Composite Electromagnetic Scattering Characteristics of Actual Terrain and Moving Rockets

. Tis research involved studying the characteristics of composite electromagnetic scattering from a rocket moving above actual terrain using the fnite-diference time-domain method. Te angular distribution curve of the composite scattering coefcient was obtained, and the infuences 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 coefcient were investigated. Results show that the composite scattering coefcient oscillates with the scattering angle and increases in the direction of mirror refection. 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 infuence of diferent terrain undulations on the composite scattering coefcient is noticeable, it follows no fxed pattern.


Introduction
In the increasingly complex feld 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][2][3][4].Tis is particularly important in unpredictable geographical circumstances where the propagation and scattering of electromagnetic waves are infuenced 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][6][7].Te fnite-diference time-domain (FDTD) method is a widely used numerical tool for full-wave analyses of electromagnetic felds in complex media.It has a wide range of applications that deal with complex media and geometries in diferent scientifc and engineering felds-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][9][10][11].
Currently, the fnite-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 infnite-wave propagation can be modeled efectively by employing absorbing boundary conditions [18].Moreover, while FDTD demands a denser mesh to deal with the details, its meshing procedure is comparatively uncomplicated.
Te 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, specifcally inhomogeneous soils.Teir 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.Teir fndings demonstrated that the FDTD method is capable of efectively 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 Kirchhof approximation to simulate the polarimetric scattering characteristics of a rough, twodimensional surface successfully.Tey also verifed the high efciency of the method by using parallel computations, providing a new direction for the study of threedimensional scattering problems.Yang et al. [21] introduced an iterative hybrid approach that integrates a fast, multistage, multipole algorithm with the Kirchhof approximation to analyze composite electromagnetic scattering from a three-dimensional target above a two-dimensional, randomly rough surface.Te researchers utilized a truncation rule and a fast far-feld approximation to decrease the computational burden.Tey evaluated the hybrid method rigorously and found that it has exceptional accuracy and efciency, surpassing those of conventional methods by considerable margins.Smith et al. [22] utilized ultralowfrequency (ULF) electromagnetic waves and investigated their efectiveness in detecting targets submerged in the ocean.Tey used FDTD model simulations to confrm the efective 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 ofers an efective solution for addressing electromagnetic scattering problems in two dimensions.Tis method involves the utilization of Laguerre polynomials in decomposing the computational space into multiple subdomains.Tis approach enhances both the computational efciency 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] ofers 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 fve regions in China.Tey 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.Te FDTD is subsequently used to investigate the composite electromagnetic scattering characteristics of the actual terrain in conjunction with a specifc type of moving rocket positioned above it.We employed numerical computations to obtain the angular distributions of the composite scattering coefcients.Tis study further investigated the infuence on the composite scattering coefcient 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.

