Using CGM and FDFD Techniques to Investigate the Radar Detection

Using CGM and FDFD Techniques to Investigate the Radar Detection of TwoDimensional Airplanes in Random Media for Beam Wave Incidence Hosam El-Ocla1,2 and Mohamed Al Sharkawy1,3 1

2

Department of Computer Science Lakehead University Thunder Bay, Canada

Department of Computer Engineering University of Bahrain Bahrain E-mail: [email protected]

Department of Electronics and Communications Arab Academy for Science, Technology, and Maritime Transport Alexandria, Egypt E-mail: [email protected] 3

Abstract A previous study proved that the radar cross sections (RCS) of targets differ greatly with the illumination region curvature, assuming a plane wave incidence. In this work, we present numerical results to investigate the effect of target configuration on the backscattering enhancement in the laser radar cross section (LRCS), aiming to enhance the efficiency of the radar detection of objects in a random medium. Two different techniques – the Current Generator Method (CGM) and the Finite-Difference Frequency-Domain (FDFD) Method – are used to illustrate and verify the calculated results. The target configuration includes the illumination region and the target’s complexity with E-wave incidence. We consider concave and convex illumination regions of partially convex targets with a large size. Keywords: Beam; concave; convex; double passage; radar cross section; FDFD; random medium; scattering

S

1. Introduction

everal methods to formulate scattered waves have been presented over the years, such as in [1-5]. As a result of the wave’s propagation and scattering in random media, a backscattering enhancement phenomenon is produced [6-8]. This phenomenon results in the double-passage effect, where the radar cross section (RCS) in the random medium is twice that in free space, and sometimes more than twice, depending on the random medium’s inhomogeneities [8, 9]. In past years, this effect was studied using a method that solved the scattering problem as a boundary-value problem for targets with IEEE Antennas and Propagation Magazine, Vol. 56, No. 5, October 2014

finite size [10]. It has been shown that the target’s configuration, the random media, and the polarization are the parameters that affect the radar cross section for plane-wave incidence [11, 12]. Our results, as in [11], were in excellent agreement with those obtained for a circular cylinder in free space in [13]. Specifically, assuming a plane-wave incidence, the radar cross section, and accordingly the enhancement in the radar cross section, were greatly different depending on the illumination region and the target’s complexity. Target detection would therefore not be accurate when scattering data depend on the target’s configuration. ISSN 1045-9243/2012/$26 ©2014 IEEE

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The nature of the incident waves is expected to have implications on the scattered waves. In this regard, the laser radar cross section (LRCS) [14] was calculated for typical convex targets, such as cones and spheres [15, 16]. Other studies were carried out for targets with convex illumination regions of partially convex cross section [17]. Turbulence regions in the atmosphere cause plenty of aircraft accidents, due to the spectral density of the refractive-index variations [18, 19]. This concerns those who fly in civil and military aircraft, particularly when we consider the misleading radar detection data that is collected as a result of turbulence vortex effects [20, 21]. In applications such as a radar network where an aircraft may be sensed from different locations, the laser radar cross section should be independent of both the incidence angle and the object’s complexity in a random medium such as turbulence.

random medium molecules. Based on this assumption, depolarization could be ignored. The time factor exp ( iωt ) is assumed and suppressed in the following CGM section.

2. Scattering Problem Using CGM The geometry of the problem is shown in Figure 1. A random medium is assumed as a sphere of radius L around a target of mean size a  L . The target is also to be described by the dielectric constant ε ( r ) , the magnetic permeability µ , and

