Composite Electromagnetic Scattering Characteristics of Actual Terrain and Moving Rockets

Peng, Guanhongye; Ren, Xincheng; Guo, Lixin; Zhu, Xiaomin; Yang, Pengju; Zhao, Ye; Zhang, Bohang; Wu, Guoshan

doi:https://doi.org/10.1155/2024/2878468

International Journal of Antennas and Propagation

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Abstract Introduction Discussion Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2024 | Article ID 2878468 | https://doi.org/10.1155/2024/2878468

Composite Electromagnetic Scattering Characteristics of Actual Terrain and Moving Rockets

Guanhongye Peng,¹Xincheng Ren,¹Lixin Guo,²Xiaomin Zhu,¹Pengju Yang,¹Ye Zhao,¹Bohang Zhang,¹and Guoshan Wu¹

Academic Editor: Raffaele Solimene

Received15 Jan 2024

Revised09 Apr 2024

Accepted04 May 2024

Published21 May 2024

Abstract

This research involved studying the characteristics of composite electromagnetic scattering from a rocket moving above actual terrain using the finite-difference time-domain method. The angular distribution curve of the composite scattering coefficient was obtained, and the influences of factors such as the angle of incidence, the frequency of the incident electromagnetic wave, the content of moisture in soil, the dielectric constant of the rocket-shell material, the height of the rocket, and the undulation of the terrain on the composite scattering coefficient were investigated. Results show that the composite scattering coefficient oscillates with the scattering angle and increases in the direction of mirror reflection. It also decreases with increasing angle of incidence, frequency of the incident wave, and altitude of the rocket, while it increases with increasing soil moisture and dielectric constant of the rocket-shell materials. Although the influence of different terrain undulations on the composite scattering coefficient is noticeable, it follows no fixed pattern.

1. Introduction

In the increasingly complex field of electromagnetic detection and surveillance in the environment, it is especially critical to understand accurately and analyze thoroughly the composite electromagnetic scattering characteristics of targets above various terrains [1–4]. This is particularly important in unpredictable geographical circumstances where the propagation and scattering of electromagnetic waves are influenced by a multitude of factors, including the surface relief of the terrain, the dielectric characteristics of the medium situated beneath the topographic surface, and the geometric characteristics and parameters of the target entity and the dielectric properties of the material, etc. [5–7]. The finite-difference time-domain (FDTD) method is a widely used numerical tool for full-wave analyses of electromagnetic fields in complex media. It has a wide range of applications that deal with complex media and geometries in different scientific and engineering fields—including geophysical problems such as oil and gas exploration and environmental monitoring—where multiphysics issues such as the scattering of electromagnetic waves by complex materials and structures must be modeled using accurate calculations [8–11].

Currently, the finite-element method (FEM) [12, 13], the method of moments (MoM) [14, 15], and the FDTD method [16, 17] are the primary numerical techniques used to investigate the composite electromagnetic scattering properties of a target and its surroundings. In practical applications, the FEM requires precise meshing, and the generation of meshes for problems involving complex geometries becomes extraordinarily complicated and time-consuming. For comparison, MoM is a frequency-domain approach that necessitates separate operations for every individual frequency, rendering it unsuitable for the analysis of problems involving broad bandwidths or nonlinearity. In contrast, FDTD handles open-region problems in a more natural manner, and infinite-wave propagation can be modeled effectively by employing absorbing boundary conditions [18]. Moreover, while FDTD demands a denser mesh to deal with the details, its meshing procedure is comparatively uncomplicated.

