Deterministic Sea Wave Reconstruction and Prediction Based on Coherent S-Band Radar Using Condition Number Regularized Least Squares

Abstract

Coherent S-band radar is a remote sensing observation device with high spatial-temporal resolution and can be used to achieve deterministic sea wave reconstruction and prediction (DSWRP) technology. However, coherent S-band radar can observe nonlinear details of the sea surface due to its high resolution, which makes the propagation operator matrix an ill-conditioned overdetermined matrix. To solve this problem, this paper proposes a DSWRP scheme using condition number regularized least squares (CN-RLS) for coherent S-band radar. First, the space-time velocity information was obtained from the radar echo. Second, the CN-RLS method solved the phase-resolved model coefficients. Finally, the deterministic wave field was predicted according to the solved model coefficients. The proposed scheme was verified by simulation data and the real radar dataset observed by the coherent S-band wave-measuring radar onboard the ship XIANGYANGHONG-18 in the East China Sea in April 2024. The predicted wave elevation of the proposed method was compared with the wave elevation observed based on the X-band wave-measuring radar, and the root mean square error (RMSE) and correlation coefficient (CC) were 0.22 m and 0.76, respectively, which show that the proposed method could effectively implement the DSWRP technology.

1. Introduction

DSWRP technology refers to the reconstruction of the instantaneous wave height within the observation range and the prediction of the phase-resolved wave in the specific space coordinate in the future, which plays an important role in maritime safety operations [1,2,3,4]. The key to DSWRP technology is accurately observing the wave field, estimating the model parameters, and then constructing the instantaneous wave elevation of the desired location based on the wave propagation model [5,6,7]. The traditional wave observation devices, such as buoys, use fixed-point measurement methods and cannot provide large-area detection. As a new type of wave observation equipment, microwave radar can provide a higher spatial-temporal resolution and larger coverage of wave detection, and it is widely favored [8].

When estimating the model coefficients of noncoherent microwave radar, 3D-FFT processing is required on the backscattered image received by the radar [9,10]. Fueile [11] proposed a parameter estimation method for an ocean wave model based on the regularized least squares (RLS) method in his research, and carried out simulation verification of the proposed scheme. To take into account both the time variation of the radar scanning process and the ill-posed problem of matrix inversion in least squares, Al-Ani et al. [12,13] proposed a single-frequency least squares model parameter estimation method that can receive mixed spatial-temporal data generated by rotating radar antennas. However, the modulation transfer function (MTF) between the image amplitude spectra and the sea wave amplitude spectra requires frequent calibration and is susceptible to weather conditions, which poses challenges for DSWRP techniques adopting noncoherent radar [13,14].

Unlike noncoherent radar, the signal transmitted by coherent microwave radar has all determined phase angles, and this salient feature provides the ability to obtain radial Doppler velocities from sea echoes [15,16,17,18,19]. According to wave theory, there is a deterministic relationship between the wave model parameters and the radial velocity of coherent microwave radar, so the DSWRP technique utilizing coherent microwave radar does not need calibration [20]. Nwogu et al. [21,22] proposed a scheme to realize deterministic wave field reconstruction in physical space instead of Fourier space, namely, the velocity potential function method. The simulation results show that the sea surface elevation can be effectively reconstructed by using the proposed scheme, but the sea surface elevation cannot be predicted. Connell et al. [23] proposed using the LS method to solve the wave motion model parameters using the frequency shift measured by the quasi-coherent microwave radar, and then realize the DSWRP by the superposition scheme. Støle-Hentschel et al. [24] applied the X-band radar with coherent reception to illuminate the sea surface toward the dominant wave direction and they used the acquired signal amplitude and phase-estimated Doppler velocity to reconstruct the sea surface respectively, which verified that the sea surface reconstructed by coherent reception radar amplitude and phase were consistent. Subsequently, Zhang et al. [25] proposed using the spatial sampling method to estimate the model coefficients of the linear wave model by L-curve regularized least squares (L-curve RLS), and then reconstruct and predict the instantaneous sea wave surface displacement. The L-curve RLS is a kind of Tikhonov regularization (ridge regression) method, which uses the L-curve method to find the optimal regularization parameter of the Euclidean norm. However, Tikhonov regularization is sensitive to outliers and cannot effectively select important features, which leads to a decrease in the interpretivity of the model.

