }}_{\mathit{s}}=180°$	$\mathit{F}=400 \mathbf{km}$ ${\mathit{\alpha }}_{\mathit{s}}=30°$ ${\mathit{\alpha }}_{\mathit{i}}=30°$ ${\mathit{\varphi }}_{\mathit{s}}=180°$	Relative Errors
30°	$\left|{\theta }_w\right|$	4.207°	4.207°	0%	4.207°	4.207°	0%	4.207°	4.207°	0%
	$h$	0.402 m	0.346 m	16.1%	0.287 m	0.270 m	6.3%	0.028 m	0.046 m	39.1%
90°	$\left|{\theta }_w\right|$	90°	90°	0%	90°	90°	0%	90°	90°	0%
	$h$	0.234 m	0.203 m	15.3%	0.171 m	0.159 m	7.6%	0.014 m	0.025 m	44%
150°	$\left|{\theta }_w\right|$	175.8°	175.8°	0%	175.8°	175.8°	0%	175.8°	175.8°	0%
	$h$	0.402 m	0.346 m	16.1%	0.287 m	0.270 m	6.3%	0.028 m	0.046 m	39.1%

Table 4

Four combinations of $a, b, \mathrm{and} c$.

	$\mathit{a}$	$\mathit{b}$	$\mathit{c}$
1	0.005	0.005	0.005
2	0.01	0.005	0.005
3	0.005	0.01	0.005
4	0.005	0.005	0.01

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