the electric conductivity ν . For simplicity ε ( r ) is expressed as

ε= (1) In this paper, we probe the effect of target configuration ( r ) ε 0 1 + δε ( r ) , on the backscattering enhancement in the laser radar cross section. It is intended to propose an object detection technique where ε 0 is assumed to be constant and equal to the free-space that eliminates, or at least reduces, this effect on the laser radar permittivity, and δε ( r ) is a random function with cross section, to improve radar imaging in the sense of being independent of target aspects, to have a more accurate laser δε ( r ) = 0 , radar cross section. In doing this, we use a beam-wave incidence with different sizes, and examine its effects on the back(2) scattering enhancement (BE), assuming different illumination δε ( r ) δε ( r ′ ) = B ( r,r ′ ) , regions of partially convex objects of various complexities. In our previous work [11, 12], we mainly focused on the plane B ( r ,r ′ )  1 , wave as an ideal incident wave in the far field. In this work, we aim to investigate the effect on the backscattering enhancement (3) of having a beam wave incident with finite size, and to compare kl ( r )  1 . this with the case of plane-wave incidence, considering various configuration parameters of the object. Two different techniques Here, the angular brackets denote the ensemble average, and were used to investigate this problem: the Current Generator B ( r, r ′ ) and l ( r ) are the local intensity and local scale size of Method (CGM) [2, 10], and the Finite-Difference Frequencythe random-medium fluctuation, respectively, and Domain (FDFD) Method [22, 23]. The CGM is an exact method that calculates the scattered waves from the whole surface= of k ω= (ε 0 µ0 ) 2π λ is the wavenumber in free space. µ the object, including the shadow region. It produces results that agreed with the free-space data in [13]; however, it needs quite a long computation time for objects of large sizes. As a result, we used the FDFD method as an approximate method that required less time. The FDFD method is considered to be a flexible technique that can be efficiently used to define composite structures. Moreover, if it is to be compared with the Finite-Difference Time-Domain (FDTD) technique, the FDFD method does not require a lot of attention to the time parameters needed by the time-domain waveform source. Furthermore, the FDFD method can easily handle both non-homogenous and dispersive media. The targets were of large sizes, up to five wavelengths in free space, to accommodate the radar detection requirements of large targets such as aircraft. In the meantime, the target size was sufficiently larger than the beamwidth, and this was different from the case of a plane-wave incidence, where waves always cover the target. This scattering problem was solved two-dimensionally while considering horizontal polarization (E-wave incidence). Scalar waves were postulated based on the conditions of an isotropic medium with slight fluctuations in intensity strength and a fairly realistic large local scale size of 92

Figure 1. The description of the problem of wave scattering from a conducting cylinder in a random medium. IEEE Antennas and Propagation Magazine, Vol. 56, No. 5, October 2014

and ν are also assumed to be constants: µ = µ0 , ν = 0 . For a practical turbulent atmosphere, the condition in Equation (3) may be satisfied. This implies that the medium particles would have a great size and smooth surface without sharp edges when kl  1 , so the waves won’t scatter producing vector components. Having a weak fluctuation intensity, that is B  1 , would also remove or reduce the amount of re-incident waves that may transverse in the backward direction. We can therefore assume the forward-scattering approximation and the scalar approximation [24]. Consider the case where a directly incident beam wave is produced by a line source f ( r ′ ) along the y axis. The beam wave incidence maintains a specific width around the target. The line source is located at rt beyond the turbulence in the far field. An electromagnetic wave radiated from the field source propagates in the random medium, illuminating a conducting target, and it induces a current on its surface. A scattered wave from the target is produced by the surface current, and propagates back to the observation point, which coincides with the source point. Here, let us designate the incident wave by uin ( r ) , the scattered wave by us ( r ) , and the total wave by = u ( r ) uin ( r ) + us ( r ) . The target is assumed to be a conducting cylinder with boundary contour described by [25] r= a 1 − δ cos 3 (θ − φ )  ,

(4)

where φ is the rotation index, δ is the concavity index, and ( r ,θ ) are the cylindrical polar coordinates. We can deal with this scattering problem two dimensionally under the condition of Equation (3), as will be explained shortly; we therefore represent r as r = ( x, z ) . Assuming a horizontal polarization of the incident beam, we can impose the Dirichlet boundary condition for the wave field u ( r ) on the cylinder’s surface, S. That is,

us ( r ) = ∫ J E ( r2 ) G ( r r2 ) dr2 . S

This can be represented by us ( r ) = ∫ dr1 ∫ dr2 G ( r r2 ) YE ( r2 r1 ) uin ( r1 rt )  . (8) S

in the random medium, G ( r r ′ ) , we can express the surface current wave as J E ( r2 ) = ∫ YE ( r2 r1 ) uin ( r1 rt ) dr1 ,

(5)

S

us ( r )

2

= ∫ dr01 ∫ dr02 ∫ dr1′ ∫ dr2′ S

S

YE ( r01 r1′ ) YE∗ ( r02

S

S

  kx′ 2    kx′ 2  r2′ ) exp  −  1   exp  −  2     kW     kW  

G ( r r01 ) G ( r r02 ) G∗ ( r r1′ ) G∗ ( r r2′ ) In our representation of

us ( r )