Despite the significant progress in the above studies, implementing DSWRP techniques remains a challenging problem. When coherent microwave radar observes the actual sea surface, it is inevitably disturbed by nonlinear components, such as breaking waves, which seriously affect the accuracy of DSWRP. This article proposes a deterministic wave field prediction method based on condition number regularized least squares (CN-RLS) for coherent S-band radar. First, the space-time velocity information is obtained from the radar echo, and then the phase-resolved model coefficients are solved by the CN-RLS method. Finally, the deterministic wave field is predicted according to the solved model coefficients. The new scheme was verified by adopting the simulation data and the dataset collected by the coherent S-band wave-measuring radar and X-band wave-measuring radar onboard the East China Sea XIANGYANGHONG-18 in April 2024.

The remaining sections of this paper are organized as follows. Section 2 explains the principles and methods of DSWRP. The performance of the proposed method is verified by simulations in Section 3. Section 4 analyzes the performance of the proposed method with real radar experimental data and results. The results are discussed in Section 5. Finally, Section 6 is the conclusion.

2. Principle and Method

2.1. Wave Propagation Model

According to linear wave theory, the undulating wave surface is viewed as a series of sine waves with different directions, different frequencies, and different amplitudes, which are superimposed in random phases. As shown in Figure 1, the sea surface elevation $\eta$ at space-time coordinates $(r,\beta ,t)$ is [26]

$$\eta (r,\beta ,t)=\sum _{i=1}^N\sum _{j=1}^M{A}_{ij}cos[k_{i}rcos(\beta -{\varphi }_j)-{\omega }_{i}t+{\epsilon }_{ij}]$$

(1)

where N and M denote the number of wavenumbers $k_i$ (or angular frequencies ${\omega }_i$) and the number of wave propagation directions ${\varphi }_j$, respectively. $A_{ij}=\sqrt{2S({\omega }_i,{\varphi }_j)\Delta \omega \Delta \varphi }$ represents the amplitude at ${\varphi }_j$ in the direction of the angular frequency ${\omega }_i$; $S({\omega }_i,{\varphi }_j)$ is the directional wave spectra; $\Delta \omega$ and $\Delta \varphi$ denote the angular frequency resolution and angle resolution; $k_i$ and ${\omega }_i$ satisfy the dispersion relation ${\omega }_i^2=g{k}_i$; g is the acceleration due to gravity; $\beta$ is the radar azimuth; and ${\epsilon }_{ij}$ is a uniformly distributed random phase on $(0,2\pi )$. Denote ${\mathrm{\Omega }}_{ij}=k_{i}rcos(\beta -{\varphi }_j)-{\omega }_{i}t$; then, (1) can be abbreviated as

$$\eta (r,\beta ,t)=\sum _{i=1}^N\sum _{j=1}^M(a_{ij}cos{\mathrm{\Omega }}_{ij}+b_{ij}sin{\mathrm{\Omega }}_{ij})$$

(2)

where $(a_{ij},b_{ij})=(A_{ij}cos{\epsilon }_{ij},-A_{ij}sin{\epsilon }_{ij})$ denotes the pair of model coefficients. The horizontal velocity u and vertical velocity w of water particles are expressed as

$$u(r,\beta ,t)=\sum _{i=1}^N\sum _{j=1}^M(a_{ij}{\omega }_{i}cos{\mathrm{\Omega }}_{ij}+b_{ij}{\omega }_{i}sin{\mathrm{\Omega }}_{ij})$$

(3)

$$w(r,\beta ,t)=\sum _{i=1}^N\sum _{j=1}^M(a_{ij}{\omega }_{i}sin{\mathrm{\Omega }}_{ij}-b_{ij}{\omega }_{i}cos{\mathrm{\Omega }}_{ij}).$$

