A SAR Echo Simulation Method for Ship Targets in the Sea Based on Model Segmentation and Electromagnetic Scattering Characteristics Simulation

Highlights

What are the main findings?

• By using 3D modeling software and electromagnetic simulation software, it is possible to realize the segmentation of complex targets and the simulation of their electromagnetic scattering characteristics quickly, thereby obtaining the RCS data of targets.
• Given the known target RCS data, the physical working process of SAR can be simulated rapidly and completely through software algorithm, thereby enabling the simulation of SAR echo signals.

What are the implications of the main findings?

• This study demonstrates that SAR echo simulation for ship targets in the sea can be fully achieved through software simulation and numerical calculation. In environments with complex conditions and limited software and hardware resources, SAR echo simulation can be accomplished quickly and accurately at a low cost.

Abstract

The simulation of synthetic aperture radar (SAR) echo signals usually relies on complex hardware equipment and a large amount of scene data, which results in high costs and low efficiency. In order to simulate SAR echo signals of ship targets in the sea quickly and accurately in complex environments at a lower cost, this paper proposes a SAR echo simulation method based on model segmentation and electromagnetic scattering characteristic simulation. This method first implements the simulation of sea models under different sea conditions based on PM wave spectrum model and the Monte Carlo method, and segments them according to the requirements of simulation resolution. Then, it uses Python API 3.11 in Blender 4.5 to segment the ship model automatically and optimize the visible surface elements and mesh for each sub-model. Next, it uses Lua API in Feko to simulate the electromagnetic scattering characteristics of each sub-model of the sea and the ship target automatically, and obtains the required radar cross section (RCS) data of the ship target in the sea after processing. Finally, SAR echo simulation is realized through dual-channel technology. To further verify the simulation result, the chirp scaling (CS) algorithm is used for imaging processing. The results show that this method can realize SAR echo simulation of various ship targets under different sea conditions in a quick, accurate and cost-effective manner without the need for any hardware equipment.

1. Introduction

The ocean plays a crucial role in global transportation and economic exchange. With the increasing frequency of marine resource exploitation and global maritime activities, the monitoring of ship targets in the sea has become a key technical requirement for safeguarding maritime rights and maintaining maritime security. Synthetic aperture radar (SAR) is an active microwave remote sensing technology. By the coherent processing of scattered signals obtained from the movement trajectory of radar platform, SAR can synthesize a larger antenna aperture effectively, thereby breaking through the physical limitation of antenna size and achieving higher resolution beyond that of traditional real-aperture radars. SAR has the ability to capture images day and night. Its working wavelength is usually between 3 and 75 cm (equivalent to X, C, S, L and P bands), which enables SAR signals to penetrate clouds, vegetation and certain surface materials effectively [1]. Therefore, SAR plays a significant role in applications such as Earth observation, environment monitoring and marine management.

Obtaining real SAR echo signals of ship targets in the marine environment is not only costly and inefficient, but also subject to various limitations such as weather and equipment conditions. Therefore, establishing a high-precision simulation system to simulate SAR echo data in actual scenarios is of great significance. By simulating SAR echo signals, various complex marine scenarios can be realistically simulated, including the characteristics of sea clutter under different sea conditions, the movement patterns of different ship targets, and different SAR imaging mechanisms [2,3]. This provides a virtual testing environment for the design and optimization of SAR systems, helping to analyze the system performance deeply and reducing the cost and risks of actual development.

In recent years, various SAR echo simulation technologies have been developed, which can be broadly classified into two categories: those based on hardware and those based on software. Hardware-based simulation methods aim to reproduce SAR echo signals through physical devices, such as radar target simulators and hardware-in-the-loop (HIL) systems. These methods can generate highly realistic echo signals by modeling the signal transmission, delay, Doppler effect and system response directly in real time. In 2014, Yuan developed an FPGA-based SAR echo simulator to address the high computational burden of traditional software-based echo generation. By implementing the SAR signal generation algorithm on a dedicated digital signal processing board, the simulator is capable of producing echo signals in a significantly accelerated manner with reduced energy consumption [4]. In 2015, Xu proposed a real-time SAR echo simulator based on multi-FPGA parallel computing architecture. The system employs a pipeline structure and multi-channel parallel processing to accelerate echo generation, enabling real-time simulation for both point targets and natural scenes [5].

Although hardware-based SAR simulators provide high real-time performance and fidelity, they are typically optimized for fixed or small-scale test scenarios, such as single-point targets or static environments. When applied to large-scale, dynamic marine scenes with multiple moving vessels or complex sea clutter, these systems become difficult to adapt without substantial redesign, limiting their flexibility and scalability across different application domains.

In contrast to hardware-based simulators, software-based SAR simulation techniques provide greater adaptability and lower implementation cost by numerically modeling the radar signal formation process. Broadly speaking, software approaches can be classified into image-level simulation and echo-level simulation. Image-level methods focus on generating SAR images directly by modeling scene reflectivity or scattering characteristics without simulating SAR raw echo signal explicitly. These methods are highly efficient in computation, but at the cost of sacrificing the physical consistency with actual SAR signal generation [6].

By contrast, echo-level simulation methods seek to reproduce complete SAR signal acquisition processes, including electromagnetic wave propagation, target scattering and coherent signal collection at the sensor, followed by processing with conventional imaging algorithms. A fundamental and widely adopted framework is the time-domain point target method, which has been systematically formulated in the literature, notably in Synthetic Aperture Radar Processing by Franceschetti. In this method, the observed scene is discretized into a collection of point scatterers, and the raw echo is synthesized by coherently summing the time-delayed and phase-modulated contributions from each scatterer according to the radar geometry [7]. This approach provides high physical fidelity and strong modeling flexibility, but suffers from extremely high computational complexity when applied to large-scale scenes. In 2005, Cumming presented a frequency-domain framework for SAR signal modeling in Digital Processing of Synthetic Aperture Radar Data, where SAR system is characterized by a two-dimensional transfer function in range–Doppler or wavenumber domain. Based on this framework, echo signals can be generated efficiently via inverse transforms [8]. This method offers high computational efficiency, but is generally less flexible in handling complex motion and scattering scenarios compared with time-domain approaches.