The FDTD method has demonstrated considerable adaptability in addressing electromagnetic scattering phenomena, and it has been utilized extensively and validated in various studies. For example, Moss et al. [19] employed a three-dimensional FDTD method and the Monte Carlo method to simulate the electromagnetic scattering of targets in continuous random media, specifically inhomogeneous soils. Their research focused on analyzing the impact of the physical parameters of the medium on the radar-scattering cross section and on validating the accuracy of the composite scattering model. Their findings demonstrated that the FDTD method is capable of effectively simulating the electromagnetic scattering from targets in complex media. Further, Li et al. [20] applied the FDTD method with a combination of sinusoidal and pulsed-wave excitations and used the Kirchhoff approximation to simulate the polarimetric scattering characteristics of a rough, two-dimensional surface successfully. They also verified the high efficiency of the method by using parallel computations, providing a new direction for the study of three-dimensional scattering problems. Yang et al. [21] introduced an iterative hybrid approach that integrates a fast, multistage, multipole algorithm with the Kirchhoff approximation to analyze composite electromagnetic scattering from a three-dimensional target above a two-dimensional, randomly rough surface. The researchers utilized a truncation rule and a fast far-field approximation to decrease the computational burden. They evaluated the hybrid method rigorously and found that it has exceptional accuracy and efficiency, surpassing those of conventional methods by considerable margins. Smith et al. [22] utilized ultralow-frequency (ULF) electromagnetic waves and investigated their effectiveness in detecting targets submerged in the ocean. They used FDTD model simulations to confirm the effective penetration and scattering capabilities of ULF waves at various frequencies and depths, and their results supplied crucial data for the design of oceanic detection systems. He et al. [23] propose the Laguerre-FDTD method for domain decomposition, which offers an effective solution for addressing electromagnetic scattering problems in two dimensions. This method involves the utilization of Laguerre polynomials in decomposing the computational space into multiple subdomains. This approach enhances both the computational efficiency and accuracy of the method, which results in a substantial reduction in computational time and resource consumption when dealing with complex electromagnetic problems, compared to traditional methods.

Methods generally employed for modeling rough surfaces include the Monte Carlo method [24], which relies on random sampling. However, the outcomes obtained using this approach often exhibit a certain degree of randomness, which limits its ability to represent the true undulations of rough surfaces correctly. In comparison, a digital elevation model (DEM) [25] offers accurate topographic elevation information derived from authentic geographic-survey data, which enable the use of an exact replica of the undulating features of actual landscapes. Numerous scholarly investigations have recently assessed the precision of the Copernicus DEM and have compared it with an assortment of open-source DEMs [26, 27]. For example, in the research they performed, Li et al. [28] used ICESat-2 altimetry data to compare the Copernicus DEM, the NASA DEM, and the AW3D30 DEM for five regions in China. They found that the Copernicus DEM provided the most precise depiction of the topography, followed by the AW3D30 DEM, and lastly the NASA DEM.

Herein, we utilize the four-component model proposed by Wang and Schmugge to simulate soil permittivity. We also employed a DEM to represent the undulation of an actual terrain surface. The FDTD is subsequently used to investigate the composite electromagnetic scattering characteristics of the actual terrain in conjunction with a specific type of moving rocket positioned above it. We employed numerical computations to obtain the angular distributions of the composite scattering coefficients. This study further investigated the influence on the composite scattering coefficient of various factors, including the angle of incidence, the moisture content of the soil, the frequency of the incident electromagnetic wave, the permittivity of the rocket casing material, the altitude of the rocket, and undulation conditions of the terrain.

2. Modeling

Figure 1 shows a geometric representation of the composite electromagnetic scattering from a specific type of cruise rocket located above a real terrain surface. The rocket is positioned vertically above the surface, which is underlain by an isotropic, homogeneous soil medium that extends to infinity. The cross section of the rocket comprises an isosceles trapezoid, a rectangle, and an isosceles triangle. The heights of the isosceles trapezoid, rectangle, and isosceles triangle are denoted by , , and , respectively (Figure 1). The half-width of the base of the isosceles trapezoid is denoted by , while those of the base of the triangle and the rectangle are represented by the same variable, .

Figure 2 illustrates the FDTD model utilized to compute the composite electromagnetic scattering from a specific rocket above an actual terrain surface. In this model, the connecting boundary AB is considered to be a planar surface that extends to an absorbing boundary layer. The region below the connecting boundary AB represents the total field area, while the region above it represents the scattered-field area. The incident wave is introduced into the former region by imposing an equivalent electromagnetic flux on the connecting boundary. The output boundary CD is positioned within the scattered-field region, and it runs parallel to the connecting boundary and extends outward to the absorbing layer. In addition, the uniaxial perfectly matched layer (UPML) absorbing boundary is established outside the FDTD computational domain. The wavelength of the incident electromagnetic wave is taken to be , while the thickness is given as 10 grids.

