A Fast Algorithm for Electromagnetic Scattering from One-Dimensional Rough Surface

In this paper, the Adaptive Modified Characteristic Basis Function Method (AMCBFM) is proposed for quickly simulating electromagnetic scattering from a one-dimensional perfectly electric conductor (PEC) rough surface. Similar to the traditional characteristic basis function method (CBFM), Foldy-Lax multiple scattering equations are applied in order to construct the characteristic basis functions (CBFs). However, the CBFs of the AMCBFMare different from those of the CBFM. In the AMCBFM, the coefficients of the CBFs are first defined. Then, the coefficients and the CBFs are used to structure the total current, which is used to represent the induced current along the rough surface. Moreover, a current criterion is defined to adaptively halt the order of the CBFs. The validity and efficiency of the AMCBFM are assessed by comparing the numerical results of the AMCBFM with the method of moments (MoM). The AMCBFM can effectively reduce the size of the matrix, and it costs less than half the CPU time used by the MoM. Moreover, by comparing it with the traditional CBFM, the AMCBFM can guarantee the accuracy, reduce the number of iterations, and achieve better convergence performance than the CBFM does. The second order of the CBFs is set in the CBFM. Additionally, the first order of the CBFs of the AMCBFM alone is sufficient for this result.


Introduction
Electromagnetic (EM) scattering from rough surface has been widely studied and applied in the research areas of marine communication, target detection, and stealth technology [1][2][3][4][5].The method of moments (MoM) [6][7][8] has high accuracy and is widely used to simulate the EM scattering from rough surface.However, the dimensions of the solution matrix of the MoM are  × , where  represents the number of unknowns.When solving this problem using the MoM, one has to simulate a sufficiently long rough surface, which results in many unknowns  and time-consumption.Several numerical methods have been developed, such as the fast multipole method (FMM) [9] and the characteristic basis function method (CBFM) [10][11][12].In the FMM, an additional algorithm is used to deal with the kernel function.In addition, the iterative method is applied to solve the matrix equation.However, the FMM is an iterative method, which is limited by the convergence of the matrix equation.In the CBFM, the Foldy-Lax multipath scattering equations [13] are applied in order to structure the characteristic basis functions (CBFs).The self-interaction is first considered in order to generate the primary characteristic basis functions (PCBFs).Moreover, the mutual coupling effect of the subcells is considered in order to build the secondary characteristic basis functions (SCBFs).In previous work of our teams [14], the CBFM is used to simulate the EM scattering from a dielectric rough surface.In this work, the orders of the SCBFs were set by us and were determined through several tries.In the previous work of others [15], the following criterion was used to determine the order of the SCBFs: where Z is the impedance matrix, V is the known vector that is excited by the incident wave, J  represents the induced current that is obtained by the CBFM, and ‖ ⋅ ‖ 2 is the L2 norm.From (1), it is readily seen that Z must be refilled and the matrix vector product is calculated in order to determine the order of the SCBFs.Therefore, determining the order of the SCBFs using (1) will be time-consuming and require more memory.

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International Journal of Antennas and Propagation In this paper, the Adaptive Modified Characteristic Basis Function Method (AMCBFM) [16] with a new current criterion is proposed.Similar to the traditional CBFM, the Foldy-Lax multipath scattering equations are also used to structure the CBFs.The PCBFs, such as the CBFM, are first obtained.However, the coefficients of the PCBFs are defined.Then, the primary total current of each cell is obtained by combining the coefficients and the PCBFs.Then, the coefficients are taken with the PCBFs in order to get the first order of the SCBFs.The coefficients of the first order of the SCBFs and the first total current of each cell are obtained.Like the primary total current, the first total current is used to get the second order of the SCBFs.Higher orders of the SCBFs are similarly obtained.Moreover, the new current criterion that is defined in this paper is used to adaptively halt the order of the SCBFs.
The remainder of this paper is organized as follows.In Section 2, the theoretical formulations of the AMCBFM are exhibited in detail.In Section 3, the validity and efficiency of the AMCBFM are assessed and compared with the MoM and the CBFM.Section 4 presents a summary of this paper and further research plans on this topic.