Whichever category is used, a core technical challenge in software SAR simulation lies in accurate modeling of scattering characteristics of the scene and targets, because radar cross section (RCS) and its spatial and temporal variation directly determine the amplitude and phase of received echo signal. In 2024, Hua proposed a time-domain RCS modeling method based on a forward scattering center model, which constructs the unit impulse response of complex targets and convolves it with arbitrary wideband signals to generate synthetic echoes [9]. This approach improves computational efficiency significantly and allows flexible adaptation to different waveforms, though it mainly targets static objects and relies on the accuracy of the scattering center model for more complex or dynamic scenarios. In 2025, Li focused on marine ship targets, combining physical optics-based RCS computation with echo generation algorithms to simulate SAR echo under varying incident angles, distances and target materials [10]. This method produces realistic amplitude variations in simulated echo, but its applicability is mainly limited to ships and sea targets, and it provides only simplified phase information, which may restrict its use in high-fidelity coherent imaging scenarios.

Although the aforementioned methods can effectively generate SAR echo or simulate target RCS in specific scenarios, they each have certain limitations. Some approaches are only suitable for particular target types or static scenes and struggle to handle large-scale sea surfaces with multiple ships. High-fidelity electromagnetic simulation, while accurate, is computationally expensive and time-consuming, whereas fast RCS modeling methods offer efficiency but have limited capability in capturing dynamic behaviors or complex scattering structures.

This paper aims to develop a complete method for simulating the SAR echo signals of various ship motion targets under complex sea conditions. Therefore, this paper has referenced, integrated and improved various existing methods, and combined multiple software platforms to propose a SAR echo simulation method based on model segmentation and electromagnetic scattering characteristic simulation. This method aims to maximize efficiency while ensuring authenticity and accuracy of the simulation, providing a practical solution for SAR echo simulation of large-scale sea surface ship targets.

The content arrangement of this article is as follows: Section 1 is the introduction, where some existing technologies are summarized and analyzed, and the purpose and significance of this research are clarified. Section 2 is materials and methods, which is further divided into four sections. The first part focuses on the research of sea surface model simulation and segmentation methods under complex sea conditions. The second part studies a method for segmenting complex ship models rapidly. The third part focuses on the simulation methods for electromagnetic scattering characteristics of the sea surface and the ship model. The fourth part studies an efficient and accurate method for simulating SAR echo signals. Section 3 corresponds to Section 2, and it presents the results of each part of the research, as well as conducts analysis and verification. Section 4 conducts a detailed analysis and discussion on the application value, shortcomings, and subsequent improvement directions of the research content of this article. Section 5 summarizes the research work and results of this paper.

2. Materials and Methods

2.1. Simulation and Segmentation of Sea Model

Wave spectrum refers to the distribution of wave energy on sea surface in terms of frequency and direction. It is obtained through Fourier transform of the autocorrelation function of sea surface height variations and reveals the second-order statistical characteristics of waves and the distribution laws of wave energy in different wavelengths and directions. By establishing an actual sea model based on wave spectrum, the wave fluctuation characteristics under various sea conditions can be described, which helps to improve accuracy, prediction ability and practical application of the sea surface electromagnetic scattering model, and reveal the influence of different marine environments on electromagnetic wave scattering [11,12]. The commonly used wave spectrum includes the PM spectrum, the JONSWAP spectrum, the Elfouhaily spectrum, etc.

The PM wave spectrum is one of the most classic and fundamental empirical wave spectrum models. It was derived by Pierson and Moskowitz based on the analysis of a large amount of measured data from the North Atlantic. Its core assumption is that the sea surface is in a fully developed state, that is, the wind speed is constant, the wind direction is constant, the wind blowing time is long enough, and the wind blowing distance is infinite. At this time, wave energy can only be determined by wind speed and reaches statistical equilibrium [13]. The PM wave spectrum can be expressed as follows:

$$S\left(k\right)={\displaystyle \frac{\alpha }{4{\left|k\right|}^3}}\mathrm{e}\mathrm{x}\mathrm{p}\left(-{\displaystyle \frac{\beta g_c^2}{k^2{U}_{19.5}^4}}\right)$$

(1)

where $\alpha =8.10\times {10}^{-3}$ and $\beta =0.74$ are dimensionless empirical constants, $g=9.81 \mathrm{m}/{\mathrm{s}}^2$ is the gravitational acceleration, $k$ is wave number and $U_{19.5}$ is wind speed at a height of 19.5 m above sea level. Its relationship with $U_{10}$, wind speed at a height of 10 m above sea level is: $U_{19.5}\approx 1.026U_{10}$.

The JONSWAP wave spectrum is an empirical wave spectrum derived from measurements in the North Sea, designed to represent fetch-limited seas. Unlike the PM spectrum, which assumes a fully developed sea, the JONSWAP spectrum accounts for finite fetch effects by introducing a peak enhancement factor $\gamma$, which sharpens the spectral peak near the dominant wavenumber. This allows it to better capture the energy concentration in developing seas and the associated higher wave heights around the spectral peak. For fully developed sea, the peak enhancement factor $\gamma$ is 1, and the JONSWAP spectrum simplifies to the PM spectrum [14].

The Elfouhaily wave spectrum is a unified directional sea surface spectrum that combines both long gravity waves and short capillary-gravity waves. It is derived based on a combination of theoretical considerations and field observations, providing a smooth transition between low-frequency (gravity) and high-frequency (capillary) components [15].

Internationally, the sea states codes based on Douglas sea scale is commonly used to describe the level of ocean waves. It uses wind speed and wave height to describe specific changes on the sea surface. The corresponding relationship is shown in

Table 1. Sea states codes based on Douglas sea scale.

Code	${\mathbf{H}}_{\mathbf{s}}$ (m)	Characteristics
0	0	Calm (glassy)
1	<0.1	Calm (rippled)
2	0.1–0.5	Smooth (wavelets)
3	0.5–1.25	Slight
4	1.25–2.5	Moderate
5	2.5–4	Rough

[16].

