High Precise Scattering Centers Models for Cone-Shaped Targets Based on Induced Currents

Based on induced currents of cone-shaped targets, the high precise scattering center model is derived in this paper.The distribution characteristics of the induced currents and their relationships with the attributes of scattering centers are investigated in detail; the high precise scattering center model is obtained. In order to validate the scattering center model, numerical simulations of two cone-shaped targets are presented, and the accuracies of the models are validated through comparing the range profiles simulated by the models with those by a full-wave numerical method. The validation results demonstrate that this model is superior to the existing model in precisely characterizing the scattering centers induced by creeping waves.


Introduction
Scattered fields of radar targets at sufficiently high frequencies can be approximated as the sum of scattered fields from individual scattering sources, and these scattering sources are generally called scattering centers [1].The term of scattering center has been expanded for now.Scattering phenomena under cases of lower frequency, creeping waves, for example, can also be described in terms of scattering centers and these scattering centers are also significant scattering components in some conditions [2,3].The characteristics of radar returns from a target are primarily determined by the attributes of its scattering centers, such as locations and scattering amplitudes, and their dependencies on aspect angle, frequency, and polarization [4].Therefore, parametric scattering center models that concisely and precisely characterize radar returns have gained considerable attention in many radar applications, such as automatic target recognition (ATR) [5,6] and radar image interpretation [7,8].
Parametric scattering center model is the combination of a series of analytical functions that describe the changes of location and scattering amplitude of scattering centers with frequency and aspect angle of line of sight (LOS) of radar.The existing scattering center models for monostatic radar configuration include the damped exponential model [9,10], the geometrical theory of diffraction (GTD) based model [11,12], the attributed scattering center model [13][14][15], and the sliding scattering center model [16].These scattering center models are all developed for scattering problems of high frequency, such as specular reflection by large-scale smooth surface or diffraction by large-scale straight edge.The precision of these models is greatly challenged for describing the scattering problem of lower frequency, such as diffraction by small-scale edges and rescattered waves of creeping waves and travelling waves on smooth surface.And even worse, there are no generalized analytical solutions for these kinds of scattering problems.
To tackle this problem, the models of the induced currents on the target are built at first; then the scattering center model is derived by the electric field integral expression based on the current model.The modelling of the induced currents on the target is easier than the scattering center modelling, for the precise numerical results of currents can be acquired by full-wave numerical computation.The well validated fullwave numerical method, namely, the hybrid finite elementboundary integral-multilevel fast multipole algorithm (FE-BI-MLFMA) [17][18][19][20] is used in this paper.It is found that for cone-shaped target, its scattering characteristics are mainly determined by one-dimensional (1D) currents on the lines in the plane constructed by the LOS and the rotational axis of the cone.Therefore, the modelling of the induced currents on the target is greatly simplified.
Based on the built model of the 1D currents, the relationship between the characteristics of currents and the attributes of scattering centers is investigated; the formation processes of scattering centers by its contributing currents are analyzed.It is found that one parameter of current models is sometimes related to several scattering centers, and vice versa.In other words, the coupling of multiple scattering sources has been included in the current model.Moreover, the scattering phenomena of creeping waves are also included in the current model, and the corresponding scattering center can be easily acquired based on the current model.In order to validate the presented method, numerical simulations of two coneshaped targets are investigated in detail, and the accuracies of the built models are validated through comparing the range profiles (RPs) simulated by the models with the real RPs of the two targets computed by FE-BI-MLFMA, as well as the RPs simulated by the attributed scattering center model obtained by traditional method.
The remainder of the paper is organized as follows: Section 2 presents the modelling of the induced currents on a blunt-nosed cone target.Section 3 presents the derived scattering center model based on the current model.Section 4 presents the validation of the built scattering center models.Finally, Section 5 provides the conclusions of this paper.