EM Scattering from a Rough Surface Using the MoM.
Assume that there is a tapered plane wave   incident upon a PEC rough surface, as shown in Figure 1, where  = () is a Gaussian rough surface with a simulated length .  and   are the incident angle and the scattered angle, respectively.Space Ω 0 with the relative permittivity  0 and the relative permeability  0 is the incident space.Here, space Ω 0 is assumed to be the free space; i.e.,  0 = 1, and  0 = 1.  represents the scattered wave in space Ω 0 .The position vector is r = x + ẑ.
For a Horizontal (H) polarized incident wave, the surface integral equation for this scattering problem is [17] where  denotes the interface of the PEC rough surface, n is the unit normal vector of the rough surface, (r, r  ) = (/4) (1)  0 ( 0 |r − r  |) is the two-dimensional Green's function in the space Ω 0 , and   (r  ) =   (r  ) +   (r  ) represents the total wave including the incident wave and the scattered wave in space Ω 0 .
For the Vertical (V) case, the surface integral equation is as follows [17].
By using the MoM with the pulse basis functions and the point matching technique, the integral equations ( 2) and ( 3) can be discretized into the following matrix equations.
The elements of the impedance matrix A and B in detail are where  0 is the wavenumbers in space Ω 0 and Δ  = Δ √ 1 + (  (  )) 2 .The subscripts  and  are the field point and source point along the rough surface, respectively.The elements of the right vector b and V are   (r  ), and the left vectors U and I are the unknown current vectors.

Theoretical Formulations of AMCBFM.
Analyzing ( 4) and ( 5), the two equations can be rewritten as where Z is the impedance matrix with the dimensions  × , the dimensions of the right vector V are ×1, the dimensions of the unknown current vector J are  × 1, and  is the number of the discrete segments along the rough surface.
V  indicates the incident wave upon the  ℎ cell.
(1) The Primary Total Current J () . For the  ℎ ( = 1, 2, . . . . . ., ) cell, only the self-interaction is considered to generate the PCBFs J   such as the CBFM where Z  is the self-interaction impedance matrix of the  ℎ cell, and J   and V  are the PCBFs and the incident wave vector of the  ℎ cell, respectively.Using the lower upper (LU) decomposition technique, one can obtain the PCBFs J   .The lower and the upper triangular matrices of Z  are obtained and saved.
Assuming the total current J   is only from the PCBFs, it is called the primary total current J
Then, by multiplying (12) Using the LU decomposition technique, the PCBFs coeffi- and the primary total current are acquired.It is obviously found that the mutual coupling effect of the subcells is neglected, and thus J ()  is not accurate.However, the coefficients [ can be used to get the SCBFs.
(2) The First Total Current J  (1)    .Considering the mutual coupling effect of the subcells when building the SCBFs, for the  ℎ ( = 1, 2, . . . . . ., ) cell, the first order of the SCBFs J  1  is defined as where J  1  is the first order SCBFs.In addition, the coefficients [ are obtained using (13).
The first total current J  (1)    is defined as where  (1)   and  (1)  1 are the coefficients of the first order of SCBFs.
International Journal of Antennas and Propagation It is readily seen that ( 16) has 2M unknowns and M equations.
In order to solve this problem, ( 16) is multiplied by

𝑖
. Similar to the first order of SCBFs, the  ℎ order of the SCBFs is defined as where
(4) The Current Criterion.The current criterion that determines whether the higher-order SCBFs are needed is defined as [16] where ‖ ⋅ ‖ 2 is the L2 norm.It is readily seen that the order of the SCBFs can be adaptively halted.
(5) The Percentage Error.The percentage error of the AMCBFM or the CBFM is defined as where J / represents the induced current that is obtained using the AMCBFM or the CBFM, J  denotes the induced current that is calculated using the MoM, and ‖ ⋅ ‖ 2 is the L2 norm.

