This notion gives rise to a new theory of radar polarimetry.1 .... For our discussion, monostatic radars have the transmitter and receiver located ..... configuration with a bistatic angle equal to zero and with the bisecting line equal ..... is extended from a triangular surface, the monostatic target diagram, to a tetrahedral volume.
14 Bistatic Radar Polarimetry Theory
Anne-Laure Germond, Eric Pottier, Joseph Saillard CONTENTS 14.1 Introduction 14.2 Polarimetry Background 14.3 Radar Wave Polarization Theory 14.4 Radar Target Polarimetry 14.5 The Polarization Fork 14.6 The Euler Parameters 14.7 Monostatic and Bistatic Polarization Conclusions References
14.1
INTRODUCTION
The history of radar (radio detection and ranging) begins in the 1920s with the discovery that metallic objects reflect radio waves. The conventional radar system may measure the amplitude, frequency, and differential phase of the received wave for comparison with the transmitted wave to recover target information. However, the detection and the identification of radar targets will be more difficult with this type of radar because of the increasingly hostile targets environment. To overcome the effects of target environments, the use of wave polarization data can enhance target detection. This notion gives rise to a new theory of radar polarimetry.1 Since the 1950s, scientists such as G. Sinclair have been interested in the way that targets depolarize the transmitted electromagnetic waves reflected back to the transmitter. The Sinclair matrix models the modification of the transmitted polarization state compared to the scattered polarization state.2 Backscattering refers to a scattered wave directed toward the source. If the transmitter and receiver are located at the same place, the radar configuration is called monostatic. When the transmitter and the receiver are widely separated, the configuration is called bistatic. The monostatic polarimetric radar is a particular bistatic case where the bistatic angle tends to zero.3 The requirement to detect small radar cross section (RCS) or stealthy targets raises a need for bistatic radar systems. Applying polarimetric techniques to target detection and classification means that we must now examine the physics of bistatic radar polarimetry.
14.2 14.2.1
POLARIMETRY BACKGROUND HISTORY
First of all, only amplitude and frequency information of the electromagnetic wave were measured in the 1920s. Some current radar systems also allow the measurement of the relative phase, which helps to resolve the physical features of scatters and targets. N. Wiener discovered some important
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properties of polarized waves during the period 1927 to 1929. R. C. Jones and H. Mueller developed these discoveries. Later, G. Sinclair showed that the polarization state of a wave scattered by radar target is different from the transmitted wave. He expressed the change, which is related to the properties of a coherent radar target, by the 2 × 2 scattering matrix, commonly called the Sinclair matrix. Then, E. Kennaugh4 introduced a new approach to the radar theory, which was based on the studies of G. A. Deschamps,5 and developed the optimal target polarization concept for the reciprocal monostatic relative phase case. The meteorological radar is a direct application of this study. At the beginning of the 1970s, J. R. Huynen6 made a phenomenological study of the target in the particular case of the backscattering. Huynen’s target measurements on the relative phase backscattering matrix sparked a new interest in polarimetric radar developments. He published his phenomenological theory of radar targets in which he defines nine independently modeled physical parameters for the monostatic configuration case. These Huynen parameters bring out geometrical properties and physical information about the structure of the target, and they can help in target identification. These parameters are linked together by four monostatic target equations. A second polarimetric tool is the polarization fork or Kennaugh fork, which is defined with the characteristic polarization states. This increases the target information in a monostatic radar configuration. Furthermore, the location of the monostatic characteristic points can be determined by the five Euler parameters, which are extracted from the backscattering matrix. All of these previous monostatic theoretical results7 have been redeveloped in this chapter according to a bistatic configuration. First, Polish researcher Z. H. Czyz8–10 proposed a theory of the bistatic radar polarimetry with a geometrical approach of the problem, whereas the approach we developed follows an analytical approach. Currently, M. Davidovitz and W. M. Boerner have proposed a decomposition of any matrix into the sum of symmetric and skew-symmetric matrices.11 The extension of the monostatic theory to the bistatic one is based on this decomposition. W. M. Boerner is well known throughout the radar polarimetry community for his research in the area of the direct and inverse vector electromagnetic scattering problems.12,13 His theoretical contributions have encouraged a new international community of engineers and scientists. He promotes the advantages of the radar polarimetry around the world and presents regularly very encouraging results gained from using polarization vector measurements. The expansion of the polarimetry theory to the bistatic case leads to the definition of seven new bistatic parameters and the derivation of nine polarimetric bistatic target equations.14 Additionally, we propose the concept of a bistatic target diagram. The 14 characteristic polarization states of the new bistatic polarization fork and the 7 bistatic Euler parameters are also presented.15 The results concerning the particular case of the monostatic theory are recalled after the presentation of those of the bistatic theory.
14.2.2
WHY POLARIZATION IS IMPORTANT
Radar polarimetry theory starts in the 1950s, when G. Sinclair demonstrated that a target changes the polarization state of a transmitted wave. More information about target can proceed from the knowledge of the scattered wave polarization state for a given transmitted wave. During the past 15 years, basic research studies on the fundamentals of coherent and partially coherent radar polarimetry were carried out with applications to target detection in clutter, target and background clutter classification, and target imaging and identification. To understand the importance of bistatic radar polarization, consider that a conventional scalar radar measures only one component of the scattered wave, which is the scattered part defined by the single receiver antenna characteristics. A polarimetric radar has two receiving antennas with orthogonal polarization states. Both components of the scattered wave are measured successively to get the radar vector. First, one © 2001 CRC Press LLC
polarization state is transmitted, and two measurements made in both the transmitter polarization and orthogonal to the transmitter polarization. Then, the orthogonal polarization state is transmitted, and another two measurements are made. Polarimetric radar has additional information in the received vector radar, which allows monostatic polarimetric measurements that bring out geometric properties and physical information about the structure of the target. Furthermore, the use of the polarimetric radar allows detection of some stealthy targets that completely depolarize the polarization state of the transmitted wave. In that case, the copolarized power, when antennas at the transmission and at the reception have the same polarization state, equals zero, but the crosspolarized power is then maximum.
14.2.3
MONOSTATIC
AND
BISTATIC RADAR
Whereas many radar systems utilize the same site for transmission and reception, this is not the only configuration that can be employed. A system with widely separated antennas used for transmission and reception is called bistatic radar. In the bistatic case, the radiation source and the receiver are at different locations. The target cross section is not only a function of its orientation and frequency, it is also function of the bistatic angle described by the location of the target toward the transmitter and the receiver.16,17 Bistatic radar appeared at the beginning of the 1930s.18–20 Because of the difficulty of the integrating transmitters and receiver with the same antenna, the first radars had bistatic configurations. When duplexer technology appeared in 1936, the monostatic system took the place of the bistatic system radar for practical reasons.21 For our discussion, monostatic radars have the transmitter and receiver located at the same place. Bistatic radars have large distances between the transmitter and the receiver relative to the transmitter-target and target-receiver ranges.22 The geometry of a monostatic configuration and bistatic configurations are illustrated in Figures 14.1 and 14.2. The bistatic radar system has some advantages, because the wide separation of the receiver and transmitter eliminates any coupling. Target Transmitted signal Backscattered signal Transmitter Receiver
FIGURE 14.1 Monostatic radar systems have the transmitter and receiver in essentially the same place. Target
Transmitted signal
β
Scattered signal Receiver
Transmitter
FIGURE 14.2
Bistatic radar systems have a large angle β between the transmitter and receiver.