Parametric Expressions of Scattering Centers
The induced currents on a cone-shaped target are investigated in this section.The cone-shaped target is a perfect electric conductor (PEC).The geometry of the cone and the relative position of the target and radar are shown in Figure 1.The body-fixed Cartesian coordinate system (, , ) is used to describe the geometry, its origin is at the bottom center of the target, and the rotationally symmetric axis of the cone is located along the -axis, as shown in Figure 1.The geometrical parameters of the target are set as  = 0.732 m,  = 0.061 m, ℎ = 3.721 m, and  = 10.4 ∘ .The radar parameters are set as follows: the frequency range is f = 1∼2 GHz, with sampling interval of Δ = 12.5 MHz.Suppose that the distance between the radar and the target satisfies the far-field condition ( 0 > 2 2 /, where  0 is the distance between the radar and the target and  is the size of the target).In far-field condition, the incident wave can be taken as plane wave; then the induced currents on the target illuminated by the plane waves from the monostatic radar are computed by FE-BI-MLFMA.The characteristics of currents on the flank and the bottom are investigated separately in the following.

Characteristics of Induced Currents on the Flank.
As the cone is a rotationally symmetrical object, the change of the LOS in azimuth angle (denoted by ) does not affect the characteristics of induced currents.Therefore, without loss of generality, we set the change of LOS to be in the  plane, and the elevation angle of the LOS is denoted by .Then, the wave vector of incident wave can be expressed as ⃗   = (2/)(− cos ẑ − sin x).
When the incident electric field is under vertical polarization, the magnetic field can be expressed as ⃗   = −  φ = −  ŷ.The induced currents on the cone are presented in Figure 2, where the used parameters of the incident waves are  = 1 GHz,  = 8 ∘ .Figure 2(a) shows the amplitude of the induced currents and Figure 2(b)1 is the intersection line of the flank of the cone with the plane constructed by the axis of cone and the direction of the incident electric field, and 2 is the intersection line of the flank with the plane constructed by the axis of cone and the direction of the incident magnetic field.In our geometry setting, the generatrix in  plane of lit region is 1 and the generatrix in  is 2.
As shown in Figure 2(a), the maximal values of currents are on 1 and the minimal values of currents are on 2.The conclusion agrees with the analytical results based on physical optics method (PO) [21][22][23][24].The normals of the flank on 1 and 2 are n1 = cos x + sin ẑ and n2 = cos ŷ + sin ẑ.Based on PO, induced current can be approximately expressed as J = 2n × H  , so the currents on 1 and 2 can be derived as J 1 = 2  (− cos ẑ + sin x) and J 2 = 2  (sin x).The incident electric field is generally set as 1 V/m in computation; then the incident magnetic field is   = 1/, where  is the wave impedance of free space.
Optical currents on 1 computed by PO method are compared with the values computed by FE-BI-MLFMA (called full-wave currents for short), as presented in Figure 3 ( = 8 ∘ ). Figure 3 of the currents.The range of horizontal axis indicates the distance between a point on 1 and the top of the cone Figure 3(a) shows that there are significant differences between the optical currents and the full-wave currents.The full-wave currents along 1 are more complex than that of the optical currents.Near the two ends of 1, the amplitude of the full-wave currents is obviously larger than that of the optical currents.Within the internal region of 1, the amplitude fluctuates as a combination form of periodicity and linearity.The generatrix in  plane in shadow region is denoted by 1  .The induced currents on 1  are shown in Figure 4.The fluctuation feature and the values of the induced currents on 1  are similar to the case of 1, which means the scattering components of creeping waves are the denominated scattering contributions when the elevation angle of the LOS is small ( = 8 ∘ < ).The range of horizontal axis indicates the distance between a point on 1  and the top of the cone.
Based on the characteristics of the induced currents on 1 and 1  under different LOSs, the full-wave currents are described by the following expressions:  (, , ) =   (, , ) exp (  (, , )) (1) where (1) describes the complex value of induced current; its amplitude and phase are indicated by   and   .Equation ( 2) is proposed according to the amplitude characteristics of the induced currents on 1 and 1  under different LOSs.Equation ( 3) is proposed according to the phase characteristics of the induced currents on 1 and 1  . indicates the position of the current element on 1 ( = 1) or 1  ( = 2).  (⋅) and   (⋅) are the amplitude and the phase of the currents, respectively.  ( = 1, 2, . . ., 9) indicates the parameters to be estimated.  is irrelevant to frequency, but  2 ,  3 ,  4 ,  5 ,  8 , and  9 are linear-circular correlated with .
To show characteristics of the currents on the whole conical surface, we slice the cone along line 1 and unfold the flank to a fan surface, as shown in Figure 5.The currents on the fan and the bottom are shown in Figure 6.The halfangle of the fan is  = /, where  is the outer radius of the fan.
Based on PO method, the two-dimensional currents can be expressed as J(, ) = 2n() × H  = 2  (sin x − cos  cos((/))ẑ), where (, ) is the coordinates of the individual current on the fan.The component of J(, ) along vertical direction (V = − cos ẑ + sin x) can be expressed as (4).Therefore, the change of currents with  can be described by the currents on 1 and the dependant function ().
Equations ( 4) and ( 5) describe the fact that the change of induced currents with  can be expressed by the currents on 1 and the dependent function (⋅).