(4)

According to Figure 1, the radial water particles velocity $v_r$ observed by radar is given by

$$\begin{array}{cc}\hfill v_r(r,\beta ,t)& =ucos(\beta -{\varphi }_j)sin\theta +wcos\theta \hfill \\ \hfill & =\sum _{i=1}^N\sum _{j=1}^M[a_{ij}{\alpha }_{ij}(r,\beta ,t)+b_{ij}{\gamma }_{ij}(r,\beta ,t)]\hfill \end{array}$$

(5)

where $\theta$ is the radar incidence angle, ${\alpha }_{ij}(r,\beta ,t)={\omega }_i[cos(\beta -{\varphi }_j)cos{\mathrm{\Omega }}_{ij}sin\theta +sin{\mathrm{\Omega }}_{ij}cos\theta ]$, and ${\gamma }_{ij}(r,\beta ,t)={\omega }_i[cos(\beta -{\varphi }_j)sin{\mathrm{\Omega }}_{ij}sin\theta -cos{\mathrm{\Omega }}_{ij}cos\theta ]$.

Therefore, (5) can be rewritten into the following matrix form:

$$\mathit{v}=\mathit{P}\mathit{x}$$

(6)

where $\mathit{x}$ and $\mathit{P}$ are the model coefficient vector to be estimated and the wave propagation matrix, respectively, which can be expressed as [25]

$$\left\{\begin{array}{c}\mathit{P}=\underset{\mathit{P}\in {\mathbf{R}}^{Z\times 2N_e{M}_e}}{\underbrace{\left[\begin{array}{cccc}\cdots & \stackrel{\alpha column}{\overbrace{\cdots }}& \stackrel{\gamma column}{\overbrace{\cdots }}& \cdots \\ \cdots & {\alpha }_{ij}(r_l,{\beta }_h,t_q)& {\gamma }_{ij}(r_l,{\beta }_h,t_q)& \cdots \\ \cdots & \cdots & \cdots & \cdots \end{array}\right]}}\hfill \\ \mathit{x}=\left[\begin{array}{c}\cdots \\ a_{ij}\\ b_{ij}\\ \cdots \end{array}\right],\mathit{x}\in {\mathbf{R}}^{2N_e{M}_e\times 1}\hfill \end{array}\right..$$

(7)

In (7), $l=1,2,\cdots ,L$; $h=1,2,\cdots ,H$; and $q=1,2,\cdots ,Q$, where Q, H, and L denote the numbers of time points, azimuth angles, and range bins, respectively. $\mathit{P}$ is an overdetermined matrix, that is, the number of observations in the spatial-temporal dimension ($Z=LHQ$) is larger than $2N_e{M}_e$, where $M_e$ and $N_e$ are the numbers of wave directions and wavenumber components, respectively. Arrange the space-time radial water particles velocity $v_r$ in the form of column vector $\mathit{v}={\left[\begin{array}{ccc}\cdots & v_r(r_l,{\beta }_h,t_q)& \cdots \end{array}\right]}^T\in {\mathbf{R}}^{Z\times 1}$. By taking the solved $\mathit{x}$ into (1), the wave surface elevation can be reconstructed and predicted.

2.2. CN-RLS Method

The $\mathit{P}$ is an overdetermined matrix of full rank ($rank\left(\mathit{P}\right)=2N_e{M}_e$); thus, (6) only has the least squares (LS) solution and no conventional solution. In practical coherent microwave radar observations, due to its high spatial resolution, it can detect some nonlinear details of the sea surface, which can interfere with the observed column vector $\mathit{v}$. Therefore, Formula (6) is also an ill-conditioned system of equations. The regularization least squares method can effectively solve this problem, and the corresponding regularization solution is [27]

$${\mathit{x}}_{\alpha \mathit{H}}={({\mathit{P}}^T\mathit{P}+\alpha \mathit{H})}^{-1}{\mathit{P}}^T\mathit{v}$$

(8)

where $\alpha$ is the regularization parameter and $\mathit{H}$ is the regularization matrix. The idea of regularization is to modify the normal matrix ${\mathit{P}}^T\mathit{P}$ so that the condition number of solving equations is greatly reduced to weaken the ill-posedness of the system of equations.