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14.3 14.3.1
RADAR WAVE POLARIZATION THEORY MONOCHROMATIC ELECTROMAGNETIC WAVES
The solution of the Maxwell equations shows that the electrical field vector E of a monochromatic plane electromagnetic wave is normal to the direction of propagation k , and the magnetic field vector H is such that the trihedral (E, H, k) is direct. So, for an electromagnetic plane wave, the H magnetic vector is proportional to the electrical wave E. The magnetic field vector gives no more information. The electrical field is only a function of the z and t, the propagation equation is 2
2 ∂ E ∂ E --------2- – ε 0 µ 0 --------2- = 0 ∂t ∂z
(14.1)
The solution of the previous equation is E ( z,t ) = E 1 t – -z- + E 2 t + -z- c c
(14.2)
with the first term associated to a propagation following the sense of the positive z at the speed c, and the second term as a propagation in the sense of the negative z at the same speed. In the following, only the term describing the z growth is taken into account, because the studied electromagnetic wave is assumed to be progressive. The combination of two sinusoidal waves along two directions mutually orthogonal to the direction of propagation defined for the positive z, can be written E ( z, t ) = E x ( z,t )x + E y ( z,t )y
(14.3)
E ( z,t ) = E ox cos ( wt – k 0 z + δ x ) with x E y ( z,t ) = E oy cos ( wt – k 0 z + δ y ) where
f = wave frequency w = pulsation of the wave k0 = wave number in the empty space δx and δy = absolute phase of the two components of the E electrical wave
14.3.2
THE ELLIPSE
OF
POLARIZATION
The two components Ex and Ey of the wave are linked by the following relationship: Ex Ey E 2 Ex- 2 ------ cosδ + ------y- = sin 2 δ – 2 ------------- E 0x E 0y E 0x E 0y
(14.4)
with δ = δy – δx The path described in the course of the time by the projection, onto an equiphase plane, of the extremity of the electrical field vector is an ellipse of equation given by the Equation (14.4). The polarization of a monochromatic progressive plane wave is graphically represented in Figure 14.3. © 2001 CRC Press LLC
FIGURE 14.3 The polarization ellipse form moves around from the segment (linear polarizations) to the circle (circular polarizations) according to the associated polarization state.
The wave polarization state is totally described by five geometrical parameters of the polarization ellipse: • The ϕ angle represents the ellipse orientation. Its definition domain is [–π/2,π/2]. • The τ angle defines the ellipticity. Its definition domain is [–π/4,π/4]. • The polarization sense is determined by the rotation sense along the propagation axis. The polarization is right elliptical if the ellipse rotates clockwise when the evolution of the wave is the propagation sense; otherwise, the polarization is left elliptical. • The square of the magnitude A of the ellipse is proportional to the power density of the received wave at the observation point. • The α angle represents the absolute phase of the ellipse. The definition domain is [–π,π].
14.3.3
JONES VECTOR
The electrical field of a monochromatic plane wave of any polarization is defined in Equation (14.3). Because the components of the electrical wave oscillate at the same frequency when the wave is monochromatic, the temporal information can be omitted, and Equation (14.3) is simplified to
E( z) = e
2π – j ------z λ
Ex e Ey e
jδ x jδ y
jωt
,E ( z,t ) = Re { E ( z )e }
(14.5)
As the wave is planar, the electrical field E(z) is the same at any point of the wave plane. It is possible to suppress the spatial information. The study of the wave can become restricted to the plane wave corresponding to z = 0, for example. The electrical field vector becomes
E( 0) =
E 0x e E oy e
jδ x
(14.6) jδ y
The vector E(0) is called the Jones vector of the wave. The amplitude and phase of the complex components of the electrical field are totally defined by this vector. In the general case, the Jones vector is expressed in any basis (x,y) so that © 2001 CRC Press LLC
Ex
E =
Ex e
=
Ey
Ey e
jδ x
(14.7) jδ y
The geometric characteristic of the polarization ellipse can be extracted from the Jones vector. The phase difference will be δ = δy – δx
(14.8)
2 Ex Ey -2 cosδ tan2ϕ = -------------------------2 Ex – Ey
(14.9)
2 Ex Ey -2 sin δ sin 2 τ = -------------------------2 Ex + Ey
(14.10)
The ϕ orientation is
The τ ellipticity is
The sense of polarization is τ < 0: right elliptical polarization τ > 0: left ellliptical polarization
(14.11)
The magnitude of the wave is A =
2
Ex + Ey
2
(14.12)
Any polarization state, represented by its Jones vector, can be expressed in any orthogonal polarization basis so that E = Ex x + Ey y
(14.13)
The general expression of a Jones vector, associated to any elliptical polarization state, with ν the absolute phase, is given by
E ( x,y ) = Ae
14.3.4
– jv
cosϕ – sinϕ cosτ sinϕ cosϕ jsinτ
(14.14)
THE STOKES VECTOR
For each complex Jones vector, there exists a real equivalent representation, which is the Stokes vector g(E) given by g(E) = © 2001 CRC Press LLC
T*
*
2[V] (E ⊗ E )
(14.15)
where ⊗ is the Kronecker product, T means transpose and * conjugate, and [V] is defined by the following matrix: 1 1 0 V = ------2 0 1
1 0 0 –1
0 1 1 0
0 –j j 0
(14.16)
So the components of g(E) are equal in any basis (A,B) to 2
2
g1
g ( E ( A, B ) ) =
2
EA + EB
g0
2
EA – EB
=
(14.17)
*
g2
2Re ( E A E B )
g3
– 2Im ( E A E B )
*
The components g1, g2, g3 of the Stokes vector g(E(A,B)) correspond to the Cartesian coordinates of a point located at the surface of a sphere of radius equal to the value of the component g0. That is the most important interest of the Stokes vector. Moreover, when a wave is totally polarized, the real components of the Stokes vector are linked together by the following relation: 2
2
2
2
g0 = g1 + g2 + g3
(14.18)
Physically, g0 expresses the total intensity of the polarized wave, with g1 the part of the horizontally or vertically polarized wave, g2 the part of the ±45° linearly polarized wave, and g3 the part of the right or left circular polarized wave. Another variable, called the polarization ratio, defines the wave polarization state following so that E ρ E ( x,y ) = -----y Ex
14.3.5
(14.19)
THE POINCARÉ SPHERE
The Stokes vector allows the representation of any polarization state of a totally polarized wave at the surface of a sphere, called the Poincaré sphere, shown in Figure 14.4. So, each polarization state is uniquely represented by a point at the surface of the Poincaré sphere. The last three normalized components of the Stokes vector g(EP) can be expressed in two different ways with q cos 2ϕ cos 2τ p u p = sin 2ϕ cos 2τ sin 2τ vp where ϕ, τ = the spherical angles γ, δ = the Deschamps parameters © 2001 CRC Press LLC
cos 2γ = sin 2γ cos 2δ sin 2γ sin 2 δ
(14.20)
v 2ϕ 2τ
O
P
2γ 2δ
u
q FIGURE 14.4 the Poincaré sphere is a helpful tool for visualizing the wave polarization state associated with the τ ellipticity angle and the ϕ orientation angle.
14.3.6
COMPARISON
OF
DIFFERENT POLARIZATION STATES
The hermitian product of two Jones vectors is defined as *
〈 A ( x,y ) ,B ( x,y )〉 = A ( x,y ) B ( x,y ) T
(14.21)
The orthonormal basis (eH, eV) is how the different polarization states are studied. The parallelism and orthogonality notions are only valid for waves that go on the same direction. Two electromagnetic waves are considered orthogonal if the hermitian product of the two associated Jones vectors is null, so that ϕ = ϕA + π --〈 A ( x,y ) ,B ( x,y )〉 = 0 ⇒ B 2 τ = – τ A B
(14.22)
Two electromagnetic waves are considered parallel if the two associated Jones vectors are proportional and ϕ = ϕA A ( x,y ) = kB ( x,y ) , k ∈ C ⇒ B τB = τA
(14.23)
The link between the different polarizations is represented in Figure 14.5. Different representations of the polarization of an electromagnetic wave have been shown. The Jones vector and the Stokes vector, which are associated with a polarization state, and also the Poincaré sphere were defined.