Modeling
Figure 1 shows a geometric representation of the composite electromagnetic scattering from a specifc type of cruise rocket located above a real terrain surface.Te rocket is positioned vertically above the surface, which is underlain by an isotropic, homogeneous soil medium that extends to infnity.Te cross section of the rocket comprises an isosceles trapezoid, a rectangle, and an isosceles triangle.Te heights of the isosceles trapezoid, rectangle, and isosceles triangle are denoted by h 1 , h 2 , and h 3 , respectively (Figure 1).Te half-width of the base of the isosceles trapezoid is denoted by d 1 , while those of the base of the triangle and the rectangle are represented by the same variable, d 2 .
Figure 2 illustrates the FDTD model utilized to compute the composite electromagnetic scattering from a specifc 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.Te region below the connecting boundary AB represents the total feld area, while the region above it represents the scattered-feld area.Te incident wave is introduced into the former region by imposing an equivalent electromagnetic fux on the connecting boundary.Te output boundary CD is positioned within the scattered-feld 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.Te wavelength of the incident electromagnetic wave is taken to be λ � 10 m, while the thickness is given as 10 grids.Te COP-DEM is ofered 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% confdence level (LE90).Te evaluation results exhibited a mean inaccuracy of −1.29 m, with a standard deviation of 0.85 m.
Te Copernicus DEM product handbook states that the absolute horizontal accuracy of the COP-DEM is infuenced by two factors.Te frst is the positional accuracy of the individual DEM view, which depends upon the quality of the TerraSAR-X image base product.Te accuracy of this product is estimated to be better than 0.3 m.Te 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 quantifed 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 confdence level of 90%.
In this study, we used the COP-DEM to refect 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 infuence on the soil's permittivity.Models that are now prevalent in the feld include the Wang and Schmugge model [30], the GRMDM model [31], and the Topp model [32].Te Wang and Schmugge model is particularly noteworthy for its exceptional accuracy and for its applicability in estimating the dielectric properties of soils.Tis is primarily due to its comprehensive consideration of multiple parameters, including soil density, moisture content, and temperature.It thus extends beyond the oversimplifed 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.
Te moisture compression point of soil is defned by the four-component model proposed by Wang and Schmugge.Tis model considers the dry soil composition in terms of sand content (S%) and clay content (C%), with the constraint that the S + C ≤ 100 then defnes 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: Defne the parameters as shown in the following equation: Typically, the density of rocks within a soil is denoted by ρ r � 2.6 g/cm 3 , and the density of dry soil is represented by ρ b .Te cumulative porosity of the soil is thus given by the following equation:  Te value of ρ b 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 where In these equations, the permittivity of ice is ε i � 3.2 − j0.1, the permittivity of rock is ε r � 5.0 − j0.1, while ε a � 1.0 is the permittivity of air.Te permittivity ε w of pure water at the operating frequency can be determined using Debye's formula, as given by the following equation: where ( 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 S � 51.5% and clay content C � 13.5%.

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 fgures: a trapezoid, a rectangle, and a triangle.Te trapezoid has height h 1 � 3λ and base width d 1 � 2λ.Te rectangle has height h 2 � 6λ.Te triangle has height h 3 � λ, and its base width is the same as the half-width of the rectangle, d 2 � λ.
Te permittivity can be obtained from the Kramers-Kronig dispersion relation and the defnition of the direct-transfer probability [34].Te real part of the 4 International Journal of Antennas and Propagation permittivity corresponds to the material's capacity to retain charge, whereas the imaginary part represents its ability to dissipate electromagnetic energy.Te formula for the permittivity is Te real part of the permittivity is denoted as ε 1 , while the imaginary part is denoted as ε 2 .Tese values can be computed using equations ( 15) and ( 16), respectively. and In these equations, the symbol ω ′ is used to represent the (complex) frequency of the radiation, while P represents the principal component of ε 1 (ω).Te quantities ], c, and m, respectively, denote the dimensions of the valence, conduction bands, and the mass of the carriers.Te term BZ refers to the path within the frst Brillouin zone.Moreover, E (c,v) (k) and ψ (c,v)k represent the energy and the intrinsic wavefunction at each location along the path parametrized by k.Lastly,  e and p → 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.Te selected materials comprise the three metallic elements, iron (Fe), nickel (Ni), and titanium (Ti), along with the three intermetallic compounds aluminum-magnesium alloy (Mg 2 Al 3 ), titanium-nickel alloy (TiNi), and coppernickel alloy (Cu 3 Ni).In this work, we represented the crystalline forms of the elements Fe, Ni, and Ti as cubic structures characterized by the lattice parameters, a � 5.000 Å, b � 5.000 Å, c � 5.000 Å, α � β � c � 90 °, and V 0 � 125.000Å3 .Te 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 fles from the materials project resource library [35,36] 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].Tis 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 structuraloptimization process, we set the truncation energy (ENCUT) to 480 eV, the optimal energy-convergence criterion (EDIFF) to 1 × 10 −5 eV, and the interatomic energyconvergence 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.Te calculation procedure encompasses the infrared (IR), visible (VR), and ultraviolet (UV) energy spectra.Figure 5 presents the simulated permittivity values for the corresponding materials.
Te solid line in Figure 5 represents the real component of the material's permittivity.Tis component indicates the material's ability to store energy and refects the speed at which electromagnetic waves propagate within the medium.A higher value of the real component indicates a greater capacity for polarization.Te dashed line in Figure 5 corresponds to the imaginary component of the material's permittivity.Tis component represents the material's loss factor, which characterizes the dissipation of energy from an International Journal of Antennas and Propagation 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 E � h • f, 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.