2.1. Modeling an Actual Terrain and the Permittivity of Its Soil

A DEM provides a digital portrayal of the elevation relief of Earth’s surface. With the increasing deployment of optical stereo-mapping satellites and interferometric radar satellites, the collection of global and regional DEM datasets is expanding. There are now several globally available open-source DEM datasets, including ETOPO, GTOPO30, GMTED2010, SRTM DEM, NASA DEM, ASTER GDEM, AW3D30, TanDEM-X DEM, and Copernicus DEM. The Copernicus digital elevation model (Copernicus DEM or COP–DEM) is a freely accessible dataset provided by the European Space Agency. It is globally available in two resolutions: 30 m and 90 m. The COP-DEM is a digital surface model (DSM) that accurately depicts the elevation of the Earth’s surface, encompassing various features such as buildings, infrastructure, and vegetation. The COP–DEM dataset is an edited version of a DSM taken from the WorldDEM product. It was created by sampling high-resolution DEMs and using subsequent manual editing to improve its uniformity in the representation of water bodies and other inaccurate terrain characteristics.

The COP–DEM is offered in three distinct resolution variants: EEA-10 (resolution: 10 m), GLO-30 (resolution: 30 m), and GLO-90 (resolution: 90 m) (Figure 3). Validation of the vertical accuracy of the COP–DEM dataset using ICESat reference data is presented in the product handbook [29]. A total of approximately 11 million ICESat points were chosen as valid data points, with an absolute vertical precision of 2.17 m at the 90% confidence level (LE90). The evaluation results exhibited a mean inaccuracy of −1.29 m, with a standard deviation of 0.85 m.

The Copernicus DEM product handbook states that the absolute horizontal accuracy of the COP–DEM is influenced by two factors. The first is the positional accuracy of the individual DEM view, which depends upon the quality of the TerraSAR-X image base product. The accuracy of this product is estimated to be better than 0.3 m. The second factor is the absolute vertical error of the DEM data when projected onto a horizontal displacement. A determination of the absolute horizontal accuracy can be obtained from the absolute vertical error, which is quantified as the arithmetic mean of LE90ABS being <4 m. Furthermore, the discrepancy in the horizontal distance between a true surface position and the corresponding point in the DEM was found to be <6 m, with a confidence level of 90%.

In this study, we used the COP–DEM to reflect the topographic relief of the landscape accurately. In addition, we assumed that the underlying soil is uniform and isotropic, even though research on the dielectric characteristics of soils has demonstrated that the amount of water per unit volume has a considerable influence on the soil’s permittivity. Models that are now prevalent in the field include the Wang and Schmugge model [30], the GRMDM model [31], and the Topp model [32]. The Wang and Schmugge model is particularly noteworthy for its exceptional accuracy and for its applicability in estimating the dielectric properties of soils. This is primarily due to its comprehensive consideration of multiple parameters, including soil density, moisture content, and temperature. It thus extends beyond the oversimplified assumption of considering the volumetric water content of the soil as the sole determinant of permittivity. Instead, it provides a thorough characterization of the permittivity by incorporating additional variables such as soil texture and temperature.

The moisture compression point of soil is defined by the four-component model proposed by Wang and Schmugge. This model considers the dry soil composition in terms of sand content () and clay content (), with the constraint that the then defines the moisture compression point of a soil by the following equation:

An empirical formula for the critical humidity of a soil body can then be written as follows:

Define the parameters as shown in the following equation:

Typically, the density of rocks within a soil is denoted by = 2.6 g/cm³, and the density of dry soil is represented by . The cumulative porosity of the soil is thus given by the following equation:

The value of is determined based on equations (5) and (6):where

After establishing the parameters as indicated above, the permittivity can be computed based on the soil moisture.

When ,where

In these equations, the permittivity of ice is , the permittivity of rock is , while is the permittivity of air. The permittivity of pure water at the operating frequency can be determined using Debye’s formula, as given by the following equation:whereand

Given the operating frequency, soil temperature, and soil composition, the equivalent permittivity of the soil can be calculated by utilizing these equations. Table 1 [33] provides the sand and clay contents for various types of soils.

