Electromagnetic Scattering by Multiple Columns Partially Buried in a Ground Plane

Wideband electromagnetic scattering from multiple objects partially buried beneath rough earth soil surfaces is an important topic during recent years due to its extensive applications in several fields, such as ground remote sensing, ground penetrating radar applications, and target identification. Due to the advantages of finite-difference time-domain (FDTD)method in calculating the wideband electromagnetic scattering from rough surface in the presence of multiple objects, the FDTD method under ultrawideband (UWB) Gaussian pulse wave incidence is utilized in the present study for analyzing the frequency response of rough soil surfaces with periodically distributed multiple rectangular cross-section columns buried partially. In this paper, the dielectric property of the actual land surface is expressed using the four-component model, and the actual rough land surface is simulated utilizing Monte Carlo method with exponential correlation function. The emphasis of the present study is on analyzing thewideband response signatures of composite backscattering coefficient varyingwith frequency on the basis of extensive numerical simulations, in particular for calculating and discussing in detail the influence of the root-mean-square height and the correlation length of rough soil surface, soil moisture, the length and the width of the rectangular cross-section column, separation distance, burial depth, and tilt angle on the composite backscattering coefficient.


Introduction
During recent years, more and more attentions are paid to the electromagnetic (EM) wave scattering from rough surface and composite scattering from rough surface with target because of their spread application [1][2][3][4][5] in environmental remote sensing, target tracking and recognition, radar detection, wireless communication, and microwave probing of composite materials and their surface roughness.In particular, the short pulses and ultrawideband radar are widely used in target identification and remote sensing [6].Ultrawideband radar needs the wideband RCS of target to achieve long-range detection and tracking stability [7].The methods of solving scattering problems are plenty, such as the Kirchhoff approximation (KA) [8], small perturbation method (SPM) [9], small slope approximation (SSA) [10], the Method of Moment (MoM) [11], the Finite-Difference Time-Domain (FDTD) [12], and finite element method (FEM) [13].In previous studies, the composite scattering from the rough sea surface and single target under harmonic wave irradiation have been reported and various methods are used in these studies [14][15][16][17][18][19][20].However, the scattering from rough land surface such as soil surface, sand surface, and periodic distributed target under pulse wave irradiation was considered rarely [21,22].
As a numerical method, FDTD can calculate the electromagnetic scattering problems under pulse wave and has superiority of the time-domain numerical algorithm [23][24][25].In this paper, we consider the characteristics of composite wideband electromagnetic scattering from rough soil surface and the partially buried periodic distributed rectangular cross-section columns under Gaussian pulse wave irradiation using FDTD, and the frequency response curves of back composite scattering coefficient are obtained.The results of numerical calculation have a very great importance for achieving the remote sensing information of the earth under the complex electromagnetic environment.