Finite-Difference Time-Domain (FDTD) Methods
Based on the principles of the fnite-diference approach in the time-domain (FDTD), the diference equation for a transverse magnetic (TM) wave in a two-dimensional electromagnetic-feld problem can be represented mathematically as follows:  International Journal of Antennas and Propagation where m represents the location of the FDTD grid node for the feld component on the left side of the equation.Te 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: and where ε 1 , μ 1 , and σ 1 denote the dielectric parameters of the medium inside the computational domain.Te quantities s x and s y are uniaxial parameters in the x and y directions, respectively, which are given by where the parameters σ x , K x are calculated as follows, respectively. and Optimal absorption occurs when n � 4. Te other parameters in these equations are σ max � (n + 1)/( �� ε r √ 150πδ) and k max � 5-11, where d is the thickness of the UPML layer.Once the FDTD calculation attains stability, we record the near-feld computational outcomes at the output boundary.Te far-area dispersed feld can be derived subsequently by employing the time-harmonic feld-extrapolation technique based on the equivalence principle.
Equation (26) shows the computation of the composite scattering coefcient for the combination of the target and the soil surface: Te normalized radar-scattering cross section in equation ( 26) is given by In equation ( 27), the variables r, E s , and E i , respectively, represent the distance from the origin to the point being observed, the electric feld of the scattered wave in the far zone, and the electric feld of the incoming wave.Te variable L is defned as the sampling length of the soil surface.

Validation of FDTD Methods
To assess the validity of the Finite-Diference Time-Domain (FDTD) algorithm in this study, the composite scattering coefcients of an exponentially rough surface and the target positioned above it were computed using the time-harmonic feld FDTD method.Tese 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 ] � 0.1 GHz, the angle of incidence to be θ i � 30 °, the length of the random rough surface to be L � 160λ, the correlation length to be l � 1.5λ, the root-mean-square of the height heave to be δ � 0.1λ, the complex permittivity of the medium below the rough surface to be ε r � 71.2717 − j92.4644, and the target to be a combination of a rectangle and a trapezoid where the height of the rectangle is h 1 � 4λ, the width is h 2 � 2λ, the height of the trapezoid is h 3 � 3λ, and the width of the upper base of the trapezoid is L 1 � 4λ and that of the lower base is L 2 � 2λ.
Te angular distributions obtained using these two algorithms exhibit a high degree of similarity (Figure 6).Tese results provide empirical evidence supporting the validity and accuracy of the method employed in this research.

Numerical Results and Discussion
In the subsequent numerical computations, unless otherwise specifed, the frequency of the incident electromagnetic wave is 30 MHz and the angle of incidence is θ s � 30 °. DEM dataset at a particular terrain was selected wherein the soil moisture content is assumed as m � 40% and its permittivity is then given by ε � 18.94 − j2.89.Te rocket casing is composed of nickel monomers, with a permittivity of ε Ni � 29.04 − j8.98.Te rocket is positioned at a height of h � 20λ above the ground.In the simulation, the number of rough surfaces is quantifed as 20 [40,41].

Variation of the Composite Scattering Coefcient with the
Angle of Incidence.Figure 7 shows the computed variation of the scattering coefcient with the angle of incidence.Tese results were achieved by considering three diferent incident angles: θ i � 30 °, θ i � 60 °, and θ i � 75 °.An International Journal of Antennas and Propagation oscillatory-like change in the scattering coefcient with respect to the scattering angle is evident in Figure 7.In addition, scattering enhancement can be observed in the direction of specular refection, namely, at the angles θ s � 30 °, θ s � 60 °, and θ s � 75 Te fndings presented in Figure 7 show that the composite scattering coefcient is substantially infuenced by the angle of incidence.Specifcally, it decreases as the angle of incidence increases within the extended range of scattering angles, particularly −60 °< θ s < 60 °.Within narrower ranges of scattering angles-specifcally −80 °< θ s < − 60 °and 60 °< θ s < 80 °-the composite scattering coefcient increases as the angle of incidence increases.Tis phenomenon can be attributed to the pronounced coupling efect resulting from abrupt changes in the topographic relief along a specifc direction.