14.4
RADAR TARGET POLARIMETRY
We discussed the polarization of an electromagnetic wave in the previous section. This section describes the link between the polarization state of a wave scattered by a target and the transmitted wave. The polarization transformation of the backscattered wave can be modeled by two matrices: the scattering and the Kennaugh matrices. The objective is to extract information relative to the characteristic of the target from these matrices. © 2001 CRC Press LLC
Ε(x,y) = (A,α,ϕ,τ )
Ε ll(x,y) = (B, β, ϕ, τ )
Initial polarization
Parallel polarization
Ε(x,y) = (A, α, ϕ + π/2, − τ )
+ = (A, α, ϕ, − τ ) Ε(x,y)
Orthogonal polarization
Conjugate polarization
T
FIGURE 14.5 Link between different polarizations.
14.4.1
SCATTERING MATRIX: THE FIRST IMPORTANT CONCEPT
Definition When a target is illuminated by an electromagnetic wave, the polarization of the scattered wave is generally different from that of the transmitted wave. The nature of the depolarization depends on the geometry of the target. G. Sinclair proved that the target acts as a polarization transformer and defined this change by the 2 × 2 complex scattering matrix, which links the Jones vectors of the transmitted and the received waves together. Two orthogonally polarized signals are transmitted successively to measure the components of the scattering matrix.
d
i
E = [ S ] ( A,B ) E ,
d
EA d
EB where
i
=
S AA S AB E A
(14.24)
S BA S BB E iB
Ei = the incident Jones vector and Ed the diffused Jones vector
The polarization state A is transmitted first. Two receiver measurements are made that include the copolarized component SAA and the crosspolarized component SBA. Then, the orthogonal polarization state B is transmitted, and two new measurements are obtained. These measurements fill the complex scattering matrix. Describing the target behavior means that we must know the scattering matrix. However, this matrix depends on the position of the target, but also on the frequency of the transmission, which makes life interesting. © 2001 CRC Press LLC
The two bases at the transmitter and at the receiver are linked by the local basis of the target. Two conventions exist in the literature, the BSA convention and the FSA convention, and they define the link between the local basis of the target and the reception basis differently.23 The BSA Convention BSA is the abbreviation for backscatter alignment. The sense of propagation of the scattered wave is opposite to the sense of the propagation vector defining the direct reception basis, as shown in Figure 14.6. For the particular case of the backscattering, the bases are identical. For the particular case of the forward scattering, the ki, kd, vi, and vd vectors belong to the same incident plane, and hi and hd are of same direction but of opposite sense. The FSA Convention FSA is the abbreviation for forward scatter alignment. The sense of propagation of the scattered wave and the sense of the propagation vector, which defines the basis at the reception, are identical. The transmission and the reception bases linked by the FSA convention are visualized in Figure 14.7. For the particular case of the backscattering, the vectors ki and kd on one hand, and hi and hd on the other hand, are of same direction but of opposite sense, whereas vi and vd vectors are equal. For the particular case of the forward scattering, the vectors ki, kd, vi and vd belong to the incident plane, and the hi and hd vectors are equal.
Z
vi
hi ki
kd
180 θi
θd
hd vd
ϕd
y
ϕi
x
FIGURE 14.6 Backscatter alignment (BSA) convention.
Z
vi
hd k d
hi ki 180
θi θ d
vd
ϕd
x
ϕi
y
FIGURE 14.7 Forward scatter alignment (FSA) convention for defining transmitter, target, and scattered waves. © 2001 CRC Press LLC
Choice of Convention In the following, all the results are expressed in the BSA convention, which is the usual convention for the radar community. Furthermore, for the particular case of the monostatic configuration, the scattering matrix is symmetric. This specificity has been used when the monostatic radar polarimetry theory has been elaborated. However, it is important to notice that the physical values such as the voltage or the intensity are invariant, whatever the convention used. The transformation of the scattering matrix from a convention to the other one expresses itself as follows:
[S]
BSA
FSA = 1 0 [S] 0 –1
(14.25)
Decomposition of Any Bistatic Scattering Matrix The complex scattering matrix of a target links the Jones vectors of the transmitted and the received waves. The main basic difference between the monostatic and the bistatic matrix is that the bistatic scattering matrix [Sbi] is no longer symmetric in the antenna coordinate system, the BSA convention. When the backscattering is symmetric, [Sbi] can be broken down into a sum of two matrices: a symmetric one and a skew-symmetric one.11 The symmetric matrix models a monostatic configuration of a target, and the skew-symmetric matrix models additional information resulting from the bistatic configuration. However, it is important to notice that the elements of the symmetric part depend on the target but also on the location of the radar system and, consequently, on the bistatic angle.
[ S bi ] ( A,B ) =
S AA S AB
[ S bi ] ( A,B ) =
S AA S AB
S BA S BB
= [ S s ] ( A,B ) + [ S ss ] ( A,B )
S
S
S AB S BB AB + S BA with S SAB = S---------------------, 2
+
0
SS
S AB
SS
– S AB 0
(14.26)
S AB – S BA SS S AB = --------------------2
Examples of Monostatic and Bistatic Scattering Matrices of Canonical Targets With the help of a special program, we can calculate the bistatic signatures of two canonical targets: a rectangular flat plate and a dihedral. The simulations are made for targets of length equal to 30 cm at the frequency of 10 GHz. The signature is studied in the basis (Ox, Oy, Oz), with O the center of the target and (Oz) the direction normal to the plane of the local surface of the target. The direction of the electromagnetic waves is identified by the angles θ and ϕ, θi and ϕi for the incident wave and θd and ϕd for the scattered wave. The Rectangular Flat Plate The signature of a rectangular flat plate is calculated such that the angles ϕi and ϕd are equal to 180°, as shown in Figure 14.8. The transmitter and the receiver are located in the (xOz) plane. The definition domains of angles are given by © 2001 CRC Press LLC
Z
θd 0
180- θi θd
o X
ϕi = ϕd=180
y FIGURE 14.8 Definition of angles for the flat plate.
θ ∈ [90°,270° ] i θ d ∈ [ – 90° ,90° ] ϕ i ∈ [ 0°,180° ] ϕ d ∈ [ 180°,360° ]
(14.27)
So, the monostatic configuration corresponds to θ d + 180° = θ i
(14.28)
Figure 14.9 specifies the locations of the backscattering mechanism, and those of the particular bistatic configurations, which correspond to the case of the bisecting line, merged into the axis (Oz). Figures 14.10 through 14.13 show the evolution of the real and imaginary parts of the copolarized element SAA and the crosspolarized element SAB of the scattering matrices, which model the plate for different configurations of the radar system. The second copolarized and crosspolarized elements are not presented, since they are identical two by two to those presented respectively. For the copolarized elements, the maximum of diffusion is obtained when the configuration is the particular bistatic case defined by the following relation θd + θi = 180°. As for the crosspolarized elements, they are null whatever the direction of the transmission and the reception. So, for the particular bistatic cases presented before, the plate does particular bistatic configurations + 90 o
θd 0o -
backscattering - 90
o
90 o
θi
180 o
FIGURE 14.9 The locations of the backscattering mechanism of those particular bistatic configurations corresponding to the case of the bisecting line merged into the axis (Oz). © 2001 CRC Press LLC
FIGURE 14.10 A plate—real part of the SAA copolarized element.
FIGURE 14.11 A plate—imaginary part of the SAA copolarized element.
FIGURE 14.12 A plate—real part of the SAB crosspolarized element.
FIGURE 14.13 A plate—imaginary part of the SAB crosspolarized element.
not depolarize. It scatters the same polarization state as those of the transmitted electromagnetic wave. The monostatic configuration defined by a normal incidence is such that it is a specific configuration with a bistatic angle equal to zero and with the bisecting line equal to the normal axis. The theoretical backscattering matrix is then equal to [ S plate ] = 1 0 0 1
(14.29)
The Rectangular Sides Dihedral The signature of a dihedral of dimension L = H = 0.30 m, with an aperture angle of 2α = 90° is calculated at the frequency of 10 GHz. Figure 14.14 shows the angles used in the computation. The transmitter and the receiver describe the same plane, which is horizontal so as to study the response of the vertical dihedral, but also the response of an oriented dihedral of β angle presented in the Figure 14.15. © 2001 CRC Press LLC
z L
L
θi = θd = 90o
ϕi = 180o
β
y P
H
−α side
+α
1
sid
e2
ϕd = 0o
axis of rotation
X
FIGURE 14.14 dihedral.