Suppose the eigenvalues of the normal matrix ${\mathit{P}}^T\mathit{P}$ are ${\lambda }_1,{\lambda }_2,\cdots ,{\lambda }_{2N_e{M}_e},0<{\lambda }_i<{\lambda }_{i+1}$; the corresponding unit orthogonalized eigenvector is ${\mathit{e}}_1,{\mathit{e}}_2,\cdots ,{\mathit{e}}_{2N_e{M}_e}$; and the condition number is $cond\left({\mathit{P}}^T\mathit{P}\right)={\lambda }_{2N_e{M}_e}/{\lambda }_1$. The condition number of the identity matrix is 1, but for a non-invertible matrix, the condition number is ∞. A larger condition number indicates that the system being solved is more sensitive to small perturbations [28]. In such cases, it is often difficult to obtain reasonable solution coefficients. The ill-posedness of the normal matrix not only shows that the value of $cond\left({\mathit{P}}^T\mathit{P}\right)$ is large but also has s eigenvalues with small inputs relative to the largest eigenvalue ${\lambda }_{2N_e{M}_e}$, that is, ${\lambda }_1<{\lambda }_2<\cdots <{\lambda }_s\ll {\lambda }_{2N_e{M}_e}$. In this article, these smaller eigenvalues are modified according to the regularization idea to weaken the ill-posedness of the system of equations. Construct the following vectors:

$${\mathit{y}}_i={\beta }_i{\mathit{P}}^T\mathit{P}{\mathit{e}}_i={\beta }_i{\lambda }_i{\mathit{e}}_{i}i=1,2,\cdots ,s$$

(9)

where ${\beta }_i>0$, and the modified normal matrix can be expressed as

$$\mathit{B}={\mathit{P}}^T\mathit{P}+{\mathit{y}}_1{\mathit{y}}_1^T+{\mathit{y}}_2{\mathit{y}}_2^T+\cdots +{\mathit{y}}_s{\mathit{y}}_s^T.$$

(10)

According to regularity, the eigenvectors of matrix $\mathit{B}$ are still ${\mathit{e}}_1,{\mathit{e}}_2,\cdots ,{\mathit{e}}_{2N_e{M}_e}$, while the eigenvalues are ${\lambda }_i+{\beta }_i^2{\lambda }_i^2(1\le i\le s)$ and ${\lambda }_i(s+1\le i\le 2N_e{M}_e)$.

From the above analysis, we know that s and ${\beta }_i$ determine the ill-conditioning of matrix $\mathit{B}$. Assuming the expected condition number of matrix $\mathit{B}$ is K, the condition index ${\displaystyle \frac{{\lambda }_{2N_e{M}_e}}{{\lambda }_i}}(i=1,2,\cdots ,2N_e{M}_e)$ is calculated. Therefore, we have ${\displaystyle \frac{{\lambda }_{2N_e{M}_e}}{{\lambda }_1}}>{\displaystyle \frac{{\lambda }_{2N_e{M}_e}}{{\lambda }_2}}>\cdots >{\displaystyle \frac{{\lambda }_{2N_e{M}_e}}{{\lambda }_s}}>K>{\displaystyle \frac{{\lambda }_{2N_e{M}_e}}{{\lambda }_{s+1}}}>{\displaystyle \frac{{\lambda }_{2N_e{M}_e}}{{\lambda }_{s+2}}}>\cdots >{\displaystyle \frac{{\lambda }_{2N_e{M}_e}}{{\lambda }_{2N_e{M}_e}}}=1$, which can determine the value of s. Small eigenvalues ${\lambda }_1,{\lambda }_2,\cdots ,{\lambda }_s$ are modified by (9), where ${\beta }_i(i=1,2,\cdots ,s)$ can be represented as

$${\beta }_i={\displaystyle \frac{\sqrt{{\lambda }_{s+1}-{\lambda }_i}}{{\lambda }_i}}.$$