Variation of the Composite Scattering Coefcient with Soil
Moisture Content. Figure 8 shows the variation of the composite scattering coefcient with variations in the soil moisture content at the fxed temperature T � 20 °.Tis fgure illustrates the variations for three distinct values of the Te oscillatory variation of the scattering coefcient with respect to the scattering angle is clear in Figure 8. Te scattering is also enhanced in the direction corresponding to specular refection.Tis pattern is observed consistently in all subsequent numerical calculations and will not be reiterated in further discussions.Te impact of the soil moisture level on the composite scattering coefcient is notable; it increases as the soil moisture increases across the whole range of scattering angles.Te result is particularly obvious in the ranges −80 °< θ s < 10 °and 40 °< θ s < 80 °, while it is comparatively smaller in the range 10 °< θ s < 40 °.
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.

Variation of the Scattering Coefcient with the Frequency
of the Incident Wave. Figure 9 shows the composite scattering coefcient as a function of the frequency of the incident wave.Tis fgure presents the results calculated for the three specifc frequency values, f 1 � 30 MHz, f 2 � 40 MHz, and f 3 � 50 MHz.
Figure 9 demonstrates the notable impact of the frequency of the incident wave on the composite scattering coefcient.Specifcally, the composite scattering coefcient decreases as the frequency increases within a substantial portion of the range of scattering angles; i.e., −80 °< θ s < − 70 °and −10 °< θ s < 80 °.However, there is a smaller portion of the scattering angle range (−70 °< θ s < − 10 °) where the composite scattering coefcient remains relatively unchanged with frequency.In addition, when −10 °< θ s < 80 °, the oscillation amplitude of the angular distribution curves for the scattering coefcient is larger for f 1 � 30 MHz than for either f 2 � 40 MHz or f 3 � 50 MHz, and the corresponding scattering coefcient exhibits a higher peak at the angle of incidence θ s � 30 °. Tis occurs because θ s � 30 °is the mirror refection direction for the angle of incidence θ i � 30 °.Other peaks are due to resonances between the wavelength of the electromagnetic wave and the structure of the terrain at those locations.In addition, a clear observation can be made based on Figures 10 and 11, indicating that the composite scattering coefcient of the intermetallic compound is generally greater than that of the metal monomer in the case of the rocket material.International Journal of Antennas and Propagation 5.5.Variation of the Scattering Coefcient with the Altitude of the Rocket.Figure 12 shows the angular distributions of the composite scattering coefcients for a cruise rocket as it moves from having its trapezoidal base in contact with the ground to the heights h 1 � 5λ, h 2 � 20λ, and h 3 � 45λ.Te data presented in Figure 12 illustrate a clear inverse relation between the composite scattering coefcient and the altitude of the rocket.When h 1 � 5λ and h 3 � 45λ, the oscillation frequency of the composite scattering coefcient is lower across the entire range of scattering angles.However, the quasi-amplitude of the angular distribution curve corresponding to h 1 � 5λ is greater than the quasi-amplitude of the angular distribution curve corresponding to h 3 � 45λ.Te frequency of oscillation of the composite scattering coefcient is greater across the entire range of scattering angles when h 2 � 20λ.In addition, the quasi-amplitude of the angular distribution curve at this altitude is similar to that observed when h 3 � 45λ.Because the base of the rocket International Journal of Antennas and Propagation 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 electromagneticscattering efect.