FIGURE 14.15 Rotation of the dihedral.
Definition of the angles for the
In the monostatic case, the scattering matrix of a vertical dihedral is linked to that of an oriented dihedral by the following relation: T
[ S oriented ] = [ U rotation ] [ S vertical ] [ U rotation ]
with
[ U rotation ] =
(14.30)
cosβ sinβ – sinβ cosβ
where β represents the orientation angle around the line of sight of the radar. The theoretical scattering matrix of a vertical dihedral in a monostatic configuration is [ S vertical ] = 1 0 0 –1
(14.31)
The backscattering matrix of an oriented dihedral around the line of sight is [ S oriented ] = cos2β sin2β sin2β -cos2β
(14.32)
The simulations visualize the scattering of the target in any point of the horizontal plane, situated inside the aperture of the dihedral, such that the ϕi angle belongs to the interval 135° to 225°, and the θi and θd angles equal 90°. Figure 14.17 indicates the location that corresponds to the backscattering defined by ϕi = (ϕd + 180°), and on the other hand corresponds to particular bistatic configurations for which the bisecting line is directed toward the direction such that ϕd equals zero. Figures 14.18 through 14.21 show the evolution of the real and imaginary parts of the copolarized and the crosspolarized elements of the bistatic scattering matrices of a vertical dihedral. The real and imaginary parts of the second copolarized element are not presented, since they are equal in absolute value but opposite in sign. As for the crosspolarized elements, they are null. The modulus of the copolarized element is a maximum for monostatic configurations. The vertical dihedral is very directing in the elevation plane. However, for monostatic configurations, the vertical dihedral is not guiding in the azimuth plane. © 2001 CRC Press LLC
+ 45ο
P
ϕd
0ο
− 45ο 135ο ϕ i=180o , ϕd =0o
particular bistatic configuration
backscattering
ϕi =170o , ϕd =10 o
FIGURE 14.16
FIGURE 14.18 A vertical dihedral—real part of the SAA copolarized element.
FIGURE 14.20 A vertical dihedral—real part of the SAB crosspolarized element.
ϕi
180ο
225ο
FIGURE 14.17
FIGURE 14.19 A vertical dihedral—imaginary part of the SAA copolarized element.
FIGURE 14.21 A vertical dihedral—imaginary part of the SAB crosspolarized element.
The vertical dihedral scatters almost all the power in the direction of the transmission. So, the real and imaginary parts of the copolarized elements tend to zero as soon as the configuration becomes bistatic. After the study of the elements of the scattering matrix of a vertical dihedral, we present the horizontal plane response, of a 22.5° oriented dihedral, as shown in Figure 14.22. © 2001 CRC Press LLC
22.5o
22.5o
P
ϕi=180o , ϕd= 0 o
FIGURE 14.22
ϕi=170o , ϕd=10 o
The 22.5° oriented dihedral.
Figures 14.23 through 14.26 present the curves of the SAA copolarized element and the SAB crosspolarized element calculated in the horizontal plane. The SAA copolarized element and the SAB crosspolarized element are identical. The SBB copolarized element is very closed to –SAA and tends to the equality of –SAA when the configuration is monostatic for a normal incidence. The phase between both copolarized elements is then equal to 180°. The maximum of the modulus for the four elements of the scattering matrix are identical and are obtained for the monostatic configuration of normal incidence. The response of the dihedral becomes very directed and the scattering of the target decreases very quickly as soon as the value of ϕd is different from 0°.
FIGURE 14.23 A 22.5° oriented dihedral—real part of the SAA copolarized element.
FIGURE 14.24 A 22.5° oriented dihedral—imaginary part of the SAA copolarized element.
FIGURE 14.25 A 22.5° oriented dihedral—real part of the SAB crosspolarized element.
FIGURE 14.26 A 22.5° oriented dihedral—imaginary part of the SAB crosspolarized element.
© 2001 CRC Press LLC
For a rotation of 22.5°, the theoretical monostatic scattering matrix, when the direction of transmission is the (Ox) axis, becomes 2 [ S 22.5°oriented ] = ------- 1 1 2 1 –1
(14.33)
The form of the matrix implicates the equality of the amplitude of the four elements of the scattering matrix and a phase opposition between the diagonal elements. These theoretical results are confirmed by the look of the curves obtained by the simulation software, associated to the 22.5° oriented dihedral for monostatic and quasi-monostatic configurations situated around the incidence direction defined by ϕi , equal to 180°. The next situation studied is the 45° oriented dihedral. Figures 14.27 through 14.30 show the evolutions of the SAA and SAB elements of bistatic scattering matrices that model the 45° oriented dihedral.
FIGURE 14.27 A 45° oriented dihedral—real part of the SAA copolarized element.
FIGURE 14.28 A 45° oriented dihedral—imaginary part of the SAA copolarized element.
FIGURE 14.29 A 45° oriented dihedral—real part of the SAA copolarized element.
FIGURE 14.30 A 45° oriented dihedral—imaginary part of the SAA copolarized element.
© 2001 CRC Press LLC
The two copolarized elements are equal, and the two crosspolarized also, so only two are presented. The copolarized elements are very poor facing the crosspolarized one, both for the monostatic configuration of normal incidence and for the quasi-monostatic associated. The theoretical monostatic scattering matrix of a 45° oriented dihedral takes the following form: [ S 45°oriented ] = 0 1 10
(14.34)
The copolarized elements are null. For the particular monostatic case, the crosspolarized elements are of same phase and of maximum value for this angle of rotation when ϕi equals 180°. For any bistatic configuration, the crosspolarized elements are null.