2

(9)

, we use an approximate

solution for the fourth moment of the Green’s function in the random medium, M 22 , given as [10, 11, 26] M 22 = G ( r r1′ ) G ( r r01 ) G∗ ( r r2′ ) G∗ ( r r02 ) = G ( r r1′ ) G ( r r2′ ) G ( r r01 ) G ( r r02 ) ∗

(10)

∗

+ G ( r r1′ ) G∗ ( r r02 ) G ( r r01 ) G∗ ( r r2′ ) . In wave propagation through a continuous random medium, we may assume that the Green’s function has approximately a complex Gaussian probability distribution in an isotropic random medium [10, 27]. Here, we obtain an analytical form for the second moment of the Green’s function, M11 , as G ( r r01 ) G∗ ( r r02 ) = G ( ρ , z ρ01 , z01 ) G∗ ( ρ , z ρ02 , z02 )

where rt represents the source-point location, and it is assumed

= M11 ( ρ , z ρ01 , ρ02 , z01 , z02 ) . (11)

that rt = ( 0, z ) . We consider uin ( r1 rt ) to be represented as   kx 2  uin ( r1 rt ) G ( r1 rt ) exp  −  1   , =   kW  

S

Here, YE is the operator that transforms incident waves into surface currents on S, and depends only on the scattering body. The current generator can be expressed in terms of wave functions that satisfy the Helmholtz equation and the radiation condition, as formulated in detail in [10-12]. The average intensity of the backscattered wave for light beam incidence is therefore given by

u ( r ) = 0 , where u ( r ) represents E y [24].

Using the current generator YE and the Green’s function

(7)

We solve M11 using the Helmholtz equation as [12] (6)

where W is the beamwidth. The beam expression [24] is useful only as an approximation, because its validity is limited since it defines the incident wave only around the cylinder. The scattered wave is then given by

IEEE Antennas and Propagation Magazine, Vol. 56, No. 5, October 2014

in M11 ( ρ , z ρ01 , ρ02 , z01 , z02 ) = M11 ( 0, ρ , z ) m ( 0, ρ0d )

(12) where in M11 ( 0, ρ , z ) = G0 ( ρ , z ρ01 , z01 ) G0∗ ( ρ , z ρ02 , z02 ) . (13)

93

Here, ρ= ρ01 − ρ02 . For a two-dimensional problem, 0d m ( ρ d , ρ0d ) can be expressed as  2  k m ( 0, ρ= 0 d ) exp −  4  z − z0

D= t ( ρ , z z0 )

∫ 0

z

 z ′ − z0

∫ Dt  z − z0

z0







 

ρ0d , z ′ z0  dz ′ , (14)

z′   D  ρ , z − , z ′  dz ′ , 2  

(15)

z′  z′  z′      D  ρ , z − , z′=  2  B  0, z − , z ′  − B  ρ , z − , z ′   , (16) 2  2  2      where D is called the structure function of the turbulence [10]. Equation (3) establishes a condition and assumes that the randomness intensity, B, is low enough such that the medium has a fairly small number of particles, and that leads to having large separations, ρ , among particles. In [28], it was proven that D agrees better with the two-dimensional isotropic relation for larger ρ among particles than for smaller ρ . It was concluded that a random medium can be considered as a two-dimensional turbulence in the enstrophy inertial range. This was derived and compared with calculations based on wind data from 5754 airplane flights. As a result, three-dimensional problems can be analyzed two-dimensionally under the condition of Equation (3) in the absence of vortex stretching the nonlinear inertial force in the direction of the y axis of the cylinder that is aligned with the line source.

3. Two-Dimensional FDFD Formulation The FDFD solution provided in this paper follows the analysis in [23]. However, in the present formulation, only the E y component is used in the resulting matrix equation, instead of using E y , H x , and H z , as presented in [29]. This allows for additional memory reduction, and hence the possibility of analyzing much larger problems. Starting from Maxwell’s equations for the total electric and magnetic fields,

E iy = E0

(

E ys +

(

94

(19)

 ∂H xs 1 ∂H zs  ε 0 − = − 1 E iy . (20)   ε yi  ∂z jωε yx ∂x  

1 jωε yz

The magnetic-field components H x and H z can be expressed in terms of the y component of the scattered electric field as = H xs

∂E ys

1 jωµ xz

µ  +  0 − 1 H xi , ∂z  µ xi  (21)