(11)

Therefore, the eigenvalues of $\mathit{B}$ are ${\lambda }_{s+1}(1\le i\le s)$ and ${\lambda }_i(s+1\le i\le 2N_e{M}_e)$, where the condition number $cond\left(\mathit{B}\right)={\displaystyle \frac{{\lambda }_{2N_e{M}_e}}{{\lambda }_{s+1}}}\ll cond\left({\mathit{P}}^T\mathit{P}\right)$, greatly reducing the ill-conditioning of the original equation. Usually, matrices with a condition number less than 10 are considered well-conditioned [29,30], so in this work, K was taken as 10. The solution $\widehat{\mathit{x}}$ of the model coefficients can be obtained by solving the modified method equation $\mathit{B}\mathit{x}=\mathit{v}$ according to the conjugate gradient method [31]. Figure 2 shows a typical example of solving the equation system $\mathit{A}\mathit{x}=\mathit{y}$, where the matrix $\mathit{A}$ was a randomly generated overdetermined matrix of size 50 × 40, and $\mathit{x}$ was a randomly generated column vector of size 40 × 1. The matrix $\mathit{A}$ was reordered to arrange $\mathit{y}$ in ascending order. Some outliers (red dots in the figure) were added to the data $\mathit{y}$ as interference sources. The CN-RLS and L-curve RLS methods were used to solve $\widehat{\mathit{x}}$ and estimate $\widehat{\mathit{y}}$, respectively. The statistical parameters are provided in the figure. From these results in Figure 2, it can be observed that the interference points in the observed data had a relatively small impact on the CN-RLS method, while they had a larger interference on the L-curve RLS method. This indicates that the CN-RLS method could effectively suppress the influence of outliers on solving ill-conditioned equation systems.

To analyze the effect of the value of K on the solution of the ill-conditioned matrix, Figure 3 depicts the distribution of the correlation coefficients with K values for different condition numbers, which ranged from 10 to 300. The condition number ($cond\left({\mathit{P}}^T\mathit{P}\right)$) of the ill-conditioned matrix was adjusted by changing the size of the matrix $\mathit{A}$ in Figure 2. From the results in Figure 3, it can be seen that as the value of K increased, the correlation coefficient first maintained a high level (>0.99) and then decreased, and the larger the condition number was, the more the K values maintained a high correlation coefficient level. Therefore, the K value of 10 was reasonable in this work.

2.3. Method

Figure 4 shows the flow chart of the DSWRP technology, and the specific implementation process is elaborated as the following steps:

Step 1: Read the Doppler spectra data of the radar echo signal.

Step 2: Calculate the radial velocity $v_r$ of the water particles based on the Doppler spectra of the radar echoes, and organize it in the form of column vector $\mathit{v}$. The Doppler frequency shift $f_d$ is calculated by adopting the spectral moment method [32] and $v_r={\displaystyle \frac{f_d\lambda }{2}}$, where $\lambda$ is the wavelength of the radar electromagnetic wave.

Step 3: The propagation operator matrix $\mathit{P}$ is constructed using the information about the wave direction, radar azimuth, wave angular frequency, and incidence angle, as shown in (7).

Step 4: Construct the normal matrix $\mathit{B}$ and solve the model coefficients $\widehat{\mathit{x}}$ based on the scheme proposed in Section 2.2.

Step 5: By substituting the model coefficients $\widehat{\mathit{x}}$ obtained in Step 3 into (1), the reconstruction and prediction of the wave surface elevation can be achieved.