The Monte Carlo method conducts inversion simulation based on the wave spectrum model, thereby obtaining actual wave fluctuations. It is the most commonly used two-dimensional linear sea surface modeling method. This method regards waves as a superposition of a series of harmonics with different wavelengths, periods, and initial phases. Firstly, it converts the white noise to frequency domain through Fourier transformation. Then, it uses wave spectrum to filter the result. Finally, it obtains the expression of filtered rough sea surface height fluctuations through inverse Fourier transformation [17,18]:

$$h(\overrightarrow{r},t)={\sum }_{k}A(\overrightarrow{k},t)\mathrm{e}\mathrm{x}\mathrm{p}((i\overrightarrow{k}·\overrightarrow{r})$$

(2)

$$A(\overrightarrow{k},t)=\gamma (\overrightarrow{k})\sqrt{2S(\overrightarrow{k},\phi )\delta k_x\delta k_y}{e}^{-i\omega t}+{\gamma }^{\ast }(-\overrightarrow{k})\sqrt{2S(\overrightarrow{k},\mathrm{\pi }-\phi )\delta k_x\delta k_y}{e}^{i\omega t}$$

(3)

where $\overrightarrow{k}=\left(k_x,k_y\right)$, $\delta k_x=2\mathrm{\pi }/L_x$, $\delta k_y=2\mathrm{\pi }/L_y$, $L_x$ and $L_y$ respectively represent the total length of the sea surface in $x$ and $y$ directions. $\gamma (\overrightarrow{k})$ is a complex Gaussian random sequence, $\ast$ denotes taking complex conjugate, $S(\overrightarrow{k},\phi )$ represents two-dimensional sea spectrum. To ensure that the sea surface height $h(\overrightarrow{r},t)$ is a real number, the Hermitian form of equation $A(\overrightarrow{k},t)$ should satisfy following condition:

$$A\left(k_x,k_y\right)=A^{\ast }\left(-k_x,-k_y\right),A\left(k_x,-k_y\right)=A^{\ast }\left(-k_x,k_y\right),$$

(4)

Finally, through IFFT, it obtains that:

$$h=\mathrm{I}\mathrm{F}\mathrm{F}\mathrm{T}\left[A\right]$$

(5)

Throughout the entire process, FFT and IFFT are required. Therefore, the spatial geometric length of the sea surface needs to be discretized, and at the same time, the Nyquist theorem must be satisfied:

$$f_s={\displaystyle \frac{1}{\Delta x}}\ge 2·{\displaystyle \frac{K_{cut}-K_{min}}{2\mathrm{\pi }}}$$

(6)

where $f_s$ represents the sampling frequency in $x$ direction, $\Delta x$ represents the sampling interval of $x$. Similarly, the sampling in $y$ direction also needs to comply with the Nyquist theorem. $K_{cut}$ and $K_{min}$ respectively represent cutoff wavenumber and minimum wavenumber, and their values determine bandwidth of the band to be extracted on the wave spectrum. Due to different energy intervals of wave spectrum corresponding to different wind speeds, it is necessary to ensure that the energy distribution range of the wave spectrum under current wind speed is as much as possible included within the wave number range $K_{cut}-K_{min}$ [19].

To compare the differences among the PM spectrum, the JONSWAP spectrum and the Elfouhaily spectrum, taking 3-level sea condition as an example, the sea model simulations were conducted using these three models respectively. The time consumption, RMS (root mean square) height, skewness, and kurtosis were measured as indicators. Among them, the RMS height $\sigma =\sqrt{〈{\eta }^2〉}$, which is equivalent to the standard deviation of sea surface height and characterizes the intensity of sea surface’s undulation. The skewness $S_k={\displaystyle \frac{〈{\eta }^3〉}{{\sigma }^3}}$, which is used to measure whether the distribution is symmetrical. When the skewness is equal to 0, the sea surface is completely symmetrical. When the skewness is greater than 0, the wave peaks on sea surface are sharper and the wave troughs are flatter. When the skewness is less than 0, the troughs on the sea surface are deeper. The kurtosis $K_u={\displaystyle \frac{〈{\eta }^4〉}{{\sigma }^4}}$, which is used to measure the sharpness of the distribution. When the kurtosis is equal to 3, the sea surface conforms to a Gaussian distribution. When the kurtosis is greater than 3, more peaks appear on the sea surface. When the kurtosis is less than 3, the sea surface becomes smoother [20,21]. The simulation results and analysis can be found in the first part of Section 3.

In order to facilitate the subsequent simulation of electromagnetic scattering characteristics, the complete sea model needs to be divided into multiple sub-blocks, and each sub-block should be triangulated. Then, it should be saved as a STL format file. STL format is the most common file format in 3D printing field, which is used to record triangular face element data of the model surface. The principle of converting ordinary matrix data into STL format is shown in Figure 1.

The STL format file stores two matrices, P and T. Matrix P converts the data from the original two-dimensional matrix H into a one-dimensional matrix, while matrix T is composed of multiple three-dimensional vectors. Each vector records the sequence of three vertex data of a certain triangular face element in matrix P. In this paper, a local data matrix H is extracted from the complete sea model and converted into matrix P each time. Then, an autonomous design combination algorithm is used to generate matrix T, so that every four (2 × 2) adjacent sampling points in matrix H form a square that is divided into two triangular face elements in an “upper right—lower left” manner. Finally, it is exported in STL format to achieve the segmentation of the entire sea model.

2.2. Ship Target Model Segmentation

The segmentation of the ship target model is achieved through Blender. Blender is a powerful and lightweight 3D graphics and image software that offers a wide range of functions such as 3D modeling, materials and textures, animation system, rendering engine and visual effects. It also has a complete Python API that enables quick execution of various operations within the software, making it highly suitable for large-scale repetitive model segmentation tasks [22].

The main steps for model segmentation in Blender are shown in Figure 2.

• Import model file and create a copy to avoid operating on the initial model directly, as shown in Figure 2a;
• Set segmentation units and construct a rectangular prism cutter based on the requirements of simulation resolution, as shown in Figure 2b;
• Create a Boolean modifier, adjust the solver parameters, perform a Boolean intersection operation on the cutter and the model copy, as shown in Figure 2c;
• Utilize ray detection method to optimize the segmentation results. Emit rays from the location of SAR towards the model, remove completely invisible elements and clean up isolated edges and vertices, as shown in Figure 2d;
• Export the segmentation results in STL format and delete the temporary objects.

By using Python script programs, the above operations can be carried out automatically. Firstly, construct a temporary rectangular cutter. Then, determine the boundary values around entire model manually and build a segmentation grid according to the requirements of simulation resolution. Next, realize the above model segmentation function by writing a script program and adjusting construction position of the cutter through grid coordinates continuously. Finally, the automatic segmentation function of the entire model is achieved.