∂E ys

µ  1 H zs = − +  0 − 1 H zi . jωµ zx ∂x  µ zi  In Equations (20) and (21), the permittivity and permeability parameters are indexed in such a way that these equations will be used for the perfectly matched layer (PML) that will be used to truncate the computational domain, and the non-perfectly-matched-layer regions as well, as defined in [30]. Following the same procedure as in [30], Equations (20) and (21) can be reduced to one equation using the central finite-difference approximation, which can be written in terms of the y component of the scattered electric field as

+ d (i ,k ) E ys ( i ,k +1) + e(i ,k ) E ys ( i +1,k ) = f( i ,k ) , (22)

a=

)

(18) ∇× H + H

,

optical wavenumber, θ i is the incident angle with respect to the x axis of the global coordinate system, and r0 is the beam radius at z = 0 . Having defined the incident beam, one can write the y component of the scattered electric field in the form

∇ × E i + E s = − jωµ H i + H s ,

s

 

2

where the coefficients a, b, c, d, and e are defined as

and then by separating the total field into incident and scattered field components, we obtain

i

) 

where E0 is the magnitude of the incident electric field, k is the

(17)

∇ × H tot = + jωε E tot ,

)

(

 k x sin θ i + z cos θ i −  kr0 e 

a(i ,k ) E ys ( i −1,k ) + b(i ,k ) E ys ( i ,k −1) + c(i ,k ) E ys ( i ,k )

∇ × E tot = − jωµ H tot ,

(

The superscripts i and s are used to denote the incident and scattered fields, where the incident field is the field that would exist in the computational domain with no scatterers. In this paper, the FDFD formulation for a two-dimensional case is given, and the method presented is applied to problems that belong to this case, without loss of generality. The incident beam is given by

) = + jωε ( E

i

+E

s

).

b=

1 2

2

2

2

( ∆x ) ω ε yz (i,k ) µ

, 1 zx (i − , k ) 2

1

( ∆z ) ω ε yx(i,k ) µ

, 1 xz (i , k − ) 2


where

c =1 − a − b − d − e , d=

e=

1

( ∆z )2 ω 2ε yx(i,k ) µ

1 xz (i , k + ) 2

= γ (z)

1 2

( ∆x ) ω ε yz (i,k ) µ 2

µ=

,

,

,

(27)

2 z ( 3 − n )( 2 − n )(1 − n )  L  +

3− n

−

n z   1− n  L 

2

n z 1 n ,  − 2− n  L  3 3− n

(28)

where r1 = ( ρ , 0 ) , r2 =

( − ρ , 0 ) , r0 = ( 0, 0 ) , and rt = ( 0, z ) . B ( r, r ′ ) = B0 , and L is a rough size of the

Usually, random medium models are described by media of effective or mean permittivity with random fluctuations that are generated from a prescribed correlation function. Based on this, one can express the dielectric constant based on Equation (1), with a correlation function B ( r r ′ ) implemented in this work as  x −x 2+ z −z 2 ′ ∆n1 exp  − 1 2 2 1 2 B (r r ) =  ς 

2

 ,  

(23)

where ζ is the correlation length. According to Equation (1), the random medium can be defined through the relationship between the refractive index, n0 , and the relative permittivity (i.e., n0 = ε r µr ), where the refractive index, n0 , consists of N random scatterers embedded in the medium. Since the FDFD technique is used to model the proposed configuration, one can express the spatial distribution function of the refractive index in terms of the correlation function as follows: N

n ( r= ) n0 + ∑ B ( r, r ′ ) .

(24)

i =1

4. Numerical Results Although some of the incident-wave rays become sufficiently incoherent in the propagation through random medium particles, we should pay attention to the spatial coherence length (SCL) of the incident waves around the target. The degree of spatial coherence is defined as G ( r1 rt ) G ( r2 rt ) Γ ( ρ, z ) = . 2 G ( r0 rt ) ∗

(25)

According to [11, 12, 26], Γ ( ρ , z ) can be given as 2

lz 2

1 zx (i + , k ) 2

 ε0  f (= − 1 E iy . i , k )    ε yi 

Γ ( ρ, z ) = e − µγρ ,

π B0 L3

(26)