For the numerical calculations performed in this study, we calculated the permittivity of the soil using the sand content and clay content .

2.2. Model of the Rocket and the Permittivity of Its Casing Material

As noted above, the cross-sectional shape of the rocket considered in this study comprises three geometric figures: a trapezoid, a rectangle, and a triangle. The trapezoid has height and base width . The rectangle has height . The triangle has height , and its base width is the same as the half-width of the rectangle, .

The permittivity can be obtained from the Kramers–Kronig dispersion relation and the definition of the direct-transfer probability [34]. The real part of the permittivity corresponds to the material’s capacity to retain charge, whereas the imaginary part represents its ability to dissipate electromagnetic energy. The formula for the permittivity is

The real part of the permittivity is denoted as , while the imaginary part is denoted as . These values can be computed using equations (15) and (16), respectively.and

In these equations, the symbol is used to represent the (complex) frequency of the radiation, while P represents the principal component of . The quantities ν, c, and m, respectively, denote the dimensions of the valence, conduction bands, and the mass of the carriers. The term refers to the path within the first Brillouin zone. Moreover, and represent the energy and the intrinsic wavefunction at each location along the path parametrized by k. Lastly, and represent the standard momentum operator and the polarization vector, respectively.

In this study, we focused on the investigation of six aerospace materials frequently employed in the manufacture of rockets. The selected materials comprise the three metallic elements, iron (), nickel (), and titanium (), along with the three intermetallic compounds aluminum-magnesium alloy (), titanium-nickel alloy (), and copper-nickel alloy (). In this work, we represented the crystalline forms of the elements Fe, Ni, and Ti as cubic structures characterized by the lattice parameters, , , , , and . The metal atoms were situated at the center of the cube. In addition, the crystal structures of intermetallic compounds can be obtained by downloading the cif files from the materials project resource library [35, 36]. The tetragonal phase I4₁/amd space group (space group No. 141) shows the crystal structure of the alloy. The lattice parameters in the original file are , , , , and . The dominant geometry in this structure comprises three Mg atoms and eight Al atoms, resulting in an 11-coordination geometry. The crystal structure of the TiNi alloy belongs to the monoclinic phase P2₁/m space group (space group No. 11), for which the lattice parameters in the original file are , , , , , and . In the structure of this alloy, each Ti atom is combined with an Ni atom in a 7-coordination geometry. The Cu₃Ni alloy has a crystal structure that falls within the orthorhombic Cmmm space group (space group No. 65). The lattice parameters in the original file are , , , , and . Its structure primarily comprises Ni atoms arranged in an 8-coordinated geometric pattern, forming connections with four identical Ni atoms and four identical Cu atoms. Figure 4 illustrates the structures of these three metallic elements and the three intermetallic compounds. In particular, Figure 4(a) depicts the structural model for the metallic elements, Figure 4(b) shows the structural model of Mg₂Al₃, Figure 4(c) displays the structural model of TiNi, and Figure 4(d) exhibits the structural model of Cu₃Ni.

We determined the permittivity of each of these materials using the Vienna Ab initio Simulation Package (VASP). VASP relies on a combination of techniques that utilize plane-wave basis sets and ultra-soft pseudopotentials [37, 38]. This software has been employed extensively in the area of materials, simulations, and calculations due to its notable precision and simplicity of operation. To optimize structural convergence for modeling the target materials, we employed a network of 6 × 6 × 1 k-points. For the structural-optimization process, we set the truncation energy (ENCUT) to 480 eV, the optimal energy-convergence criterion (EDIFF) to 1 × 10⁻⁵ eV, and the interatomic energy-convergence criterion (EDIFFG) to 2 × 10⁻² eV/Å. To acquire additional data points for determining the permittivities of the target materials more precisely, we employed a network of 7 × 7 × 3 k-points. The calculation procedure encompasses the infrared (IR), visible (VR), and ultraviolet (UV) energy spectra. Figure 5 presents the simulated permittivity values for the corresponding materials.