2.3. Simulation of Electromagnetic Scattering Characteristics of Sea Surface and Ship Targets

The electromagnetic scattering characteristics simulation of sea surface and ship targets is achieved through Feko. Feko is a powerful three-dimensional full-wave electromagnetic field simulation software, renowned for its hybrid solution technology and ability to handle problems involving large-scale and complex structures. It also includes a comprehensive Lua API, enabling the automation of electromagnetic scattering characteristic simulation tasks for a large number of models through script programming.

Feko possesses a wide range of solution methods which are applicable to various different simulation scenarios, such as the Method of Moments (MOM), the Multi-Layer Fast Multipole Method (MLFMM), as well as the Physical Optics Method (PO), the Geometrical Optics Method (GO) and the Large Element Physical Optics Method (LEPO), which are adapted to high-frequency conditions.

The PO is based on integral equations and is grounded on surface currents. It assumes that each point on the scattering body is largely independent of other points, and the resulting effects can be disregarded. By decomposing the target into many triangular sub-element surfaces and solving the scattering fields of each sub-element independently based on the integral of incident field, the sum of all the element solutions is thus obtained. Since the summation of scattering fields of the triangular sub-element surfaces is achieved by converting the area integral into an integral-free calculation, the computational load is reduced and the calculation speed is improved, making it very convenient to solve scattering fields of large-scale electrical objects [23,24]. Figure 3 shows the schematic diagram of electromagnetic scattering model of the PO, where $\overrightarrow{E^i}$ and $\overrightarrow{H^i}$ are incident fields, $\overrightarrow{E^s}$ and $\overrightarrow{H^s}$ are scattering fields, $\widehat{n}$ is unit normal vector of the target, and $\overrightarrow{J_s}$ and $\overrightarrow{J_{ms}}$ are surface currents and surface magnetic flux densities of the target.

From Stratonovich-Zhilin Integral Equation of Electromagnetic Field, it can be obtained that:

$$\overrightarrow{E^s}={\int }_s\left[j\omega \mu \left(\widehat{n}\times \overrightarrow{H}\right)G_0+\left(\widehat{n}\times \overrightarrow{E}\right)\times \nabla G_0+\left(\widehat{n}\times \overrightarrow{E}\right)\nabla G_0\right]ds$$

(7)

$$\overrightarrow{H^s}={\int }_s\left[j\omega \epsilon \left(\widehat{n}\times \overrightarrow{E}\right)G_0-\left(\widehat{n}\times \overrightarrow{H}\right)\times \nabla G_0+\left(\widehat{n}\times \overrightarrow{H}\right)\nabla G_0\right]ds$$

(8)

where $\nabla G_0$ represents the gradient of free-space Green’s function,

$$G_0={\displaystyle \frac{e^{-ikr}}{4\mathrm{\pi }r}}$$

(9)

where $r$ represents the distance from the source point to the field point. The calculations of target surface current $\overrightarrow{J_s}$ and surface magnetic flux density $\overrightarrow{J_{ms}}$ are shown as follows:

$$\overrightarrow{J_s}=\widehat{n}\times \overrightarrow{H}, \overrightarrow{J_{ms}}=-\widehat{n}\times \overrightarrow{E}$$

(10)

$${\rho }_s=\epsilon \widehat{n}·\overrightarrow{E}, {\rho }_{ms}=\mu \widehat{n}·\overrightarrow{H}$$

(11)

where ${\rho }_s$ and ${\rho }_{ms}$ represent the density of target surface charges and surface magnetic charges. By rearranging the above equation, it can be obtained that:

$$\overrightarrow{E^s}={\int }_s\left[j\omega \mu \overrightarrow{J_s}{G}_0-\overrightarrow{J_{ms}}\times \nabla G_0+{\displaystyle \frac{{\rho }_s}{\epsilon }}\nabla G_0\right]ds$$

(12)

$$\overrightarrow{H^s}={\int }_s\left[j\omega \epsilon \overrightarrow{J_{ms}}{G}_0+\overrightarrow{J_s}\times \nabla G_0+{\displaystyle \frac{{\rho }_{ms}}{\mu }}\nabla G_0\right]ds$$

(13)

In the process of grid division using the PO, grid edge length is limited by wavelength. When the incident frequency is high and the target is large, the number of grids to be divided is extremely large, and the requirements for memory and time in the solution process are still relatively high. The LEPO improves it by correcting the phase of basis function and using multiple wavelengths to divide the target. This reduces the number of divided grids significantly, thereby reducing memory and time required for the calculation, which is conducive to the simulation of electromagnetic scattering characteristics of large-scale ship targets [25,26]. The corrected expression of the basis function is as follows:

$$f_n\left(r\right)=\left\{\begin{array}{c}{\displaystyle \frac{l_n}{2{A^{+}}_n}}{p_n}^{+}{e}^{-jk\left({p_n}^{+}-{p_{nc}}^{+}\right)},r\in {T_n}^{+}\\ {\displaystyle \frac{l_n}{2{A^{-}}_n}}{p_n}^{-}{e}^{-jk\left({p_n}^{-}-{p_{nc}}^{-}\right)},r\in {T_n}^{-}\end{array}\right.$$

(14)

In FEKO, simulations were conducted for flat plate, dihedral angle and ship target using multiple methods. The results are shown in

Table 2. Performance and efficiency test results of several typical methods.

Model	Size	Algorithm	Memory (MB)	Time Consumption (h)
Flat plate	$5\lambda$	MOM	823	1.86
Flat plate	$5\lambda$	MLFMM	2.9	0.91
Flat plate	$5\lambda$	PO	163	0.16
Dihedral angle	$6\lambda$	MOM	1121	2.17
Dihedral angle	$6\lambda$	MLFMM	3.3	1.31
Dihedral angle	$6\lambda$	PO	216	0.36
Ship	$3660\lambda$	LEPO	21,460	26.3

:

From the results in Table 2, it can be seen that the memory required for the MOM and MLFMM operations is much greater than that of the PO. Moreover, as the target size increases, the operation time will multiply exponentially, and it may even be impossible to calculate. For the LEPO derived from the PO, the computing time is significantly reduced, the memory usage is reasonable, and there is no significant difference in accuracy compared to other methods. Therefore, the engineering requirements can be met [27].