Here, we assume range of the random medium (see Figure 1). B0 and L were assumed to be constant in our calculations. The positive index n denotes the thickness of the transition layer from the random medium to free space, and n = 8 3 was assumed in Section 3 as in [10]. It should be noted that the coherence attenuation index, α , defined as k 2 B0 Ll 4 , should be greater than two as a validity condition of Equation (10) [10]. In the following calculations, we considered kB0 L = 3π . Therefore, α was 15π 2 , 44π 2 , and 59π 2 for spatial coherence lengths of 3, 5.2, and 7.5, respectively. We considered this variety of α to represent media with different normalized local scale sizes of the random medium ( kl = 20π , 58π , and 118π , shown in Figure 2). The spatial coherence length is defined as the 2k ρ at

which Γ= e −1 ≅ 0.37 . As in Equation (12), Equation (25) is formulated. Figure 2 shows a relationship between the spatial coherence length and kl . We used the spatial coherence length to represent one of the random medium effects on the laser radar cross section. On the basis of the assumption of coherence completion of the waves in the propagation of distance 2a , we define an effective illumination region (EIR), where EIBbr is that surface that is illuminated by the incident wave and restricted by both 2kW and the spatial coherence length. It is accordingly anticipated that the target configuration including δ and ka together with the spatial coherence length and kW will generally affect the effective illumination region. As a result, the enhancement factor of the laser radar cross section will be influenced in a way that will be clarified in the numerical results.

4.1 Backscattering Enhancement In this section, we describe the numerical results for a normalized laser radar cross section (NLRCS), which is defined as the ratio of the laser radar cross section in random media, σ b , to the laser radar cross section in free space, σ 0 . Highfrequency diffraction by an edge terminating a perfectly conducting curved surface is important for many practical configurations of antennas and scatterers [31]. An airplane has a circular cross section at the fuselage away from the wings, so we assumed δ = 0 as in the F117 airplane. Wings of different

95

Figure 2. The degree of spatial coherence of an incident wave about the cylinder.

Figure 3b. The normalized laser radar cross section as a function of the target size with kW = 2 for a spatial coherence length SCL = 5.2 .

Figure 3a. The normalized laser radar cross section as a function of the target size with kW = 2 for a spatial coherence length SCL = 3 .

Figure 3c. The normalized laser radar cross section as a function of the target size with kW = 2 for a spatial coherence length SCL = 7.5 .

96


airplane models were attached to the fuselage with different curvature slopes in the xz plane, resulting in a concave-toconvex cross section and, accordingly, we assumed different values of δ . In other words, when the wing curvature with the fuselage was small, close to being flat, as in the Q-170 and X-47A airplanes, δ would tend to be 0.1. δ became higher, at about 0.2, when the wing curvature with the fuselage was greater, such as with the F16 and F117 airplanes, and δ would be about 0.3 or more with the F18 and civil airplanes. The diffractions by the discontinuous points located on the surface curvature between the airplane cylinder and wings need both quite a large modal number of the current generator, YE [1012] used in Equation (9), and, as well, a large number of sampling integration points: otherwise, YE would fail to achieve convergent results. Consequently, the values for the curvature complexity δ were relaxed and assumed to be up to 0.2, as was considered in [25, 32], only to guarantee convergent results and less computing time. The numerical results obtained were generated using the current-generator method and compared with the FDFD results. From the computed data, one could notice good agreement between the two methods, with slight differences between the FDFD and CGM results. This was attributed to the complex contributions from the inflection points on the curvature between the concave and convex regions of the object. These contributions were smoothly considered with enough integration samples to achieve the convergence, using the CGM. However, these points were modeled and approximated with the FDFD method neglecting the scattering rays out of these concavity points.

dwindled dramatically to a certain level that depended on the SCL, and then kept oscillating steadily with ka . This oscillating behavior was attributed to the limitation of having a finite size of spatial coherence length around the object in the random medium [11, 17]. Reflected waves located within the spatial coherence length were correlated, resulting in in-phase accumulation, while those waves outside the spatial coherence length were uncorrelated, leading to their being out of phase. These in- and out-of-phase waves resulted in such an oscillating radar cross section. The severity of these oscillations lessened with wider spatial coherence length, since the waves acted as being more in phase. Having greater spatial coherence length also improved the behavior of the normalized laser radar cross section in the sense of being closer to two, as shown in Figure 4. Obviously, this behavior of the normalized laser radar cross section was almost the same regardless of the target complexity δ in the high-frequency band, that is, at ka ≥ 7 . As a result and at this ka ≥ kW , where kW = 2 in Figure 3, the surface of the target acted as if it was a plane surface, and that is why δ did not have much effect on the normalized laser radar cross section.