14.4.2
THE KENNAUGH MATRIX
Like the scattering matrix links the two Jones vectors, the Kennaugh matrix relates the transmitted and the received Stokes vectors. The Kennaugh matrix proceeds from the scattering matrix with *
T
[ K ] ( A, B ) = [ V ] ( [ S ] ( A,B ) ⊗ [ S ] ( A,B ) ) [ V ]
(14.35)
Like the scattering matrix links the two Jones vectors, the Kennaugh matrix relates the transmitted and the received Stokes vectors by d
i
g ( E ) ( A,B ) = [ K ] ( A,B ) g ( E ) ( A,B )
(14.36)
If we insert the decomposition of the bistatic scattering matrix into the definition of the Kennaugh matrix recalled in Equation (14.35), that matrix can be broken down as the sum of [Ks], [Kss], and [Kc] as follows: *
T
[ K bi ] ( A,B ) = [ V ] ( ( [ S s ] ( A,B ) + [ S ss ] ( A,B ) ) ⊗ ( [ S s ] ( A,B ) + [ S ss ] ( A,B ) ) ) [ V ] [ K bi ] ( A,B ) = [ K s ] ( A,b ) + [ K ss ] ( A,B ) + [ K c ] ( A,B )
(14.37)
with T
[ K s ] ( A,B ) = [ V ] ( [ S s ] ( A,B ) ⊗ [ S s ]
*
( A,B )
T
[ K ss ] ( A,B ) = [ V ] ( [ S ss ] ( A,B ) ⊗ [ S ss ] T
[ K c ] ( A,B ) = [ V ] ( [ S s ] ( A,B ) ⊗ [ S ss ]
*
*
)[V]
( A,B )
)[V]
( A,B )
) [ V ] + [ V ] ( [ S ss ] ( A,B ) ⊗ [ S s ] ( A,B ) [ V ] )
T
*
(14.38)
• [Ks] is symmetric and corresponds to an equivalent monostatic Kennaugh matrix. • [Kss] is a diagonal matrix. • [Kc] is a skew-symmetric matrix. As the Kennaugh matrix is no longer symmetric, it will be described by 16 parameters, which correspond to the bistatic parameters. We have chosen to keep the definition of the Huynen © 2001 CRC Press LLC
parameters only depending on the equivalent “symmetric” part of the bistatic scattering matrix and to determine seven new bistatic parameters A, I, J, K, L, M, N as follows:
A0 ( A,B ) + B0 ( A,B ) + A ( A, B )
C ( A,B ) + I ( A,B )
C ( A,B ) – I ( A,B )
A0 ( A,B ) + B ( A,B ) – A ( A, B )
[ K bi ] ( A,B ) =
H ( A,B ) – N ( A,B )
E ( A,B ) – K ( A,B )
F ( A,B ) – L ( A,B )
G ( A,B ) – M ( A,B )
H ( A,B ) + N ( A,B )
F ( A,B ) + L ( A,B )
E ( A,B ) + K ( A,B )
G ( A,B ) + M ( A,B )
A0 ( A,B ) – B ( A,B ) – A ( A,B )
D ( A,B ) + J ( A,B )
D ( A,B ) – J ( A,B )
– A0 ( A,B ) + B0 ( A,B ) – A ( A,B )
(14.39)
where 1 2 A0 ( A,B ) = --- ( S AA + S BB ) 4
A ( A,B ) = S AB
1 2 s 2 B0 ( A,B ) = --- ( S AA – S BB ) + S AB 4
1 2 s 2 B ( A,B ) = --- ( S AA – S BB ) – S AB 4
ss 2
1 2 2 I ( A,B ) = --- ( S BA – S AB ) 2
1 2 2 C ( A,B ) = --- ( S AA – S BB ) 2
*
*
J ( A,B ) = Im ( S BA S AB )
D ( A,B ) = Im ( S AA S BB ) s
*
K ( A,B ) = Re [ ( S AB ) ( S AA + S BB ) ]
s
*
L ( A,B ) = Im [ ( S AB ) ( S AA + S BB ) ]
s
*
G ( A,B ) = Im [ ( S AB ) ( S AA – S BB ) ]
s
*
N ( A,B ) = Re [ ( S AB ) ( S AA – S BB ) ]
E ( A,B ) = Re [ ( S AB ) ( S AA – S BB ) ] F ( A,B ) = Im [ ( S AB ) ( S AA – S BB ) ] G ( A,B ) = Im [ ( S AB ) ( S AA + S BB ) ] H ( A,B ) = Re [ ( S AB ) ( S AA + S BB ) ]
ss
*
ss
*
ss
*
ss
*
(14.40)
J. R. Huynen identifies the monostatic Kennaugh matrix with the parameters A0, B0, B, C, D, E, F, G, H, called Huynen parameters, which are associated to a geometrical target characteristic. Then, the other parameters are null so that I = 0 ,J = 0,K = 0,L = 0,M = 0,N = 0,A = 0 s
ss
S AB = S AB and S AB = 0 © 2001 CRC Press LLC
(14.41)
14.4.3
THE TARGET EQUATIONS
The absolute phase of the scattering matrix cannot be precisely measured, so the phase relative scattering matrix is studied instead. As the phase relative bistatic scattering matrix is no longer symmetric, the bistatic polarimetric dimension of the target is equal to seven from the four modulus and the three relative phases. The bistatic Kennaugh matrix is described by 16 parameters. So, all of these 16 parameters are linked together by (16 – 7) = 9 independent equations, which will express the interdependence of the bistatic parameters. To derive these target equations, we assume that the target is pure and consequently that the scattered wave is totally polarized. The target vector k24 is introduced like the vectorization of the scattering matrix [Sbi]. 1 k = --- Tr ( [ S bi ] [ ψ ] ) 2
(14.42)
where [ψ] = a set of (2 × 2) complex base matrices, which are a linear combination of the Pauli matrices,25 with [ ψ ] = 2 1 0 , 2 1 0 , 2 0 1 , 2 0 –j 0 1 0 –1 1 0 j 0
(14.43)
The factor of 2 arises from the requirement to keep Tr([S][S]*T) an invariant. The bistatic coherency matrix [Tbi] is generated from the outer product of the k target vector by its conjugate transpose, so that
[ T bi ] ( A, B ) =
2A0 ( A,B )
C ( A,B ) – jD ( A,B )
H ( A,B ) + jG ( A,B )
L ( A,B ) – jK ( A,B )
C ( A,B ) + jD ( A,B )
B0 ( A,B ) + B ( A,B )
E ( A,B ) + jF ( A,B )
M ( A,B ) – jN ( A,B )
H ( A,B ) – jG ( A,B )
E ( A,B ) – jF ( A,B )
B0 ( A,B ) – B ( A,B )
J ( A,B ) + jI ( A,B )
L ( A,B ) + jK ( A,B )
M ( A,B ) + jN ( A,B )
J ( A,B ) – jI ( A,B )
2A ( A,B )
(14.44)
It is interesting to notice that the monostatic 3 × 3 coherency matrix equals the first three columns and rows of the bistatic coherency matrix. It proceeds from the choice to specify 9 of the 16 parameters similarly to the Huynen parameters. For a rank one coherency matrix, all principal minors of the coherency matrix must equal zero. From these minors, we obtain a set of equations. The aim is to define the last five target equations which are, with the four monostatic target equations, generic to the all equations. Finally, the nine polarimetric bistatic target equations are given following in any basis, then 2
2
K ( B0 – B ) = – ( HI + JG )
2
2
L ( B0 – B ) = JH – IG
2 A0 E = CH – DG
2 A0 ( B0 + B ) = C + D
2 A0 F = CG + DH
2 A0 ( B0 – B ) = G + H
2 A E = JM – IN
2 A0 ( B0 – B ) = I + J
2 A F = – ( JN + IM )
2
2
(14.45)
The backscattering matrix is symmetric. So, the monostatic polarimetric “dimension” of a target is equal to five, which corresponds to the three independent modulus and the two relative phases © 2001 CRC Press LLC
of the backscattering matrix. The monostatic polarimetric dimension of the target is equal to five, the nine Huynen parameters are related to each other by (9 – 5) = 4 independent equalities that are called the monostatic target equations. The four monostatic target equations correspond to five among the bistatic target equations. They are still valid because of the decomposition of the bistatic scattering matrix. The nine parameters, which appear in the monostatic target equations, are defined by the symmetric part of the bistatic scattering matrix. If the parameters A0, (B0 + B), (B0 – B), and A equal zero, this induces the canceling of all the other parameters. For that reason, 2A0, (B0 + B), (B0 – B) and 2A are called the bistatic generators of the target structure. The four generators are located at the four tops of the bistatic target diagram. The other parameters are situated such that the square sum of two of them is equal to the product of the two generators, which are on the same line, as shown in Figure 14.31. The target diagram allows the reconstruction of the 9 target equations and, more widely, the set of the 36 dependent equalities, which link the 16 bistatic parameters together. The bistatic target diagram is extended from a triangular surface, the monostatic target diagram, to a tetrahedral volume. Indeed, for the monostatic case, if the parameters A0, B0 + B and B0 – B equal zero, it induces the canceling of the all other monostatic parameters. Six sub-matrices of 2 × 2 matrix dimension, and such that their diagonal elements are two generators, can be extracted from the bistatic coherency matrix such that 2A0 ( A,B )
C ( A,B ) – jD ( A,B )
C ( A,B ) + jD ( A,B )
B0 ( A,B ) + B ( A,B )
2A0 ( A,B )
H ( A,B ) + jG ( A,B )
H ( A, B ) – jG ( A,B )
B0 ( A,B ) – B ( A,B )
2A0 ( A,B )
L ( A,B ) – jK ( A,B )
L ( A, B ) + K ( A,B )
2A ( A,B )
,
,
,
B0 ( A,B ) + B ( A,B )
E ( A,B ) + jF ( A,B )
E ( A,B ) – jF ( A,B )
B0 ( A,B ) – B ( A,B )
B0 ( A,B ) + B ( A,B )
M ( A,B ) – jN ( A,B )
M ( A,B ) + jN ( A,B )
2A ( A,B )
B0 ( A,B ) – B ( A,B )
J ( A,B ) + jI ( A,B )
J ( A,B ) – jI ( A,B )
2A ( A,B )
(14.46)
Six equalities are determined following this rule: The product of two generators is equal to the sum of the value squared of the two parameters that link these two generators on to the tetrahedral shown in Figure 14.32. Furthermore, for each side, the multiplication of one generator by one parameter among the two that belong to the same side, but are located on the opposite line, is equal to the sum or the difference of the crossed product of the two other pairs of parameters of the side. In this way, the 24 equalities, which only depend on one generator, can be constructed. For example, the relations that take into account the A generator are written in Figure 14.33.