(a)

(b)

The solid line in Figure 5 represents the real component of the material’s permittivity. This component indicates the material’s ability to store energy and reflects the speed at which electromagnetic waves propagate within the medium. A higher value of the real component indicates a greater capacity for polarization. The dashed line in Figure 5 corresponds to the imaginary component of the material’s permittivity. This component represents the material’s loss factor, which characterizes the dissipation of energy from an electromagnetic wave during its propagation through the material [39]. A higher value of the imaginary component corresponds to greater absorption of energy and its conversion into heat or other forms of loss. As represented in Figure 5, it is evident that the same material would manifest varying permittivity when subjected to varied electron energies. In this study, we employed electromagnetic waves with a frequency of 30 MHz. From Planck’s relation , the corresponding energy is 1.24 × 10⁻⁷ eV. Consequently, the permittivity value utilized in this research for further calculations and analyses is based on the scenario where the electron energy approaches zero.

3. Finite-Difference Time-Domain (FDTD) Methods

Based on the principles of the finite-difference approach in the time-domain (FDTD), the difference equation for a transverse magnetic (TM) wave in a two-dimensional electromagnetic-field problem can be represented mathematically as follows:where represents the location of the FDTD grid node for the field component on the left side of the equation. The quantities Δx and Δy are the discrete grid widths in the x and y directions of the FDTD region, respectively. We established the absorption boundary by using an anisotropic medium represented by an UPML.

Maxwell’s equations for TM waves in a (passive) anisotropic medium are the following:andwhere , , and denote the dielectric parameters of the medium inside the computational domain. The quantities and are uniaxial parameters in the x and y directions, respectively, which are given bywhere the parameters are calculated as follows, respectively.and

Optimal absorption occurs when n = 4. The other parameters in these equations are and = 5–11, where d is the thickness of the UPML layer. Once the FDTD calculation attains stability, we record the near-field computational outcomes at the output boundary. The far-area dispersed field can be derived subsequently by employing the time-harmonic field-extrapolation technique based on the equivalence principle.

Equation (26) shows the computation of the composite scattering coefficient for the combination of the target and the soil surface:

The normalized radar-scattering cross section in equation (26) is given by

In equation (27), the variables , and , respectively, represent the distance from the origin to the point being observed, the electric field of the scattered wave in the far zone, and the electric field of the incoming wave. The variable L is defined as the sampling length of the soil surface.

4. Validation of FDTD Methods

To assess the validity of the Finite-Difference Time-Domain (FDTD) algorithm in this study, the composite scattering coefficients of an exponentially rough surface and the target positioned above it were computed using the time-harmonic field FDTD method. These results were compared with the outcomes obtained from the method of moments (MOM) computations, as depicted in Figure 6. In these calculations, the frequency of the incident wave is assumed to be ν = 0.1 GHz, the angle of incidence to be , the length of the random rough surface to be , the correlation length to be , the root-mean-square of the height heave to be , the complex permittivity of the medium below the rough surface to be , and the target to be a combination of a rectangle and a trapezoid where the height of the rectangle is , the width is , the height of the trapezoid is , and the width of the upper base of the trapezoid is and that of the lower base is .

The angular distributions obtained using these two algorithms exhibit a high degree of similarity (Figure 6). These results provide empirical evidence supporting the validity and accuracy of the method employed in this research.

5. Numerical Results and Discussion

In the subsequent numerical computations, unless otherwise specified, the frequency of the incident electromagnetic wave is 30 MHz and the angle of incidence is = 30°. DEM dataset at a particular terrain was selected wherein the soil moisture content is assumed as and its permittivity is then given by . The rocket casing is composed of nickel monomers, with a permittivity of . The rocket is positioned at a height of above the ground. In the simulation, the number of rough surfaces is quantified as 20 [40, 41].

5.1. Variation of the Composite Scattering Coefficient with the Angle of Incidence

Figure 7 shows the computed variation of the scattering coefficient with the angle of incidence. These results were achieved by considering three different incident angles: = 30°, = 60°, and = 75°. An oscillatory-like change in the scattering coefficient with respect to the scattering angle is evident in Figure 7. In addition, scattering enhancement can be observed in the direction of specular reflection, namely, at the angles = 30°, = 60°, and = 75°.