The main steps for conducting electromagnetic scattering characteristic simulations in Feko are as follows:

• Create the engineering file and configure the simulation environment, including dielectric constant, magnetic permeability, dielectric properties, etc.;
• Import model files and perform mesh subdivision. The models used in this paper are all in STL format. During model segmentation process, the mesh subdivision work has been completed simultaneously;
• Set the excitation source to be far-field excitation of a plane wave. Set the excitation parameters referring to the parameters of SAR system, including wavelength, polarization form, azimuth angle, elevation angle, etc.;
• Choose solution method as the LEPO. Set the solution range to excitation direction and adjust parameters such as solution accuracy. Then, begin simulation.

By using Lua script programs, the above operations can be executed repeatedly and quickly, enabling the automation of a large number of electromagnetic scattering characteristic simulation tasks for models. At the same time, working status can be monitored in real time through process logs, error logs, etc. Simulation process and results are completely recorded in the “.out” files, including CPU/GPU thread scheduling, electric/magnetic field intensity, phase, RCS, etc. By extracting and integrating the RCS data in each file, complete RCS simulation results of the model can be obtained.

The mesh size in the LEPO is determined based on both wavelength and geometric features of the target. In general, the facet size is chosen to be smaller than $\lambda /10$ to ensure accurate representation of induced currents. For regions with sharp edges or strong current variations, a finer discretization ($\lambda /15$–$\lambda /20$) is adopted. Additionally, geometric features are resolved with at least 3–5 elements per smallest structural dimension [28].

In order to determine the appropriate grid size, simulations were conducted for the targets at different grid resolutions, and the convergence of the test results was analyzed. Define relative error:

$$\epsilon ={\displaystyle \frac{\left|{\sigma }_n-{\sigma }_{ref}\right|}{{\sigma }_{ref}}}$$

(15)

When $\epsilon <0.02$, it can be considered as converging. The test results and analysis are presented in the third part of Section 3.

2.4. SAR Echo Simulation

The simulation of SAR echo signals mainly includes two methods: time-domain algorithm and frequency-domain algorithm. The frequency-domain algorithm first performs a Fourier transform on RCS data of the imaging scene and multiplies it by the SAR system response function. Then, through inverse Fourier transform, it returns to the time domain to generate echo signal. This method has a relatively small computational load, but it cannot introduce the velocity beamforming effect during echo generation process, and the generated echo signal has limited accuracy. The time-domain algorithm obtains echo signal by simulating the actual working process of SAR. Although it has a large computational load, the generated echo signal is more accurate [29].

When using the time-domain algorithm, it is assumed that the SAR operates in forward-looking mode. According to the SAR working principle and the SAR working parameters set during simulation process, the echo signal model of a single-point target is:

$$\begin{array}{ll}{s}_r\left(t\right)={\displaystyle \sum _{n=-\infty }^{\infty }}\sigma \omega & ·\mathrm{r}\mathrm{e}\mathrm{c}\mathrm{t}\left({\displaystyle \frac{t-n·PRT-2R\left(s;r\right)/C}{T_r}}\right)\\ & ·\mathrm{e}\mathrm{x}\mathrm{p}\left[j\mathrm{\pi }{K}_r{\left(t-n·PRT-2R\left(s;r\right)/C\right)}^2\right]·\mathrm{e}\mathrm{x}\mathrm{p}\left[-j{\displaystyle \frac{4\mathrm{\pi }}{\lambda }}R\left(s;r\right)\right]\\ & ·\mathrm{e}\mathrm{x}\mathrm{p}\left[j2\mathrm{\pi }{f}_c\left(t-n·PRT-{\tau }_n\right)\right]\end{array}$$

(16)

where $\sigma$ represents the backscatter cross section of the point target, $\omega$ indicates the bidirectional amplitude weighting of antenna, ${\tau }_n$ represents the time of the $n$th pulse emitted by SAR. By expressing one-dimensional echo signal in a two-dimensional form regarding azimuth and range directions and removing the carrier through orthogonal demodulation, the echo of a single point target can be written as follows:

$$\begin{gathered} s_r\left(s,t;r\right)=\sigma ·\mathrm{r}\mathrm{e}\mathrm{c}\mathrm{t}\left({\displaystyle \frac{t-2R\left(s;r\right)/C}{T_r}}\right)·\mathrm{e}\mathrm{x}\mathrm{p}\left[j\mathrm{\pi }{K}_r{\left(t-2R\left(s;r\right)/C\right)}^2\right]·\mathrm{r}\mathrm{e}\mathrm{c}\mathrm{t}\left({\displaystyle \frac{s}{T_{sar}}}\right) \\ ·\mathrm{e}\mathrm{x}\mathrm{p}\left[-j{\displaystyle \frac{4\mathrm{\pi }}{\lambda }}R\left(s;r\right)\right] \end{gathered}$$

(17)

The echoes of all the imaging points can be obtained by superimposition, as shown in the following equation:

$$S_r\left(n,m\right)=\sum _{k=1}^K\sigma ·\mathrm{e}\mathrm{x}\mathrm{p}\left\{j\mathrm{\pi }\left[t\left(m\right)-2R\left(n;k\right)/C\right]\right\}·\mathrm{e}\mathrm{x}\mathrm{p}\left[-j{\displaystyle \frac{4\mathrm{\pi }}{\lambda }}R\left(n;k\right)\right]$$

(18)

where $K$ represents the number of imaging points. $0<\left[t\left(m\right)-2R\left(n;k\right)/C\right]<T_r$, $\left|R\left(n;k\right)-x\left(k\right)\right|<T_{sar}$.

The traditional single-channel SAR echo simulation method can effectively reproduce the basic characteristics of static scenes, but it has certain limitations when dealing with marine scenes. Factors such as sea clutter, platform motion error, and target motion can all cause problems like image blurring and defocusing. Therefore, the multi-channel SAR technology has been extensively studied. By using multiple receiving channels with certain intervals in the flight direction, SAR can obtain more phase and amplitude information, thereby effectively suppressing sea clutter and improving the imaging performance of the target.