2A0
2A0 2A0 (B0 + B) = C
2
+D
(G
,H
) ,D 2-
B
2
=E
2
+ F2
)
B0 - B
B0 + B
(E,F)
(L,K)
(L,K
B0
4A0 A = L
)
(C
,D
)
)
(C
2
+ H2
(G
,H
(E,F)
2A0 (B0 - B) = G
2
2
+K2
B0 - B (I,J
(I,J
(M,N)
)
)
B0 + B
(M,N
)
2A ( B0 - B) = I
2A 2A ( B0 + B) = M
FIGURE 14.31 © 2001 CRC Press LLC
FIGURE 14.32
2
+N2
2A
2
+J2
2A0
(G
,H
) (L,K
(C ,D )
)
B0 - B
(E,F)
2A E = JM 2A F = - (JN
-IN +
(I, J
)
B0 + B
(M,N
)
IM)
2A
2A C = LM + KN 2A D = KM - LN
2A G = - (IL + JK) 2A H = JL - KI
FIGURE 14.33
The six last equalities, defined in Figure 14.34, are determined by the sum, or the difference, of two crossed products such that each crossed product is constructed by the way of two couples of parameters, which do not belong to the same side. They depend on no generator. The bistatic target diagram is a good helper in reconstructing all the relations obtained by canceling all the principal minors of the bistatic coherency matrix.
14.5
THE POLARIZATION FORK
14.5.1
THE POLARIMETRIC SIGNATURE
OF A
TARGET
The polarimetric signature of a target visualizes the scattered power, for a given configuration, by the target, in the copolarized and the crosspolarized channels. • When the power of the scattered wave is distributed in the two channels of reception of orthogonal polarization states, then it is said that the target depolarizes. • The PCO copolarized power is measured with a receiving antenna of polarization state identical to the polarization state of the transmitting antenna. • The PX crosspolarized power is measured with a receiving antenna of polarization state orthogonal to the polarization state of the transmitting antenna. • The evolution of the theoretical copolarized and crosspolarized powers for two canonical targets in a monostatic configuration are visualized by the sphere and the dihedral. The theoretical monostatic scattering matrix associated to a sphere is equal to [ S ] ( A,B ) = 1 0 0 1
(14.47)
The diagonal form of the matrix means that the sphere is an isotropic and non-depolarizing target.
FIGURE 14.34 © 2001 CRC Press LLC
In the following, the 2δ and 2γ angles specify the position of the point that represents the polarization state of the wave at the surface of the Poincaré sphere. Figures 14.35 and 14.36 show the evolution of the copolarized and crosspolarized power scattered by a sphere, as a function of the transmitted polarization state, expressed with the help of the Deschamps parameters 2δ and 2γ. The copolarized power is maximum for the values of the 2δ angle equal to {0°,±180°} for any value of the 2γ angle. The copolarized power is minimum for the values of Deschamps parameters such that 2δ = ±90°, 2γ = 90°. However, it is usual in the literature to express the polarization states according to the ellipticity and orientation angles. Then, the maximum of copolarized power is obtained when the ellipticity angle is null, and the minimum for an ellipticity angle equal to ±45°, and for any orientation angle. The crosspolarized power is maximum for 2δ = ±90°, 2γ = 90°, or for the value of ellipticity equal to ±45°, for any orientation angle. The crosspolarized power is null for the 2δ angle equal to {0°,±180°} for any value of the 2γ angle, which corresponds to the cancel of the ellipticity angle for any orientation angle. The theoretical monostatic scattering matrix associated to a vertical dihedral equals S ( A,B ) = 1 0 0 –1
(14.48)
The diagonal form of the matrix involves that the dihedral is a non-depolarizing target. Figures 14.37 and 14.38 show the evolution of the copolarized and the crosspolarized power. The copolarized power is maximum for the values of the 2δ angle equal to ±90°, and for any value of the 2γ angle. The minimum of the copolarized power are for 2δ angle equal to {0°,±180°} and for the value of the 2γ angle equal to 90°. As for the crosspolarized power, it is maximum for 2δ equal to {0°,±180°} and 2γ angle equal to 90°. The crosspolarized is null for 2δ angle equal to ±90° and the 2γ angle equal to {0°,±180°}.
14.5.2
PRESENTATION
OF THE
POLARIZATION FORK
The polarization fork is a representation on the Poincaré sphere of a set of points: the characteristic polarization states that form the extreme polarimetric signature of a target. These points maximize or minimize the copolarized, the crosspolarized, and the optimal powers. For a bistatic configuration,
FIGURE 14.35 Signature of a sphere (copolarized).
© 2001 CRC Press LLC
FIGURE 14.36 Signature of a sphere (crosspolarized).
FIGURE 14.37 Copolarized signature of a vertical dihedral.
FIGURE 14.38 tical dihedral.
Crosspolarized signature of a ver-
the maximum of the copolarized power is the optimal received power or less. Figure 14.39 shows the scattering due to a target. The polarization state of the transmitting antenna is called T, and the polarization state of the receiving antenna is called R. The target is modeled by the scattering matrix. The scattered wave is the polarization state equal to S.
14.5.3
THE CHARACTERISTIC BASIS
The copolarized power is measured with a receiving antenna of polarization state identical to the polarization state of the transmitting antenna.
hi = hd = h =
hx
(14.49)
hy
The copolarized power is P
CO
T
= h [ S ] ( A,B ) h
2
Target
T
T FIGURE 14.39
Polarization state of the transmit antenna
Scattering due to a target.
© 2001 CRC Press LLC
S
Polarization state of the receive antenna
R
P
CO
P
CO
S S h = ( h x h y ) AA AB x S BA S BB h y
2
2
2
= ( h x S AA + h x h y S BA + h x h y S AB + h y S BB )
2
(14.50)
Because the uniquely decomposition of the scattering matrix defined in Equation (14.26), the copolarized power can be written as P
CO
2
s
2
= ( h x S AA + 2h x h y S AB + h y S BB )
(14.51)
As the copolarized power only depends on the symmetric part of the bistatic scattering matrix, the eK and eL Jones vectors are the eigenvectors issued of the unitary basis transformation matrix, which diagonalizes the symmetric part of the scattering matrix. Because the K and L polarization states are orthogonal, they are chosen to specify the characteristic basis. The scattering matrix in the (K,L) basis can be expressed as [ S ] ( K,L ) =
S KK S KL
e
jε
– S KL S LL
S KK > S LL > 0 and ( S KK ,S LL ) ∈ ℜ
2
(14.52)
So, the characteristic basis is the specific one defined by the two Jones vectors eK and eL associated to the polarization states, which maximize the copolarized power. The Jones vectors of the characteristic points K and L construct the basis transformation matrix from the (A,B) basis to the characteristic one (K,L). Then, the coordinates of the other polarization states on the Poincaré sphere are calculated in the (K,L) basis.