Characteristics of Induced
Currents on the Bottom.Similarly, the induced currents on the bottom are investigated here.The amplitudes and phases of induced currents on the bottom are presented in Figures 7(a The currents on 1 are presented in Figure 8.The range of horizontal axis indicates the distance between a point on 1 and the point on the curved edge.From Figure 8(a), it can be seen that the amplitude of full-wave currents fluctuates as a combination form of periodicity and linearity.From  These characteristics of the currents on the bottom are similar to the case of currents on the flank.Therefore, the currents on the bottom are also described by the expression of (1), only with different parameters.The currents on 1 and 1  are indicated by the case of  = 3.Similarly, the twodimensional currents on the bottom are also expressed by (4).

Estimation of the Current Model Parameters.
The coefficients in the presented formula,  1 ∼ 9 , are estimated through matching the currents computed by the model with the currents computed by FE-BI-MLFMA.The used method for optimal estimation is genetic algorithm (GA) [25][26][27][28].The currents on the flank computed by the model with estimated parameters (the simulated currents for short) are presented in Figures 9(a The errors between the simulated currents and the real currents are examined: the correlation coefficient of the amplitudes is 92.81% and the root mean square error (RMSE) is 6.01 × 10 −5 V/m 2 ; the correlation coefficient of the unwrapped phases is 98.72% and the RMSE is 2.99 rad.
Other cases of incident angle ( = 1 ∘ ∼10 ∘ ) are also investigated.The parameters of currents are estimated with the same method.The errors between the simulated currents and the real currents are the correlation coefficient of the amplitudes is larger than 90% and the RMSE is less than 10 −4 V/m 2 ; the correlation coefficient of the unwrapped phases is larger than 95% and the RMSE is less than 4 rad.

Approach of Scattering Center Modelling
There are two classical methods for scattering center modelling.Scattering centers can be extracted directly from radar images by peak or pattern detection, or they can be described by parametric models and the unknown parameters of the model then be acquired through optimal estimation.The second method is not limited by radar resolutions and could provide more precise and elaborate descriptions of scattering centers, yet it is under the strict precondition of the accuracy and application scope of parametric model.The commonly used parametric models are proposed according to the solutions of several simple targets based on the physical optics method (PO) or geometrical theory of diffraction (GTD).Due to limitations of these asymptotic methods, the existing parametric models fail in some cases, for example, creep waves.In this paper, the parametric model is derived from the induced currents which are computed by a full-wave numerical method.All the scattering contributes are included in the derived model.
Based on the electric field integral expression under the far-field condition in the free space [29], the scattered field can be expressed as where  is wave impedance of free space,  V is the component of the vector N along V direction.
where r = sin  cos x + sin  sin ŷ + cos ẑ and r  =  sin  cos((/))x +  sin  sin((/))ŷ + (ℎ −  cos ẑ).Substituting ( 4) into (7), we can derive the results of the currents in lit region ( = 0∼).The integral part related to  can be derived based on the principle of stationary phase [30], as follows: Then, the scattered field can be expressed as where  = √2/ sin  sin  and r  is the location vectors of currents on 1.
From (10) we can see that the characteristics of the scattering fields from the flank in lit region are mainly determined by the currents on 1.Similarly, the characteristics of the scattering fields from the flank in shadow region are mainly determined by the currents on 1  , and the characteristics of the scattering fields from the bottom are mainly determined by the currents on 1 and 1  .The total fields are the supposition of the three components, as given below.
where  1 indicates the electric field induced by the currents in lit region;  2 indicates the electric field induced by the currents in shadow region;  3 indicates the electric field induced by the currents on the bottom.Based on (1) and (10), we can derive the analytic expression of the scattering center model at VV polarization.VV means the incident wave and reflected wave are all in vertical polarization.The analysis of the scattering centers corresponding to the currents on 1 is provided.The analysis processes are similar for the cases of 1  , 1, and 1  , so the repeated contents are ignored here.
Substituting the expressions of currents on 1, 1  , 1, and 1  into (10), we can derive the total scattering fields, as given by (A.1) in the Appendix.Then through the regular computation of the integral along  in (A.1), we can get 5 exponential terms as shown in (12); the five functions are the five corresponding parametric models of scattering centers contributed by the currents on 1.According to the numerical results of the parameters, the components, SC 1 , SC 3 , and SC 4 , whose amplitudes are quite small compared with the others, can be ignored in the following analysis.The components, SC 2 and SC 5 , include the contributions of four scattering centers.The four dominated scattering centers, denoted by SC 1 , SC 2 , SC 3 , and SC 4 , are illustrated in Figure 11.The relationship of the parameters of the model with the amplitudes and locations of scattering centers is listed in Table 1.
SC 1 locates at the top of the cone, which is mainly contributed by the components SC 2 of 1 and 1  ; SC 2 locates at edges of the bottom in lit region, which is mainly contributed by the components C 5 of 1 and 1; SC 3 locates at edges of the bottom in shadow region, which is mainly contributed by the components SC 5 of 1  and 1  ; SC 4 locates outside the geometry of the cone, which is mainly contributed by the components SC 2 of 1 and 1  , that is, the creeping waves in the shadow region.