Composite Scattering Model and FDTD Method
In the present study, the actual rough soil surface () is simulated utilizing Monte Carlo method [26] with 1D exponential.The scattering geometry in this paper is illustrated in Figure 1.Suppose that the upper of the coordinate axis  is homogeneous free space and that the lower is isotropic homogeneous soil medium.Periodic distributed rectangular cross-section columns are labeled as , which are partially buried in soil.The separation distance of every two adjacent columns is expressed as .ℎ is the burial depth, and  represents tilt angle, which is the intersection angle of the side length  of column and the forward of coordinate axis .
As shown in Figure 2, the calculated area of composite scattering model is built by FDTD.In the area, it is divided into scattering field region and total region by setting up adjacent boundary , at the same time, setting up extrapolated boundary  in the scattering field region.These two boundaries are planes and extended to absorption boundary.In order to absorb incident wave, we set up absorption boundary conditions which are made of anisotropic medium perfectly matched layer (PML) of thickness of ten grids outside the area, then selecting Gaussian pulse electromagnetic wave as incident wave, and total region from adjacent boundary  is introduced.
The actual rough land surface is modeled as realizations of a Gaussian random process with exponential correlation function by using Monte Carlo method [26] and the dielectric constant of soil using the four-component model established by Wang and Schmugge [27] in this work.
The power spectral density of exponential correlation rough surface is expressed as [28] where  and  are the root-mean-square and correlation length of rough soil surface, respectively. is space wave number.
The rough soil surface is simulated by Monte Carlo method, and 1D rough surface realization with length  can be expressed as where   = Δ ( = 1, . . ., ), and it represents the th discrete points of rough surface.  = 2/ is space wave number.(  ) and (  ) are Fourier's transformation pairs.(  ) is defined as where Δ is space wave number difference of wave spectrum sample adjacent spectral domain, and (0, 1) represents a Gaussian random variable that the mean is 0 and the variance is 1.When  > /2, then (  ) = ( − ) * , and this ensures that the rough surface profile obtained by inverse Fourier transform is a real number.The dielectric constant of the soil is mainly affected by the incident frequency , soil moisture   , soil surface temperature , soil type, and other factors.A four-component model is established by Wang and Schmugge, assuming the sand content in soil is , clay content in soil is , and then the humidity compression point of soil   is The critical body humidity of soil is Assuming that the rock density of soil is   and dry soil density is   , then the plot porosity of soil  is Under normal circumstances, the rock density of soil is   = 2.6 g/cm 3 , and   is determined by the following formula: When   ≤   , the effective dielectric constant of the soil is where where   = 1.0 is dielectric constant of air and   = 3.15 − 0.025 is dielectric constant of ice.  is dielectric constant of pure water and can be calculated by the Debye equation; namely, 0 = 88.045− 0.4147 + 6.295 × 10 −4  2 + 1.705 2  = 1.1109 × 10 −10 − 3.824 × 10 −12  + 6.938 In ( 11)-( 12),  denotes temperature.When   >   , the effective dielectric constant of the soil is Maxwell's equations are a set of basic equations that govern the macroscopic electromagnetic phenomena.For two-dimensional electromagnetic scattering problems, rectangular components of the electromagnetic field may be classified as independent groups, namely, TM wave and TE wave.In TM wave, as an example, FDTD difference equation has the following form [29]: where  = 40 ∘ ,  are discrete grid width along the  = 1 cm,  = 1.6 is direction of FDTD computational domain, respectively, and the coefficients , , , and  are, respectively, where the coefficients () and () are the same with (8) and the coefficients 1(), 2(), 3(), and 4() are where  0 is dielectric constant in vacuum and the value of   and   ( = , ) is where  represents the thickness of the UPML layer and  0 denotes the interface location that UPML layer is near FDTD area.When  = 4,  max = ( + 1)/√  150,  max = 5∼11, and the absorption effect is optimal [30].Near-field scattering data is obtained by the FDTD calculation.Each time field value on the boundary is recorded, and the far-zone scattered field time response is obtained using extrapolation methods of transient pulse source excitation.Then, the frequency response of scattering coefficient is obtained through substituting the scattering coefficient formula by the Fourier transform.
Here, NRCS represents the Normalization Radar Cross Section.The NRCS is expressed as [24] NRCS = lim Here,  is distance from observation point to origin,   is electric field of scattered wave, and   is electric field of incident wave.
In this work, the Gaussian pulse electromagnetic wave is given by [24] Here,  is constant and decides the pulse width, and it is set by  = 2/.Generally,  0 = 0.8.

Numerical Results
In the following calculations, the emphasis is put on the back composite scattering coefficient which varies with frequency.
It should be noted that the incident wave is Gaussian pulse electromagnetic wave of TM polarization, the width of grid is  = 0.5 cm, incident angle is   = 30 ∘ , and sampling length of rough land surface is  = 1600 cm.The final example is an ensemble average over 40 realizations of rough surfaces.The dielectric parameter of soil is calculated using models found by Wang and Schmugge [27].In this part, we consider that type of soil is silty loam.Without special explanation, in numerical simulations, parameters of rectangular crosssection column are given with  = 20 cm,  = 70 cm,  = 0 ∘ , ℎ = 5 cm, and  = 50 cm.The root-mean-square of rough surface fluctuation  is 1 cm, the correlation length of rough surface fluctuation  is 4 cm, and soil moisture  V is 0.2.
In order to obtain the characteristics of composite wideband electromagnetic scattering from the rough soil surface with partially buried periodic distributed rectangular crosssection columns, the curves of back composite scattering coefficient which varies with frequency are studied by numerical calculation under different parameters of rough surface and column.In the following, the numerical simulations are performed on the computer with two 2.60 GHz processors (Intel Xeon E5-2650), 16 GB Memory, Microsoft Windows 7 operation system, and the Fortran PowerStation 4.0 compiler.Under the configuration, the computational time and memory size for one realization are approximately 33 min and 47 MB, respectively.
Figure 3 depicts the distribution of  with the incident wave frequency  for  = 1 cm, 3 cm, and 5 cm, respectively.It is found that the influence of  on  is obvious.With the increase of , the scattering coefficient  increases, except for some high frequency domain (2.215 GHz <  < 3 GHz).In addition, the position and value for the peaks of  are nearly unchanged with .On the other hand, with the increasing incident wave frequency , the oscillation for the amplitude of curve becomes small.
To examine the scattering coefficient difference between single column case and multiple columns case, a comparison of single column case and multiple columns case is made in Figure 4.In Figure 4, the root-mean-square of rough surface is  is 1 cm.It can be readily observed that for multiple columns case obvious peaks exist, which can be attributed to the grating effect due to the scattering from periodically distributed rectangular columns.
In Figure 5, the dependency of  on  is plotted.From Figure 5, it is found that, with the increase of , the scattering coefficient  increases, and the position and value for the peaks of  are nearly unchanged with .
The distribution of  with  for different soil moisture  V with  1 = 4.9301 − 0.4510 ( V1 = 0.1 g/cm 3 ),  2 = 10.2802− 1.6464 ( V2 = 0.2 g/cm 3 ), and  3 = 26.0279− 5.2063 ( V3 = 0.4 g/cm 3 ) is depicted in Figure 6.The soil moisture represents dielectric parameter of soil.It is obvious that the influence of  V on  is small and obscure.When the frequency is less than 0.9 GHz ( < 0.9 GHz), all the curves in Figure 6 are coincident.However, it can be found that, with  V increases,  slightly become greater when the frequency is larger than 0.9 GHz ( > 0.9 GHz).
Figure 7 depicts the distribution of  with  for different breadth of column  ( = 20 cm, 60 cm, 100 cm, and 160 cm).It is found that the influence of  on  is obvious; that is, with the increase of , the number of rectangular cross-section columns decreases.And the oscillation for the amplitude of curve becomes smaller; the oscillation for the frequency of curve becomes greater.Figure 8 depicts the distribution of  with  for different length of column  ( = 30 cm, 70 cm, and 100 cm).It is found that the influence of  on  is complex and obvious; that is, at most of the frequency points, the larger  is, the smaller  is, the smaller the peak value of .Moreover, as  increases, the oscillating amplitude of the curve is smaller.
Figure 9 depicts the distribution of  with  for different separation distance of every two adjacent columns  ( = 30 cm, 50 cm, and 100 cm).It is found that the influence of  on  is obvious.At most of the frequency points, with the increase of , the number of rectangular cross-section columns decreases in the calculated area, the oscillating frequency of curve is greater, and  is larger for the same frequency point.that the influence of ℎ on  is small and obscure.Part of the curves are coincident.However, it can be found that, with the increase of ℎ,  becomes greater when frequency is larger than 1.73 GHz ( > 1.73 GHz).