14.5.4
THE COPOLARIZED POWER
Maximization and Minimization of the Copolarized Power (Figure 14.40) The determination of the characteristic polarization states is based on the scattering matrix expressed in the (K,L) basis, so the copolarized power can be expressed in this basis as P
CO
P
CO
T
= h [ S ] ( K, L ) h
2
1 - ( S 2KK + ρ *2 S KK S LL + ρ 2 S KK S LL + ρ 2 ρ *2 S 2LL ) = --------------------* ( 1 + ee )
(14.53)
The expression is identical for any monostatic or bistatic configuration and is a function of the scattering matrix elements of the polarization states that maximize or minimize the copolarized power. The polarization ratios, which are solutions of the cancellation of the derivation of the copolarized power, are S kk S KK p = 0, + ∞, + j ------, – j ------- S LL S LL © 2001 CRC Press LLC
(14.54)
Target
K
K
K scattered
Polarization state of the transmit antenna
K
Polarization state of the receive antenna
FIGURE 14.40
The four polarization ratios obtained correspond to the maximization of the copolarized power (K and L) and to the minimization of the copolarized power (O1 and O2). The associated Stokes vectors are defined by the equalities
1 1 1 1 – , g ( e L ) = 1 , g ( e o( 1,2 ) ) = --------------------------g ( eK ) = S ( KK + S LL ) 0 0 0 0
( S KK + S LL ) ( S LL – S KK ) 0
(14.55)
− + 2 S KK S LL
The fact that the K and L points are antipodal on the Poincaré sphere confirms that the (K,L) basis is orthogonal. Cancellation of the Copolarized Power (Figure 14.41) To cancel the copolarized scattered power, the polarization state of the scattered wave has to be orthogonal to characteristic of the receiving antenna, which collects the copolarized power. In fact, the polarization state of the scattered wave has to be orthogonal to the transmitted wave. The polarization states, which cancel the copolarized power, are solutions of the following relation:
Target
O1 Polarization state of the transmit antenna
FIGURE 14.41 © 2001 CRC Press LLC
T
O1
O1
Polarization state of the receive antenna
O1
P
CO
T
= h [ S ] ( K,L ) h
2
= 0
2
T
2
h [ S ] ( K,L ) h = ( S KK h x + S LL h y )e
jε
= 0
h S KK ⇒ e = ----y = ± j ------hx S LL
(14.56)
The polarization ratios correspond to those of the O1 and O2 points. Scattered Polarization States when the Polarization States K, L, O1, or O2 Are Transmitted σg ( e d ) = [ K ] ( K,L ) g ( e i )
(14.57)
Here, σ takes into account the total scattered power, because the two Stokes vectors are normalized. The scattered polarization states are calculated as σ K g ( e Kd ) = [ K ] ( K,L ) g ( e K ) with σ K = A0 ( K,L ) + B0 ( K,L ) + A ( K,L ) + C ( K,L ) ⇒ g ( e Kd ) ≠ g ( e K )
σ L g ( e Ld ) = [ K ] ( K,L ) g ( e L ) with σ L = A0 ( K,L ) + B0 ( K,L ) + A ( K,L ) – C ( K,L ) ⇒ g ( e Ld ) ≠ g ( e L )
(14.58)
However, these inequalities are transformed into two equalities when the configuration becomes monostatic. σ o1,2 g ( e o1,2d ) = [ K ] ( K,L ) g ( e o1,2 ) A0 ( K,L ) – B0 ( K,L ) with σ o1,2 = A0 ( K,L ) – B0 ( K,L ) + A ( K,L ) − + L ( K,L ) -------------------------------------A0 ( K,L )
⇒ g ( e o1,2d ) =
1 q o1,2d u o1,2d v o1,2d
=
1 – q o1,2i – u o1,2i – v o1,2i
(14.59)
The O1d and O1 on the one hand, and the points O2d and O2 on the other hand, are antipodal on the Poincaré sphere. When the polarization states O1 and O2 are transmitted, the target scatters polarization states that are orthogonal to the transmitted one. © 2001 CRC Press LLC
14.5.5
THE CROSSPOLARIZED POWER
To measure the crosspolarized power PX, the transmitting antenna and the receiving antenna have orthogonal polarization states as shown in Figure 14.42. ⊥
hd = hi x
T
x
⊥T
P = h d [ S ] ( A,B ) h i
2
P = h d [ S ] ( A,B ) h i
2
(14.60)
Because the polarization ratio is expressed like this, e = tanγ e
j2δ
(14.61)
the crosspolarized power depends on the elements of the scattering matrix and on the Deschamps parameters by the relationship S KK S LL 2 1 2 x 2 2 - sin 2γcos4δ + S KL 2 P = --- ( S KK + S LL )sin 2γ – --------------4 2 + Re ( S KL ) ( S KK – S LL )sin2γcos2δ – Im ( S KL ) ( S KK + S LL )sin2γsin2δ
(14.62)
The solutions that cancel the derivation of the crosspolarized power are as shown below. If 2γ = 0[¼] The coordinates of these points are independent of the 2δ angle value and are determined by cos 2 γ ±1 = sin 2γcos2δ 0 sin 2γsin2δ 0
(14.63)
which is about the K and L polarization states.
Target
C1scattered
C1
C1 FIGURE 14.42
Polarization state of the transmit antenna
Polarizationstate of the receive antenna
Crosspolarization measurement geometry.
© 2001 CRC Press LLC
C1
If 2γ = ¼/2 The different values of the second parameter 2δ that defined the location of the points on the Poincaré sphere are determined by 2S KK S LL sin2δcos2δ – sin2δRe ( S KL ) ( S KK – S LL ) – cos2δIm ( S KL ) ( S KK + S LL ) = 0
(14.64)
They correspond to four points C1, C2, D1, and D2, which belong to the vertical plane (OU,OV) on the Poincaré sphere as shown in Figure 14.43. If 2γ ¦ 0 and 2γ ¦ ¼/2 (Figure 14.44) After some derivation, the normalized coordinates of these points are determined to be equal to
1 q E1 ,E2 u E1 ,E2 v E1 ,E2
1 -± = -------------------------2 2 ( S KK – S LL ) --------------------------- 2
2
2
( S KK – S LL ) --------------------------2 2
2
2
( S KK – S LL ) 2 2 2 2 ------------------------------- – Re ( S KL ) ( S KK + S LL ) – Im ( S KL ) ( S KK – S LL ) 4 – Re ( S KL ) ( S KK + S LL ) Im ( S KL ) ( S KK – S LL ) (14.65)
The E1 and E2 points are linked together by
g ( e E1 ) =
1 q E1 u E1
=
1 – q E2
(14.66)
u E2
v E1
v E2 V C1
L D2
2γ=π/2
D1
U
O K
Q C2
FIGURE 14.43 Poincaré for values of the second parameter 2δ. © 2001 CRC Press LLC
Target
E1
E1
E1
Polarization state of the transmit antenna
Polarization state of the receive antenna
E1
FIGURE 14.44
The E1 and E2 exist only if the following condition is true: 2
2
2
( S KK – S LL ) ----------------------------- ≥ 2Re 2 ( S KL ) ( S KK + S LL ) 2 + 2Im 2 ( S KL ) ( S KK – S LL ) 2 2
(14.67)
The scattered polarization states, when the E1 and E2 characteristic polarization states are transmitted, are identical to those transmitted. The associated Jones vectors eE1, eE2 are the eigenvectors of the bistatic scattering matrix.
14.5.6
THE OPTIMAL POWER (FIGURE 14.45)
The last four characteristic points of the polarization fork are the transmitted polarization states, M and N, and the received polarization states M″ and N″, which maximize and minimize, respectively, the optimal scattered power. The polarization state of the transmit antenna is neither parallel nor orthogonal to the polarization state of the received antenna. These polarization states are obtained from a singular values decomposition of the bistatic scattering matrix with T
[ S diagonalized ] = [ U d ] [ S ] ( K,L ) [ U i ]
(14.68)
Target
M
M FIGURE 14.45
Polarization state of the transmit antenna
Optimal power conditions.
© 2001 CRC Press LLC
M"
Polarization state of the receive antenna
M"
where [Ui] and [Ud] are the two unitary SU(2) matrices associated respectively to the transmitter and the receiver. The eM and eN Jones vectors are the eigenvectors given by the [Ui] matrix. The eM″ and eN″ Jones vectors are the eigenvectors given by the [Ud] matrix. The coordinates of the M and N points on the Poincaré sphere are defined, as a function of the elements of the scattering matrix, by 1 2 --- ( S KK – S 2LL ) 2 ±1 = -----D Re ( S KL ) ( S KK – S LL )
q M,N u M,N v M,N
with D =
– Im ( S KL ) ( S KK + S LL )
2 1 2 --- ( S KK – S 2LL ) + [ Re ( S KL ) ( S KK – S LL ) ] 2 + [ – Im ( S KL ) ( S KK + S LL ) ] 2 4
(14.69)
The scattered polarization states, if M or N are transmitted, are respectively M″ and N″. Furthermore, the M and N and the M″ and N″ scattered polarization states are respectively orthogonal.