Validation of the Model
According to the principle of RP, the intensity and position of the peak reflect to a certain extent the scattering amplitude and downrange location of the corresponding scattering center.The accuracy of the derived scattering center model is validated through comparing the RP simulated by the model (the simulated RP for short) and the data simulated by the scattered fields computed by FE-BI-MLFMA (the real RP for short).Two cone-shaped targets are investigated here.The geometry of target I is the same as the case of Figure 1.The radiuses of bottom disks and spherical tops of target II are the same as target I.The height of target II is ℎ = 5.4 m.Two cases with different parameters setting are investigated.
Case 1 ( = 0,  = 0,  = 1∼2 GH, △ = 0.125 GHz).The simulated RP and the real RP are presented in Figure 12(a).For demonstration of the superiority of this model, the RP obtained by the attributed scattering center model through traditional method are also compared with the simulated RP, as presented in Figure 12(b).The errors of amplitude and locations of the scattering centers are listed in Table 2.
The comparison results demonstrate that the simulated RPs by this model agree well with the real RP.Superior to the exiting model, the scattering center SC 4 induced by creeping waves is precisely characterized by this model.

Figure 1 :
Figure 1: The geometry of the cone-shaped target and the radar.

Figure 3 (
Figure 3(b) shows that the two results agree well with each other, and the phase of induced current linearly changes along 1.The generatrix in  plane in shadow region is denoted by 1  .The induced currents on 1  are shown in Figure4.The fluctuation feature and the values of the induced currents on 1  are similar to the case of 1, which means the scattering components of creeping waves are the denominated scattering contributions when the elevation angle of the LOS is

Figure 4 :
Figure 4: Amplitude and phase of the induced currents on 1  in shadow region.(a) Amplitude; (b) unwrapped phase.
) and 7(b), respectively.The parameters of the incident waves are the same as those of Figure2.The line in  plane along + axis is denoted by 1, the line in  plane along − axis is denoted by 1  , and the line in  plane along + axis is denoted by 2.The maximal values of currents are on 1 and the minimal values of currents are on 2.The scattering field from the bottom is mainly contributed by the currents on 1 and 1  .
, it can be seen that the phase of induced current changes linearly along 1.The characteristics of the currents on 1  are similar to 1.
) and 10(a).For comparison, the currents computed by FE-BI-MLFMA (the real currents for short) are presented in Figures9(b) and 10(b).The used parameters in computation are f = 1∼2 GH and  = 0.125 GHz,  = 8 ∘ .The range of horizontal axis indicates the distance between a point on 1 and the top of the cone.

Figure 11 :Figure 12 :
Figure 11: Positions of four dominated scattering centers on the target.

Table 2 :
Errors of amplitude and locations of the scattering centers.