To verify the accuracy of simulation results, the chirp scaling (CS) imaging algorithm was employed for the initial imaging processing of the data. The CS imaging algorithm is one of the classic frequency-domain algorithms in SAR imaging processing, as shown in Figure 4. Its core idea is to achieve range cell migration correction (RCMC) through phase multiplicative compensation. Since no interpolation operation is required, it has high computational efficiency [30].

Simulations and imaging tests were conducted on a single-point target using single-channel and dual-channel SAR respectively. The results are shown in Figure 5 and Figure 6.

Based on the above results, an analysis and calculation were conducted, and the performance evaluation data of the single-point target are shown in

Table 3. Performance evaluation results of azimuth direction for single-channel and dual-channel.

Channel Mode	Resolution (m)	PSLR (dB)	ISLR (dB)
Single-channel	0.9563	−24.434	−20.468
Dual-channel	0.9971	−26.937	−23.845

and

Table 4. Performance evaluation results of range direction for single-channel and dual-channel.

Channel Mode	Resolution (m)	PSLR	ISLR
Single-channel	1.0811	−25.775	−20.662
Dual-channel	1.0811	−25.774	−20.662

as follows:

From the above results, it can be seen that the performance in the range direction of dual-channel SAR and single-channel SAR is basically the same. However, in the azimuth direction, dual-channel SAR can better suppress the side lobe signals and is more suitable for the simulation and imaging of ship targets in the sea. Therefore, in the subsequent part of this article, all simulations will be conducted using dual-channel SAR.

3. Results

3.1. Simulation and Segmentation of Sea Model

Simulate the sea models of 3-level sea condition by using the PM spectrum, the JONSWAP spectrum and the Elfouhaily spectrum. The results are shown in Figure 7. Simulation time, RMS height, skewness and kurtosis of each result are measured and shown in

Table 5. Measurement results of sea models of 3-level sea condition based on the three spectrum models.

Spectrum	Time Consumption (s)	RMS Height (m)	Skewness	Kurtosis
PM	40.5	0.31	0.239	2.994
JONSWAP	49.0	0.31	0.243	3.015
Elfouhaily	53.8	0.30	0.227	2.814

. To reduce the measurement errors caused by random phase, 10 simulations were conducted separately. The average values for each indicator were used as the measurement result.

By analyzing the results in Table 5, it can be found that the RMS heights of the sea models based on the three spectrum models are basically the same. In the fully developed sea, since the JONSWAP spectrum simplifies to the PM spectrum, all the results tend to be consistent, with the skewness error being only 1.65% and the kurtosis error being only 0.70%. Compared with the PM spectrum and the Elfouhaily spectrum, the skewness error is approximately 5.29%, and the kurtosis error is approximately 6.40%.

This paper is dedicated to studying various ship targets in fully developed sea. Since the RCS of ship targets is usually much larger than that of the sea surface, the error among the three spectrum models will be further reduced. Meanwhile, the simulation efficiency of the PM spectrum is approximately 17% and 25% higher than that of the JONSWAP spectrum and the Elfouhaily spectrum respectively. To better meet the requirements of real-time performance, this paper will subsequently conduct research using the PM spectrum.

The sea models of sea conditions ranging from 1 to 5 level was simulated using the PM spectrum and the Monte Carlo method, which are shown in Figure 8:

3.2. Ship Target Model Segmentation

During the process of target model segmentation, there is a small probability of errors in removing certain elements within the model, especially in some complex structures such as multi-layered element structures and cavity structures. Figure 9 presents a case where Figure 9a shows the incorrect segmentation result, while Figure 9b represents the correct one.

Set the solver mode in the Boolean modifier to “FAST” and “EXACT” respectively and perform multiple segmentation tests on the ship model with resolutions of $0.5 \mathrm{m}$, $1 \mathrm{m}$, and $2 \mathrm{m}$. The results are shown in

Table 6. Segmentation results of ship model with different resolutions and solver modes.

Resolution (m)	Solver Mode	Time Consumption (s)	Error Rate (%)
$0.5 \mathrm{m}$	FAST	767	6.81
$0.5 \mathrm{m}$	EXACT	1689	2.38
$1 \mathrm{m}$	FAST	196	4.25
$1 \mathrm{m}$	EXACT	451	1.25
$2 \mathrm{m}$	FAST	48	3.00
$2 \mathrm{m}$	EXACT	107	0.00

.

From the results in Table 6, it can be seen that selecting the solver mode as “EXACT” can effectively reduce the error rate of the model segmentation, but the time consumption will increase exponentially. Meanwhile, as the resolution increases, the model segmentation becomes more detailed, and the error rate rises accordingly.

Therefore, when the resolution requirement is not very high, the “FAST” mode can be used for processing first, and then the “EXACT” mode can be employed to handle the few erroneous parts. If there are still a few erroneous parts, we can manually delete the redundant elements and vertices in the editing mode.

After the ship target model segmentation is completed, all the segmentation results are imported into Blender and combined to form a complete model, as shown in Figure 10b. Assuming that the SAR is located directly behind the target, the downward viewing angle is $30°$, and the simulation resolution is $1 \mathrm{m}$. By comparing the ship model before and after segmentation, it can be seen that the model has been segmented along radar viewing angle at a resolution of 1 m, and all the non-visible elements have been removed, which indicates that the segmentation effect is good.

3.3. Simulation of Electromagnetic Scattering Characteristics of Sea Surface and Ship Targets

Simulate the target at different grid resolutions by using the LEPO. The results are shown in

Table 7. Simulation results of the target at different grid resolutions.

Mesh Size	Time Consumption (s)	RCS (dBsm)	$\mathbf{Relative} \mathbf{Error} (\mathit{\epsilon }$)
$\lambda /5$	38	18.2	0.095
$\lambda /8$	105	20.1	0.034
$\lambda /10$	171	20.8	0.010
$\lambda /15$	363	21.0	−

:

From the results in Table 7, it can be seen that as the grid resolution increases, the simulation results gradually converge, while the time consumption also increases exponentially. When the grid resolution is set to $\lambda /10$, the relative error $\epsilon \approx 0.01$, which is within the acceptable range. Meanwhile, the simulation efficiency is also moderate. Therefore, in all subsequent simulations, the grid resolution is set to $\lambda /10$.