The findings presented in Figure 7 show that the composite scattering coefficient is substantially influenced by the angle of incidence. Specifically, it decreases as the angle of incidence increases within the extended range of scattering angles, particularly . Within narrower ranges of scattering angles—specifically and —the composite scattering coefficient increases as the angle of incidence increases. This phenomenon can be attributed to the pronounced coupling effect resulting from abrupt changes in the topographic relief along a specific direction.

5.2. Variation of the Composite Scattering Coefficient with Soil Moisture Content

Figure 8 shows the variation of the composite scattering coefficient with variations in the soil moisture content at the fixed temperature T = 20°. This figure illustrates the variations for three distinct values of the moisture content of a sandy soil: . These moisture content values correspond to the permittivities , , and , respectively.

The oscillatory variation of the scattering coefficient with respect to the scattering angle is clear in Figure 8. The scattering is also enhanced in the direction corresponding to specular reflection. This pattern is observed consistently in all subsequent numerical calculations and will not be reiterated in further discussions. The impact of the soil moisture level on the composite scattering coefficient is notable; it increases as the soil moisture increases across the whole range of scattering angles. The result is particularly obvious in the ranges and , while it is comparatively smaller in the range . According to the computational model, the real part of the soil’s dielectric constant is enhanced with increasing soil moisture content, which enhances scattering by the ground.

5.3. Variation of the Scattering Coefficient with the Frequency of the Incident Wave

Figure 9 shows the composite scattering coefficient as a function of the frequency of the incident wave. This figure presents the results calculated for the three specific frequency values, = 30 MHz, = 40 MHz, and = 50 MHz.

Figure 9 demonstrates the notable impact of the frequency of the incident wave on the composite scattering coefficient. Specifically, the composite scattering coefficient decreases as the frequency increases within a substantial portion of the range of scattering angles; i.e., and . However, there is a smaller portion of the scattering angle range () where the composite scattering coefficient remains relatively unchanged with frequency. In addition, when , the oscillation amplitude of the angular distribution curves for the scattering coefficient is larger for = 30 MHz than for either = 40 MHz or = 50 MHz, and the corresponding scattering coefficient exhibits a higher peak at the angle of incidence . This occurs because is the mirror reflection direction for the angle of incidence . Other peaks are due to resonances between the wavelength of the electromagnetic wave and the structure of the terrain at those locations.

5.4. Variation of the Composite Scattering Coefficient with the Permittivity of the Rocket Casing

The composite scattering coefficients for the rocket casing materials are presented in Figures 10 and 11. The scattering coefficients are computed using the corresponding permittivity of , , and for titanium (), iron (), and nickel (), respectively, as the rocket casing materials. The values of the permittivity corresponding to the intermetallic compounds, namely, aluminium-magnesium alloys (), titanium-nickel alloys (), and copper-nickel alloys (), are taken as , , and , respectively. The data that are seen in Figures 10 and 11 demonstrate a positive correlation between the composite scattering coefficient and the permittivity of the rocket casing material throughout all scattering angle variations. This relationship is particularly apparent within the ranges of and .

In addition, a clear observation can be made based on Figures 10 and 11, indicating that the composite scattering coefficient of the intermetallic compound is generally greater than that of the metal monomer in the case of the rocket material.

5.5. Variation of the Scattering Coefficient with the Altitude of the Rocket

Figure 12 shows the angular distributions of the composite scattering coefficients for a cruise rocket as it moves from having its trapezoidal base in contact with the ground to the heights , , and .

The data presented in Figure 12 illustrate a clear inverse relation between the composite scattering coefficient and the altitude of the rocket. When and , the oscillation frequency of the composite scattering coefficient is lower across the entire range of scattering angles. However, the quasi-amplitude of the angular distribution curve corresponding to is greater than the quasi-amplitude of the angular distribution curve corresponding to . The frequency of oscillation of the composite scattering coefficient is greater across the entire range of scattering angles when . In addition, the quasi-amplitude of the angular distribution curve at this altitude is similar to that observed when . Because the base of the rocket is closer to the ground when its altitude is low, it is easy for secondary scattering to occur between the rocket and the ground, which enhances the compound electromagnetic-scattering effect.