3. Simulation

This study simulated the sea surface using the JONSWAP wave spectrum [33]. The expression for the JONSWAP spectrum can be written as follows:

$$S\left(\omega \right)=\alpha g^2{\omega }^{-5}exp[-{\displaystyle \frac{5}{4}}{({\displaystyle \frac{\omega }{{\omega }_p}})}^{-4}]{\gamma }^{exp[-{\displaystyle \frac{{(\omega -{\omega }_p)}^2}{2{\sigma }^2{\omega }_p^2}}]}$$

(12)

where $\gamma =3.3$ is the spectral peak enhancement factor; $\sigma$ denotes the spectral width parameter, where $\sigma =0.07$ when $\omega \le {\omega }_p$ and $\sigma =0.09$ when $\omega >{\omega }_p$; and the scaling coefficient $\alpha$ and the peak frequency ${\omega }_p$ depend on the wind fetch X and the wind speed $U_{10}$, where $\alpha =0.076{({\displaystyle \frac{gX}{U_{10}^2}})}^{-0.22}$ and ${\omega }_p=7\pi ({\displaystyle \frac{g}{U_{10}}}){({\displaystyle \frac{gX}{U_{10}^2}})}^{-0.33}$. The wind fetch was set to 80 km and the wind speed was set to $U_{10}=12$ m/s. The radar range resolution was 7.5 m, with 280 range cells, equivalent to 2100 m. The coherent accumulation time was 0.5 s, and the simulation time was 512 s, which corresponded to 1024 time points. In actual sea wave observations, the broken waves effect of the sea surface leads to nonlinear features in the observed radial velocity series of coherent microwave radar. The introduction of the broken-wave mechanism can be found in [34]. Based on the above simulation parameters, Figure 5 shows the radial velocity series obtained by simulation. Figure 5a,b represent the spatial-temporal velocity series without and with broken waves, respectively. Figure 5c shows the velocity time series of a single range cell. From the figures, it can be seen that broken waves cause intermittent distortions in the observed velocity series. Figure 6 depicts the simulated and S-band radar-observed wave spectra. The blue circle line represents the wave spectrum observed by antenna 4 at 20:11 on 8 April 2024. The green dotted lines and the black dotted lines represent the simulation results of the wave spectrum without breaking waves (corresponding to Figure 5a) and with breaking waves (corresponding to Figure 5b), respectively. It can be seen that the wave spectrum changed due to the nonlinear characteristics of the breaking waves, additional components are generated at low frequencies in the wave spectrum, and the energy of the dominant wave was reduced, which was consistent with the conclusions of our previous work [15,34]. In addition, the spectrum shape of the radar observation was consistent with the simulated spectrum shape of the case with breaking waves, which indicates that the selected wave spectrum (JONSWAP) could effectively simulate the statistical characteristics of the real ocean waves, and the impact of the breaking waves could not be ignored.

The spatial-temporal velocity series was used for model parameter estimation, and the observation region covered the whole spatial-temporal plane. The model coefficients $(a_{ij},b_{ij})$ and amplitudes $A_{ij}=\sqrt{a_{ij}^2+b_{ij}^2}$ were calculated by the method proposed in Section 2.2 and L-curve regularized least squares (L-curve RLS) [25], respectively. Figure 7 shows the estimated model parameters. The results in Figure 7 show that the estimated parameters of the proposed method were consistent with the target parameters, while the estimated parameters based on the L-curve RLS method produced additional components at low frequencies. The additional low-frequency components are nonlinear components caused by broken waves [15,34], which indicates that the proposed method outperformed the L-curve RLS method when dealing with the nonlinear interference.

The DSWRP was performed using the spatial-temporal velocity series, where we selected a temporal sample from 220 s to 290 s and a spatial area from the 100th to 180th range bins as the measurement region in the spatial-temporal plane. The reconstructed and predicted sea surface elevation was compared with the target sea surface elevation, and Figure 8 shows the absolute error of the instantaneous wave height. Figure 8a,b show the absolute spatial-temporal difference of the two methods after 50 Monte Carlo experiments, respectively, and the blue rectangular box is the measurement region. Figure 8c compares the wave surface elevation at the 140th range cell under a single simulation experiment. Figure 8d shows the results of Figure 8a,b at the 140th range cell. The figure shows that when the observed velocity series was disturbed, the absolute errors of the predicted and reconstructed sea surface elevations of the proposed scheme were lower than those of the L-curve RLS method. The statistical parameters are shown in

Table 1. Statistical parameters of simulation (L-curve RLS method/proposed method).