Figure 11a shows complete RCS simulation result of sea surface of 3-level sea condition. Assuming that the viewing angle of SAR is $30°$ and the resolution is $1 \mathrm{m}$, magnify the local area (Figure 11b) and compare it with the sea model of this area. It can be seen that the smaller the angle between the direction of incoming wave of SAR and the normal direction of the sea surface element, the greater the received scattered echo intensity. When the angle is $0°$, the echo changes from diffuse reflection to specular reflection, and at this time, the scattering intensity reaches the maximum, which is consistent with actual situation.

Figure 12 shows the RCS simulation results of ship target. Assuming that the SAR is located directly behind the ship target, with an elevation angle of $30°$ and a resolution of $1 \mathrm{m}$. Compared with original ship model in Figure 10a, it can be seen that the positions with higher scattering intensity are mainly concentrated in dihedral angles, multiple reflection cavities, edges and sharp corners of the ship model, as well as planar structures perpendicular to the incoming wave direction, which is consistent with the theoretical situation.

3.4. SAR Echo Simulation

In Section 2.4, SAR echo simulation and imaging processing for a single-point target have been carried out. To verify the accuracy of this method, the “SAR Imaging and Performance Tuning Software for Sea Surface Targets” (hereinafter referred to as the SAR Software) was utilized to conduct simulation verification using exactly the same radar and scene parameters. This software was jointly developed by a research institute of CETC and Beihang University and has now undergone engineering verification.

The echo simulation and imaging processing of single-point target were conducted respectively using the method in this paper and the SAR Software, and relevant evaluation indicators were calculated. The results are shown in

Table 8. Performance evaluation results of azimuth direction.

System	Resolution (m)	PSLR (dB)	ISLR (dB)
Ours	0.9971	−26.937	−23.845
SAR Software	1.0016	−27.121	−24.107

and

Table 9. Performance evaluation results of range direction.

System	Resolution (m)	PSLR	ISLR
Ours	1.0811	−25.774	−20.662
SAR Software	1.0535	−25.862	−20.915

:

Next, using the method described in this paper and the SAR software, SAR echo simulations and imaging processing were conducted for the ship targets under 3-level sea condition. Overall performance indicators of the images were calculated, including structural similarity (SSIM), signal-to-clutter ratio (SCR), target-to-background contrast, geometric distortion, etc. SSIM is used to evaluate structural consistency between the simulated image and the reference. Its calculation formula is:

$$\mathrm{S}\mathrm{S}\mathrm{I}\mathrm{M}\left(x,y\right)={\displaystyle \frac{\left(2{\mu }_x{\mu }_y+C_1\right)\left(2{\sigma }_{xy}+C_2\right)}{\left({\mu }_x^2+{\mu }_y^2+C_1\right)\left({\sigma }_x^2+{\sigma }_y^2+C_2\right)}}$$

(19)

SCR reflects the ability of the target to be effectively detected and identified in a complex environment. Its calculation formula is:

$$\mathrm{S}\mathrm{C}\mathrm{R}=10 {\log }_{10}{\displaystyle \frac{P_{target}}{P_{clutter}}}$$

(20)

Target-to-background contrast reflects the ability of the target to be clearly distinguished and visually identifiable in the image. Its calculation formula is:

$$\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{s}\mathrm{t}={\displaystyle \frac{{\mu }_t-{\mu }_b}{{\mu }_b}}$$

(21)

Geometric distortion is evaluated by comparing the extracted target dimensions with the ground truth model. Its calculation formula is:

$$\mathrm{D}\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}={\displaystyle \frac{\mathrm{M}\mathrm{e}\mathrm{a}\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{e}\mathrm{d}-\mathrm{T}\mathrm{r}\mathrm{u}\mathrm{e}}{\mathrm{T}\mathrm{r}\mathrm{u}\mathrm{e}}}$$

(22)

Simulation and imaging results of the method presented in this paper are shown in Figure 13. The calculation results of image performance indicators are shown in

Table 10. Image performance evaluation results.

System	SSIM	SCR (dB)	Contrast	Distortion
Ours	92%	17.3	3.7	4.5%
SAR Software	92%	18.1	4.0	0.8%

.

Extract the local range direction slice of the imaging result and compare it with the RCS simulation result. The result is shown in Figure 14:

The results in Table 8, Table 9 and Table 10 and Figure 13 and Figure 14 indicate that the SAR echo simulation method proposed in this paper is in good agreement with the theoretical analysis and actual conditions, and all the indicators fall within acceptable range of small errors. These results have verified the correctness of this method.

Next, sensitivity of the method presented in this paper to factors such as model resolution, sea condition and viewing angle of SAR will be analyzed. First, the ship model is segmented according to different resolutions, and SAR echo simulation and imaging processing were carried out. Calculate various performance indicators of the imaging results, and the results are shown in

Table 11. Image performance evaluation results for different model resolutions.

Resolution (m)	Target-Resolution Ratio	SCR (dB)	Contrast	Distortion
2	5	17.2	3.7	4.5%
1	10	17.3	3.7	4.5%
0.5	20	17.3	3.7	4.5%

:

From the results in Table 11, it can be seen that simulation effect is independent of the model resolution and the ratio of target size to model resolution. It mainly depends on the performance of the simulation system itself.

Next, SAR echo simulation and imaging processing were carried out using different viewing angles, and various performance indicators of the imaging results were calculated. The results are shown in

Table 12. Image performance evaluation results for different viewing angles.

Viewing Angle (°)	SCR (dB)	Contrast	Distortion
10	11.5	1.9	4.5%
30	17.3	3.7	4.5%
55	15.6	3.1	4.5%

:

From the results in Table 12, it can be seen that at small viewing angles, strong specular reflection of the sea surface leads to excessive clutter, thus resulting in a lower SCR. As the viewing angle continues to increase, the sea clutter gradually decreases, but the backscatter of target remains strong, thereby achieving better SCR and contrast. When the viewing angle is too large, although the sea clutter is very small, the signal strength of target also decreases, resulting in a decline in both the SCR and the contrast.

Finally, SAR echo simulation and imaging processing were conducted for ship target under sea conditions ranging from 1 to 5, and various performance indicators of the imaging results were calculated. The results are shown in

Table 13. Image performance evaluation results under different sea conditions.