5.6. Variation of the Scattering Coefficient with Topography

To replicate the characteristics of the terrain with uneven surfaces, three distinct terrains were picked for simulation purposes. They included plains, hills, and valleys (Figure 13). For each terrain, the length L is specified as 2,200 meters, and a total of n = 4,400 sampling points are chosen for the purpose of data collection. For each terrain, a total of 20 samples is acquired and subsequently subjected to calculations. The scattering coefficients are determined and subsequently subjected to statistical averaging. This methodology not only catches diverse topographical characteristics efficiently but also guarantees the extensiveness and unpredictability of data sampling, thereby establishing a dependable database for analyzing the impact of terrain alterations on the composite scattering coefficient.

Figure 14 shows the computed values of the scattering coefficient for various types of terrain, namely, valley, hill, and plain, in sequential order. The influence of the various terrains on the composite scattering coefficient is clearly evident in Figure 14. In a general sense, it can be observed that a hilly terrain exerts the most notable impact on the composite scattering coefficient, followed by a valley terrain, while a plain terrain exhibits the least influence. The magnitude of the oscillation of the composite scattering coefficient is highest for a hilly terrain, second highest for a plain terrain, and lowest for a valley terrain.

The angular distribution curves of the scattering coefficients exhibit higher oscillations for a valley terrain when compared with those of a hilly terrain or a plain terrain. Moreover, throughout the range , the angular distribution curves of the scattering coefficients for a hilly terrain exhibit a higher frequency of oscillation than those of valley or plain terrains. The angular distribution curve of the composite scattering coefficient associated with a hilly terrain exhibits notable extreme values at the mirror-image angle and other scattering angles. When the value of lies between 50° and 80°, the composite scattering coefficient associated with a valley terrain is considerably greater than that associated with hills or plains. This phenomenon is caused by the fact that as a rough surface, the height fluctuation and the incident wave length of the hilly surface meet certain resonance conditions, and the coupling effect between the incident electromagnetic wave and the actual terrain is relatively strong.

6. Conclusion

This research employs the four-component model proposed by Wang and Schmugge to simulate the permittivity of soil, uses a DEM to represent the surface undulation of an actual terrain, and utilizes the VASP to calculate the permittivity of commonly used rocket casing materials, including metallic elements and intermetallic compounds. It uses dimensions provided in the literature to represent the geometrical parameters of a specific type of rocket and employs the FDTD to investigate the composite electromagnetic scattering characteristics from a particular type of cruise rocket moving above an actual terrain. The goal of this research is to obtain angular distribution curves for the composite scattering coefficient and to establish relations between that coefficient and various factors, including the angle of incidence, soil moisture content, frequency of the incident wave, permittivity of the rocket casing material, altitude of the rocket, and undulating terrain. This study thus provides a novel viewpoint for understanding the composite electromagnetic scattering properties of a target above a rough terrain surface. Moreover, it holds considerable practical implications for target recognition within intricate situations.

In contrast to alternative numerical calculation approaches, the FDTD algorithm has demonstrated the capability of achieving elevated levels of accuracy, while jointly reducing computational time and memory requirements. Hence, this study presents an expanded approach to numerically calculate the composite electromagnetic scattering of background and target. The proposed method offers practical solutions to specific engineering challenges, particularly those involving ground surfaces as the background and electric-sized targets positioned above the ground. As a result, this research contributes to the theoretical understanding and practical resolution of relevant engineering problems.

Clearly, the simulation outcomes presented in this paper necessitate further experimental validation. In particular, the terrain surface analyzed in this study is characterized as being one-dimensional, and the soil beneath the surface is assumed to be both isotropic and homogeneous. Further, the target we analyzed is two dimensional and possesses a simple cross-sectional shape. Notably, this study has not yet addressed a composite scattering problem involving a two-dimensional background surface and a three-dimensional target. In subsequent studies, the electromagnetic scattering characteristics of complex targets and multilayer surface media can be further explored.

Data Availability

The DEM data used to support the findings of this study are available from Copernicus DEM Product Handbook (Available online: https://spacedata.copernicus.eu/en/web/guest/collections/copernicus-digital-elevation-model), and the other data are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This study was funded by the National Natural Science Foundation of China (62261054 and 62061048).

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