Numerical Simulations and Discussions
The validity and efficiency of the AMCBFM are first assessed by comparing its results with the results that were obtained from the CBFM and the MoM.Both the H and V polarized incident waves are considered.In Figure 2, the bistatic scattering coefficients that are obtained using these three methods are shown.The induced currents along the rough surface that are obtained by the AMCBFM and the MoM are given in Figure 3.The frequency of the incident wave is  = 1.5GHz, and the incident angle is   = 30 ∘ .The rms height and the correlation length of the simulated rough surface are  = 0.1 and  = 1.0, respectively.The length of the simulated Gaussian rough surface is  = 409.6with a discrete interval of Δ = 0.1.In other words, there are  = 4096 segments along the rough surface.Thus, the dimension of the MoM matrix is 4096×4096.The rough surface is equally divided into 4 cells.Thus, the dimensions of the CBFM matrix and the AMCBFM matrix are only 1024 × 1024.It is clearly determined that the dimensions of the dealing matrix have been significantly reduced in the AMCBFM and the CBFM.The orders of the SCBFs of the CBFM are set as  = 2.The current criterion of the AMCBFM is  = 0.1 and the halted order of the SCBFs is  = 1.From Figures 2 and 3, it is readily seen that the AMCBFM and the CBFM agree well with the MoM in both the H and V polarization.However, the halted order of SCBFs of AMCBFM is  = 1 through the criterion  = 0.1.The orders of the SCBFs of the CBFM are set as  = 2.Moreover, in Table 1, the CPU times of the AMCBFM, the CBFM, and the MoM are shown.It is obviously found that the AMCBFM costs less than half the CPU time of the MoM.In addition, Table 1 compares the percentage errors of the AMCBFM and the CBFM.It is readily seen that the AMCBFM with the order  = 1 is more efficient than the CBFM with the order  = 2.The results are implemented on a computer with an Intel Core i7-6700 processor, 4 GB of memory, and the Microsoft Windows 10 operating system.One may note that only one sample of the rough surface is used in Figures 2 and 3.In order to further test the efficiency of the proposed algorithm, the results obtained with  = 0.5 and  = 1.5 are shown in Table 2.These results are based on the rough surface with a simulated length of  = 409.6,a discrete interval of Δ = 0.1, and an incident angle of   = 30 ∘ .The CPU time of the MoM is 12.995 seconds for H polarization and 13.01 seconds in the V case.In the AMCBFM, the rough surface is equally divided into 4 cells.Thus, the dimensions of the AMCBFM matrix are only 1024× 1024.The current criteria of AMCBFM are set as  = 1×10 −1 ,  = 1×10 −2 ,  = 1×10 −3 ,  = 1×10 −4 , and  = 1×10 −5 .From Table 2, it is readily seen that the order of the SCBFs can be adaptively halted in the AMCBFM.Furthermore, the current criterion  = 1 × 10 −1 is sufficient for retaining the accuracy of the induced current.By comparing the CPU times between the AMCBFM and the MoM, it is easily determined that the AMCBFM with  = 1×10 −1 costs less than half the CPU time of the MoM.

Conclusions
In this paper, the AMCBFM was used to simulate the EM scattering from a Gaussian rough surface.Good agreement between the AMCBFM and the MoM was found.In addition, the AMCBFM costs less than half the CPU time of the MoM.Furthermore, compared with the traditional CBFM, the lower order of the CBFs of the AMCBFM is sufficient for guaranteeing the accuracy.Note that the scattering model is only a 2D model.In the future, this method will be expanded to solve the more realistic 3D scattering problem.
In addition, this method can also be applied to calculate the dielectric rough surface with a cylinder located above it.Finally, the AMCBFM can be combined with the compressing sensing technology in order to solve the EM scattering over a frequency band.

Figure 1 :
Figure 1: Geometric model of the EM scattering from a PEC rough surface.

Figure 2 :Figure 3 :
Figure 2: The bistatic scattering coefficients obtained by the different methods.(a) H polarization and (b) V polarization.

Table 1 :
The CPU time and the percentage error of the AMCBFM and the CBFM.

Table 2 :
The results of the AMCBFM with different error thresholds.