In Figure 3 and for ka  SCL , the normalized laser radar cross section equaled two, owing to the double passage effect. As ka expanded, the normalized laser radar cross section

In Figure 4, we considered the impact of the illuminationregion curvature represented in the incident angle θin , represented by φ in Figure 1. θin = 0 and θin = π correspond to concave and convex illumination regions, respectively. It is intended to refer to [17], where the illumination region was focused on the convex portion ( θin = π ) of the concave-convex target, and this is shown here in Figure 4b. It was evident that the normalized laser radar cross section was obviously different with the illumination region at the low ka up to a certain value, depending on the kW limit. In other words, the normalized laser radar cross section was different with θin up

Figure 4a. The normalized laser radar cross section as a function of the target size for δ = 0.1 and SCL = 3 where kW = 1.5 .

Figure 4b. The normalized laser radar cross section as a function of the target size for δ = 0.1 and SCL = 3 where kW = 2 .


97

Figure 5a. The normalized laser radar cross section as a function of the target size for δ = 0.1 and SCL = 30 where kW = 1.5 .

Figure 5b. The normalized laser radar cross section as a function of the target size for δ = 0.1 and SCL = 30 where kW = 2 .

Figure 6. The normalized radar cross section as a function of the target size with plane-wave incidence for SCL = 3 .

98


to a specific ka = 5 , 10 with kW = 1.5 , 2, respectively. These ranges of ka become greater with less randomness of the medium, where the spatial coherence length is wide enough around the target, as shown in Figure 5. With greater ka exceeding those specific values, the normalized laser radar cross section almost coincided irrespective of θin . This behavior was attributed to the effective illumination region effect, together with the stationary and inflection point contributions that differed with the concavity curvature. This was not the case with a plane-wave incidence, where the normalized radar cross section was totally different with both illumination region and complexity of the object, as in [11, 12], and as shown in Figure 6 [12]. Let us define EIR KW →∞ as the effective illumination region for a plane-wave incidence illuminating a conducting target. EIR KW →∞ = > EIR KW 2= > EIR KW 1.5 as contributions increase, particularly in the shadow region. This implies that the effective illumination region expands as more target points are involved in the contributions to the scattered waves. This leads to having more out-of-phase waves scattering depending on the directions of the scattering rays. This in turn makes the radar cross section behave differently, and obviously differently with the extension of the illumination region. As a result, we can understand that reducing the beamwidth maximizes the radar imaging capability of targets as the laser radar cross section becomes independent of the target aspects. Moreover, the normalized laser radar cross section is closer to two with smaller kW .

5. Conclusions The radar cross section and, accordingly, the backscattering enhancement in random medium such as turbulence greatly depend on the target configuration and the illumination region with plane-wave incidence. Ideally, aspects of the object with a finite size and illumination region should not have implications on the calculation of the radar cross section to have accurate radar imaging. This should be similar to the case when the object is a point target. This paper clarifies that having beam-wave incidence obviously reduces the dependence of the laser radar cross section on both the target configuration parameters and the incidence angle. Within the range of a ≥ λ , the laser radar cross section would approximately not obviously differ with the target cross section complexity, δ . This is not the case with plane-wave incidence, where the radar cross section would greatly differ with the target parameters, including size and complexity. Moreover, the backscattering enhancement is closer to two at a narrower kW , approaching the ideal case. The efficiency of target detection is therefore maximized with a smaller beamwidth, particularly in the highfrequency band. In applications such as a radar network where an airplane may be sensed from different locations, the laser radar cross section would be independent of the incidence angle using a narrow-beam-wave incidence in turbulence. This finding is valid regardless of the strength of the medium’s randomness. However, with a weak random medium, where the spatial coherence length is wider around the target, the backscattering enhancement is quite close to two. Two different IEEE Antennas and Propagation Magazine, Vol. 56, No. 5, October 2014

techniques were implemented to compute and verify the results, based on the Current Generator Method and the FiniteDifference Frequency-Domain technique. Good agreement was obtained between the two proposed techniques to solve airplane detection problems when sensed from different locations.

6. Acknowledgment This work was supported in part by both the National Science and Engineering Research Council of Canada (NSERC) under Grant 250299-02.

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