Parameters	Values
CC	0.93/0.99
MAE	0.16 m/0.04 m
RMSE	0.19 m/0.05 m

, where the proposed method outperformed the L-curve RLS method in all parameter indicators, indicating that the proposed scheme had more advantages in implementing DSWRP technology. It is worth noting that compared with the single space or time observation method, this space-time joint observation method can expand the range of the predictable region. Details of the predictable region can be found in our previous work [35].

4. Verification by Experimental Data

4.1. Experiment Description

The dataset for this experiment was obtained during the East China Sea trial on 8 April 2024, as shown in Figure 9a. The S-band coherent radar is deployed on the top deck of the XIANGYANGHONG-18, as shown in Figure 9b. In addition, an X-band radar was installed on the deck to observe the real-time sea surface height at the ship’s position. The X-band radar in this experiment was a high-precision range finder with a working frequency of 9.6 GHz, a detection range of 1–23 m, a detection accuracy of 10 mm, and a time accuracy of milliseconds. X-band radar measures the distance from the sea surface to the instrument, so the real-time elevation information of the sea surface can be obtained by subtracting the mean value. The experiment also provided attitude sensors, meteorological instruments, and a GPS positioning system. The coherent S-band radar used a LFMICW signal as the transmission signal for single-station operation. The radar adopted six antennas to operate in TDM address mode. Each antenna had a horn shape and a horizontal beam width of 30°, which covered the echo information over 180° of the sea surface. Each antenna collected 3 min of sea echo signals along its radial direction.

Table 2. Waveform parameters.

Parameters	Values
Range resolution	7.5 m
Sampling time	0.26 s
Frequency sweep period	2032 $\mathsf{\mu }$s
Operating frequency	2.85 GHz

illustrates the waveform parameters. The height of the antenna installation was 18 m, and the S-band radar used VV polarization mode. As shown in Figure 10, the sea wave moved toward antenna 4 during the experiment.

4.2. Experimental Results

Figure 11 shows the echo signals collected by antenna 4 from 20:08 to 20:11 on 8 April 2024. Figure 11a illustrates the time Doppler spectra of the 30th range bin, Figure 11b shows the radial velocity series estimated by the spectral moment method [32], and Figure 11c shows the wavenumber–frequency spectrum obtained by the 2D-FFT using Figure 11b. In Figure 11a, “spikes” appear due to intermittent broadening toward the right caused by the breaking or distortions of the sea surface [34]. The spatio-temporal velocity series in Figure 11b shows sporadic yellow spots with the characteristics of breaking waves. The corresponding wavenumber–frequency spectrum (Figure 11c) not only includes the dispersion curve representing the wave components but also features a “Group-line” caused by broken waves. These phenomena are consistent with the nonlinear characteristics described in [15].

To evaluate the proposed DSWRP scheme, a spatial-temporal sample that consisted of a time range of 70 s to 110 s and a spatial range of cells 30 to 63 was chosen as the measurement region. Two methods were used to predict the deterministic waves in other spatial-temporal areas. Meanwhile, the velocity potential function scheme [21] was adopted to reconstruct the sea surface elevation of the whole measurement region of the space-time velocity series shown in Figure 11b. The absolute errors of the prediction and reconstruction are shown in Figure 12, and the results for the 47th range cell were quantitatively analyzed, with the statistical parameters shown in

Table 3. Statistical parameters (L-curve RLS method/proposed method).

Parameters	Values
CC	0.77/0.85
MAE	0.25 m/0.12 m
RMSE	0.33 m/0.17 m

. The results of these figures and tables indicate that the proposed method performed better than the L-curve RLS method in practical radar observations.