Sea Condition	SCR (dB)	Contrast	Distortion
1	18.9	4.4	4.5%
2	18.2	4.1	4.5%
3	17.3	3.7	4.5%
4	16.7	3.5	4.5%
5	14.9	2.8	4.5%

:

From the results in Table 13, it can be seen that as sea condition level increases, the roughness of sea surface and the dynamic fluctuations intensify, resulting in a significant enhancement of the backscattering of sea clutter. Therefore, the SCR decreased and the contrast also reduced gradually. The results show that sea conditions have a crucial impact on the quality of SAR imaging and the detectability of targets.

4. Discussion

Numerous experiments have demonstrated the feasibility of the method described in this paper, while also revealing some shortcomings and limitations. The PM spectrum, as one of the classic wave spectrum models, has achieved good results in fully developed sea. However, in some specific situations, its effect is not satisfactory. For instance, the focus would be on the process of wave generation, or in coastal areas. At these points, using more advanced spectrum models yields better results.

During the process of ship model segmentation, there may occasionally be cases where the segmentation results are incorrect. In the context of this study, the error rate falls within an acceptable and relatively low range. However, as the resolution requirements keep increasing, the error rate also rises rapidly. By using the “EXACT” mode of the solver in the Boolean modifier, the error rate can be reduced effectively, but the processing efficiency will be affected significantly. The root cause of the problem might lie in the model itself, such as imperfect connections between various elements, or an unreasonable distribution structure, etc. If the model itself can be optimized, it might solve this problem effectively. If that does not work, perhaps an intelligent model can be incorporated into the Python automated segmentation script. By identifying incorrect segmentation results automatically, the solver can be promptly adjusted to “EXACT” mode and reprocessed, thereby achieving efficient and accurate model segmentation tasks. If this method can be implemented, it will completely solve this problem.

The previous experiments have shown that the PO/LEPO performs well in handling simple model structures such as sea surface and ship targets. When the model structure is very complex, this method still has some drawbacks. The PO/LEPO belongs to a high-frequency progressive method dominated by single scattering, so it will ignore some multiple scattering mechanisms. For example, multiple reflections between surfaces, multiple reflections within cavities and concave structures, re-illumination in blocked areas, etc. [31]. This may result in underestimated intensity of some strong scattering points, causing the intensity of bright spots in the SAR image to be insufficient and the overall contrast of the image to decrease. Furthermore, multiple scattering paths imply an additional propagation distance. The PO/LEPO ignores these paths, which may result in incomplete phase information of the SAR echo signals and cause the defocusing problem [32].

In response to the aforementioned issues, many scholars have already proposed improvement plans. Among them, the shooting and bouncing ray (SBR) method, which considers multi-bounce reflections by tracing ray paths based on the GO, has been widely adopted. By combining SBR with PO, higher-order interactions such as inter-facet reflections and cavity scattering can be effectively modeled, significantly improving the RCS prediction accuracy for electrically large and complex targets [33]. In addition, the physical theory of diffraction (PTD) and the uniform theory of diffraction (UTD) are commonly incorporated to compensate for the deficiencies of PO in modeling edge diffraction. These methods introduce diffraction coefficients associated with edges and wedges, enabling accurate prediction of scattering contributions in shadow regions and around sharp discontinuities. It has been demonstrated that the integration of SBR and UTD can effectively capture both multi-reflection and diffraction mechanisms [34]. Combining the above method with the one presented in this paper will effectively solve the problem that the PO/LEPO cannot handle the multiple scattering mechanism, thereby enabling the method in this paper to achieve a performance that is basically consistent with that of the “SAR Software”.

The running time and system status of each stage of the method in this article were statistically analyzed, and the results are shown in

Table 14. Results of running time and system status of each stage.

Stage	Running Time (h)	CPU Usage	GPU Usage
Sea Model Simulation	1.1	73%	0%
Ship Model Segmentation	0.5	18%	44%
RCS Simulation of Sea	13.7	85%	0%
RCS Simulation of Ship	3.3	72%	0%
SAR Echo Simulation	0.3	32%	90%

:

According to the results in Table 14, the time required to conduct a complete SAR echo simulation of a ship target in the sea is approximately 19 h, while the time needed to perform a similar simulation using the SAR Software is approximately 26 h. It can be seen that this method has excellent simulation efficiency. The test revealed that this method takes a relatively long time to process the RCS simulation of sea and ships. Due to the fact that the data volume for RCS simulation of the sea is much larger than that for ship targets, the RCS simulation of the ship is the main factor that slows down the entire method. A further analysis of this process revealed that the program spent approximately 70% of its time on reading and preprocessing the model. The main reason for this is that Feko simulates each sub-model one by one. Only when conducting electromagnetic calculations for a single model can the CPU parallel technology be utilized. Therefore, there is still some chance for optimization in this method. As mentioned earlier in this chapter, by optimizing the structure of the model itself, it might be possible to effectively increase the speed at which Feko reads the model. In addition, it is possible to attempt to implement the RCS simulation work using GPU. By leveraging GPU parallel technology, such as CUDA, it might be possible to simultaneously conduct RCS simulations for multiple sub-models, thereby enhancing the efficiency of the whole process. Theoretically, this method can at least increase the efficiency by 30–40%.

5. Conclusions

SAR echo simulation is of great significance for marine monitoring and protection. However, the simulation system is often complex in terms of hardware equipment and the acquisition cost of RCS data is high, which leads to certain difficulties in practical applications. This paper proposes a SAR echo simulation method that solely relies on software simulation and numerical calculation. The method can be divided into four parts. The first part utilizes the PM wave spectrum model and the Monte Carlo method to simulate the sea model under various sea conditions, and employs an independently designed method to segment the model. By comparing various wave spectrum models, the effectiveness of this method was demonstrated. The second part uses Blender to achieve the automatic segmentation of the ship model. The experimental results demonstrate the effectiveness of this method, and also reveal some areas that can be optimized. The third part used Feko to conduct electromagnetic scattering characteristic simulations for the sea and the ship model. By comparing various electromagnetic simulation methods, the effectiveness and applicability of this method were demonstrated, and the subsequent optimization directions were also pointed out. The final part efficiently realizes SAR echo simulation using dual-channel technology. By comparing with the SAR Software, it was proved that this method has excellent performance and is fully applicable to this research.

In conclusion, this method demonstrates excellent performance and wide applicability. It can, in a complex marine environment, using limited software and hardware resources, simulate SAR echo signals of various ship targets quickly and accurately, and has considerable application value.