International Journal of Antennas and Propagation
Figure 11 shows the relationship of  with  for different  ( = 10 ∘ , 30 ∘ , and 60 ∘ ) when ℎ = 15 cm.Obviously, the influence of  on  is complex.When frequency is less than 1.09 GHz ( < 1.09 GHz), the bigger  is, the smaller  is.If frequency satisfies the inequality (1.09 GHz <  < 2.5 GHz),  of  = 30 ∘ is smaller than others.
Hereinbefore, the results from Figures 3-11 in the case of TM polarization are obtained by plentiful calculation on the condition that other parameters are stationary.Only one point is worth illustrating; that is, the influence of some parameters on the back composite scattering coefficient is restricted by other parameters.

Conclusion
In this paper, the problems of composite wideband electromagnetic scattering from exponential correlation rough soil surface with partially buried periodic distributed rectangular cross-section columns under Gaussian pulse wave irradiation are investigated using transient FDTD.The influence of the root-mean-square and the correlation length of rough surface fluctuation, soil moisture, the length and the width of the rectangular cross-section column, separation distance, burial depth, and tilt angle on the composite scattering signatures are discussed in detail.The numerical simulations indicate that the variation of composite scattering coefficient versus frequency is oscillatory.It is also showed that the composite scattering coefficient versus frequency increases with the increase of root-mean-square of soil surface, water ratio of soil, the target section height, and the separation distance of target.However, simulation results indicate that the composite scattering coefficient versus frequency decreases with the increase of target section width.In summary, the variation of wideband scattering coefficient is very complicated and is very sensitive to incidence angle of electromagnetic wave.However, the wideband scattering coefficient under Gaussian pulse wave incidence is less sensitive to the correlation length of rough soil surface, the depth of buried objects, and the dielectric constant of target.Although the multiple columns partially buried in a randomly rough surface are a rather simplified model, the present study is potentially valuable for many applications, such as ground remote sensing, ground penetrating radar applications, underground mine detection, tunnel detection, and target identification.In particular, the simplified model of periodically distributed columns represents many realistic objects, such as tree trunks, mine props, telegraph poles.The future investigation on this topic will include electromagnetic scattering from two-dimensional rough land surface with complex multiple targets.

Figure 1 :Figure 2 :
Figure 1: Geometry of composite scattering from the rough soil surface and partially buried periodic distributed rectangular crosssection columns.

Figure 3 :Figure 4 :
Figure 3: The influence of  on .

Figure 7 :Figure 8 :
Figure 7: The influence of  on .

Figure 10
Figure 10 depicts the distribution of  with  for different burial depth ℎ (ℎ = 5 cm, 15 cm, and 25 cm).It is obvious

Figure 9 :Figure 10 :
Figure 9: The influence of  on .

Figure 11 :
Figure 11: The influence of  on .
Δ is the time increment FDTD calculation is only in a limited area.Thus, it is necessary to set absorbing boundary in truncated boundary for open domain electromagnetic scattering problem.Anisotropic perfectly matched layer (UPML) absorbing boundary is selected in this work.FDTD iteration order of UPML layers is   ,   →    →   →   and   →   ,   →   ,   , and the iterative formula of electronic field   of UPML layer of truncated conductive medium is below: (15)e the value of grade  and the left field component node labels have the same position.In(15), () denotes the electric conductivity.() represents electrical permittivity.() stands for the magnetic permeability.