To visually discuss the performance of the CN-RLS method in terms of prediction, the sea surface elevation at the ship’s position was predicted. The velocity time series of the first range bin $(r=0$ m) was taken as the measured sample. The predicted wave surface height at the position of XIANGYANGHONG-18 after 1 min was compared with the real-time elevation observed by X-band radar, as shown in Figure 13. The scatter plots of the predicted sea surface elevation by the coherent S-band radar and the observed sea surface elevation by the X-band wave-measuring radar are shown in Figure 14. The solid red and blue lines represent the linear least squares fitting results of the proposed and L-curve RLS methods, respectively. The $y_P$ and $y_L$ represent the fitting functions of the proposed and L-curve RLS methods, respectively. The statistical parameters are shown in

Table 4. Statistical parameters of experiment (L-curve RLS method/proposed method).

Parameters	Values
CC	0.67/0.76
MAE	0.26 m/0.18 m
RMSE	0.35 m/0.22 m

. The proposed method had a CC of 0.76, MAE of 0.18 m, and RMSE of 0.22 m, while the CC, MAE, and RMSE of the L-curve RLS method were 0.67, 0.26, and 0.35 m, respectively. The fitting coefficient and bias of the L-curve RLS method were 0.83 and $-0.02$ m, respectively, while the fitting coefficient and bias of the proposed method were 0.89 and $-0.01$ m, respectively. Based on the above experimental results and statistical analysis, it was shown that the proposed method could improve the prediction accuracy of the deterministic waves. The improvement of the deterministic wave field prediction accuracy provides a more reliable safety warning for the safe take-off and landing of carrier aircraft and provides security for the installation of offshore platforms, protection of ocean engineering structures, wave energy collection, and other scenarios, which is of great significance for improving maritime combat capabilities and ensuring maritime combat safety.

5. Discussion

Coherent S-band radar relies on robust wave velocity for DSWRP. When the wind speed is large enough, the sea surface inevitably generates broken waves. The high spatial-temporal resolution of coherent S-band radar makes it possible to detect these nonlinear details, which interfere with the estimated sea wave velocity. Therefore, the impact of breaking waves should be considered when implementing the DSWRP technique.

The coherent S-band radar DSWRP technique based on the CN-RLS method is proposed in this article. The normal matrix ${\mathit{P}}^T\mathit{P}$ was modified by reducing the condition number to weaken its ill-posedness, and the nonlinear interference caused by breaking waves was largely suppressed.

We discussed the performance comparison of DSWRP between the two methods under different wind speed conditions. The simulation conditions were consistent with Section 3, where the wind speed ($U_{10}$) was 4 m/s–25 m/s and the interval was 1 m/s. The CC between the two methods was analyzed using the wave spectrum obtained by simulation. Figure 15 shows the CC and RMSE variations in the two methods with wind speed. As the wind speed increased, the CC of the proposed method maintained a high level (larger than 0.97), while the CC of the L-curve RLS method decreased significantly. The RMSE of both methods increased with the increase in wind speed, but the RMSE of the L-curve RLS method was higher than that of the proposed method. These results reflect the advantages of the proposed method.

However, the method proposed in this paper is intended to be carried out under the ship’s stopped sailing state or on a fixed platform. For the ship’s sailing state or a moving platform, the influence of the ship’s speed and the ship’s six degrees of freedom should be considered [36], which needs further research in the future.

6. Conclusions

This article proposes a new method for S-band coherent radar-based DSWRP. The proposed method considers the influence of nonlinear characteristics, such as breaking waves on DSWRP, and proposes using the CN-RLS method to solve the model coefficients. From the simulation results, the CN-RLS method could reduce the condition number of the ill-posed equation. Compared with the L-curve RLS method, the CN-RLS method was less sensitive to outliers and could effectively weaken the impact of breaking waves on the DSWRP. In addition, through the analysis of the sea test data carried out in the East China Sea in April 2024, the wave surface displacement predicted by S-band coherent radar was compared with the wave surface displacement observed by X-band radar. The results show that the proposed DSWRP scheme was superior to the L-curve RLS method, and the RMSE and CC were 0.22 m and 0.76, respectively.

Future work will mainly carry out research on DSWRP technology on a mobile platform to verify the proposed scheme’s limitations and the improvement direction in more sea states.