Variation of the Scattering Coefcient with Topography.
To replicate the characteristics of the terrain with uneven surfaces, three distinct terrains were picked for simulation purposes.Tey included plains, hills, and valleys (Figure 13).
For each terrain, the length L is specifed 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.Te scattering coefcients are determined and subsequently subjected to statistical averaging.Tis methodology not only catches diverse topographical characteristics efciently 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 coefcient.Figure 14 shows the computed values of the scattering coefcient for various types of terrain, namely, valley, hill, and plain, in sequential order.Te infuence of the various terrains on the composite scattering coefcient 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 coefcient, followed by a valley terrain, while a plain terrain exhibits the least infuence.Te magnitude of the oscillation of the composite scattering coefcient is highest for a hilly terrain, second highest for a plain terrain, and lowest for a valley terrain.
Te angular distribution curves of the scattering coefcients exhibit higher oscillations for a valley terrain when 40 °< θ s < 80 °compared with those of a hilly terrain or a plain terrain.Moreover, throughout the range −30 °< θ s < 80 °, the angular distribution curves of the scattering coefcients for a hilly terrain exhibit a higher frequency of oscillation than those of valley or plain terrains.Te angular distribution curve of the composite scattering coefcient associated with a hilly terrain exhibits notable extreme values at the mirror-image angle and other scattering angles.When the value of θ S lies between 50 °and 80 °, the composite scattering coefcient associated with a valley terrain is considerably greater than that associated with hills or plains.Tis phenomenon is caused by the fact that as a rough surface, the height fuctuation and the incident wave length of the hilly surface meet certain resonance conditions, and the coupling efect between the incident electromagnetic wave and the actual terrain is relatively strong.

Conclusion
Tis 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 specifc 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.Te goal of this research is to obtain angular distribution curves for the composite scattering coefcient and to establish relations between that coefcient 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.Tis 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.Te proposed method ofers practical solutions to specifc 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.

2 International Journal of Antennas and Propagation 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.Tere are now several globally available opensource DEM datasets, including ETOPO, GTOPO30, GMTED2010, SRTM DEM, NASA DEM, ASTER GDEM, AW3D30, TanDEM-X DEM, and Copernicus DEM.Te 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.Te 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.Te COP-DEM dataset is an edited version of a DSM taken from the WorldDEM product.It was created by sampling highresolution DEMs and using subsequent manual editing to improve its uniformity in the representation of water bodies and other inaccurate terrain characteristics.

Figure 2 :
Figure 2: FDTD modeling of composite electromagnetic scattering calculations for a rocket moving above real terrain.

Figure 1 :
Figure 1: Schematic of the composite electromagnetic scattering geometry of a rocket moving above an actual terrain.
. Te tetragonal phase I4 1 /amd space group (space group No. 141) shows the crystal structure of the Mg 2 Al 3 alloy.Te lattice parameters in the original fle are a � 6.445 Å, b � 6.445 Å, c � 19.215 Å, α � β � c � 90 °, and V 0 � 798.101Å3 .Te dominant geometry in this structure comprises three Mg atoms and eight Al atoms, resulting in an 11-coordination geometry.Te crystal structure of the TiNi alloy belongs to the monoclinic phase P2 1 /m space group (space group No. 11), for which the lattice parameters in the original fle are a � 2.892 Å, b � 3.967 Å, c � 4.825 Å, α � c � 90 °, β � 105.229 °, and V 0 � 53.412 Å3 .In the structure of this alloy, each Ti atom is combined with an Ni atom in a 7-coordination geometry.Te Cu 3 Ni alloy has a crystal structure that falls within the orthorhombic Cmmm space group (space group No. 65).Te lattice parameters in the original fle are a � 2.632 Å, b � 4.121 Å, c � 8.246 Å, α � β � c � 90 °, and V 0 � 89.443 Å3 .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 Mg 2 Al 3 , Figure 4(c) displays the structural model of TiNi, and Figure 4(d) exhibits the structural model of Cu 3 Ni.

Figure 4 :Figure 5 :
Figure 4: Structural models of the metallic elements and intermetallic compounds: (a) depicts the structural model for the metallic elements, (b) shows the structural model of Mg2Al3, (c) displays the structural model of TiNi, and (d) exhibits the structural model of Cu3Ni.

Figure 6 :Figure 7 :
Figure 6: Comparison of MoM and FDTD numerical calculation results.

Figure 8 :
Figure 8: Variation of the composite scattering coefcient with soil moisture content.

Figure 9 :Figure 10 :
Figure 9: Variation of the composite scattering coefcient with the frequency of the incident wave.

Figure 11 :Figure 12 :
Figure 11: Variation of composite scattering coefcient with permittivity of metal compounds.

Figure 13 :Figure 14 :
Figure 13: Tree types of actual terrain with more considerable undulations.

Table 1 :
List of sand content and clay content of diferent soil types.