14.5.7
CONCLUSIONS
1. The bistatic polarization fork is defined by 14 characteristic polarization states, K, L, O1, O2, E1, E2, C1, C2, D1, D2, M, N, M″, and N″. All of these polarization states are obtained by maximization or minimization of either the copolarized or the crosspolarized power versus the Deschamps parameters 2δ and 2γ, or by the study of the optimal power. These 14 points are located on 4 different circles (KLO1O2), (KLE1E2), (C1C2D1D2), (MNM″N″) as shown in Figure 14.46. If the locations of the transmitter and the receiver are exchanged, a new polarization fork is obtained which corresponds to the initial one after rotation by 180° about the (KL) axis. 2. The monostatic polarization fork reduces to eight points, because K, E1, M, M″ on the one hand, and L, E2, N, N″ on the other hand, are then at the same location on the Poincaré sphere. The K and L points, which are collocated with the points, minimizing the crosspolarized power, in the monostatic theory are called the X1, X2 points. So, eight characteristic polarization states totally define the monostatic polarization fork. O1 and
V
N
C1 O1
M" E 2 L D2
D1 E1
K N Q O2
M C2
FIGURE 14.46 The bistatic polarization fork is defined by 14 characteristic polarization states, K, L, O1, O2, E1, E2, C1, C2, D1, D2, M, N, M″, and N″. © 2001 CRC Press LLC
O2 represent the copoll null, X1 and X2 the xpoll null, C1 and C2 the xpoll max, and D1, D2 the xpoll saddle. The knowledge of the global position of these points is a very good help in classifying a simple target. Because the location of C1, C2, D1, and D2 are independent of the target in a monostatic configuration, only the four next points, O1, O2, X1, X2, which are coplanar, are useful.
14.6
THE EULER PARAMETERS
The coordinates of each of the 14 characteristic points are independently determined, but their locations are linked together. Furthermore, the polarization fork and the scattering matrix are two different representations of the target model. So, seven independent elements (six geometric angles and a magnitude that uniquely define the relative phase bistatic scattering matrix) specify some angles of the polarization fork. They are called the bistatic Euler parameters. Thus, any bistatic phase relative scattering matrix [Sbi](A,B) defined in an (A,B) orthogonal basis, can be completely expressed according to seven Euler parameters as
[ S bi ] ( A,B ) = [ U ( ϕ,τ,v ) ] m
S KL -------m
1
*
S KL 2 – ------- tan α 0 m
jε
e [ U ( ϕ,τ,v ) ]
T*
2
( 1 + tan α 0 ) m with S KL = cos ( 2α E ) ---- ------------------------------------------------------------------------2 2 1 + cos ( arg ( S ) )tan 2 ( 2α ) KL 0 cot ( β E ) arg ( S KL ) = arctan --------------------cos ( 2α 0 )
(14.70)
The seven bistatic Euler parameters are {m, ϕ, τ, ν, α0, αE, βE}. • m: In this case, m2 represents the maximum copolarized received power. Then, m is called the magnitude of the target and is also linked to the radius of the Poincaré sphere in the polarization fork. • ϕ, τ, ν: The [U] matrix that diagonalizes the symmetric part of the bistatic scattering matrix depends on the three angles ϕ, τ, ν, which are the orientation, the ellipticity, and the absolute phase. T
jε
[ S bi ] ( K,L ) = [ U ( ϕ,τ,v ) ] [ S bi ] ( A,B ) e [ U ( ϕ,τ,v ) ] – jv
0 with [ U ( ϕ,τ,v ) ] = cosϕ – sinϕ cosτ jsinτ e sinϕ cosϕ jsinτ cosτ 0 e jv
(14.71)
The basis transformation defining the characteristic basis is based on the [U] matrix. Furthermore, because the [U] matrix belongs to the SU(2) matrix, the [O](3 × 3) matrix is associated the [U] matrix such that © 2001 CRC Press LLC
[ U ] = [ U2 ( ϕ ) ] [ U2 ( τ ) ] [ U2 ( v ) ] ⇒ [ O ] = [ O 3 ( 2ϕ ) ] [ O 3 ( 2τ ) ] [ O 3 ( 2v ) ]
(14.72)
The basis transformation corresponds to three rotations by 2ϕ, 2τ, 2ν about the different axis of the basis. It is interesting to notice that, because the [U] matrix diagonalizes the symmetric part of the bistatic scattering matrix, m, ϕ, τ, ν are defined identically than for a monostatic configuration. • α0: The elements of the scattering matrix are linked to α0 by the following equality: S LL 2 ------- = tan α 0 m
(14.73)
This angular parameter is linked to the characteristic polarization that cancels the copolarized power, O1 and O2, on the polarization fork by the way shown in Figure 14.47. Because the copolarized power only depends on the symmetric part of the bistatic scattering matrix, α0 is identically defined for a bistatic configuration rather than for a monostatic configuration. • 2αE and 2βE: These two angular parameters allow us to determine the cross elements of the bistatic scattering matrix 2
( 1 + tan α 0 ) m S KL = cos ( 2α E ) ---- ------------------------------------------------------------------------2 1 + cos 2 [ arg ( S ) ]tan 2 ( 2α ) KL 0 cot ( β E ) arg ( S KL ) = arctan --------------------cos ( 2α 0 )
(14.74)
Furthermore, these two angles can be represented on the polarization fork, and 2αE is constructed from the E1 and E2 points similarly to the polarizability angle, which is associated to O1 and O2. The βE angle specifies the angle between the (OQ,OV) plane and the (K L E1 E2) one, as shown in Figure 14.48.
v
o
1
2γo1 2αo Q
K
L
O U
o
2
FIGURE 14.47 Definition of the α0 angle. © 2001 CRC Press LLC
V E2
βΕ
π/2
L
Ε1 Ο 2γE1 2δE1
U
K Q
FIGURE 14.48
Definition of the βE angle.
These seven bistatic Euler parameters totally specify the relative phase bistatic scattering matrix and must have a geometrical significance in the bistatic polarization fork. The points M, N, M″, N″ are specified by the angles 2αM and βM, which are constructed on the Poincaré sphere similarly to 2αE and βE, and which are of analytical form given by 2
2
2
[ 1 + tan 2α E ] { 1 + tan 2α 0 + tan [ arg ( S KL ) ] } 2 tan ( 2α M ) = -------------------------------------------------------------------------------------------------------------2 2 { 1 + ( 1 + tan 2α 0 )tan [ arg ( S KL ) ] } 2
cot ( β M ) = [ 1 + tan 2α 0 ]cotβ E
(14.75)
For the monostatic case, four geometrical angles and a magnitude allow the construction of the polarization fork. The five independent elements, which uniquely define the backscattering matrix, link the positions of the characteristic points on the polarization fork. So, any backscattering matrix [S](A,B) defined in any (A,B) orthogonal basis can be expressed completely according to the five monostatic Euler parameters (m, ϕ, τ, ν, α0) following the form
*
[ S mono ] ( A,B ) = [ U ( ϕ,τ,v ) ] m
14.7
1
0 2
0 tan α 0
jε
e [ U ( ϕ,τ,v ) ]
T*
(14.76)
MONOSTATIC AND BISTATIC POLARIZATION CONCLUSIONS
Our study objective is to extend the radar polarimetry concept to bistatic radar systems. The theoretical approach takes into account the influence of the parameters as the polarization of the wave at reception with respect to the transmitted wave. Properly used, polarization studies can help in target recognition. First, different vector representations of the electromagnetic wave were presented. The bistatic relative scattering matrix is decomposed into two matrices: a symmetric one and a skew-symmetric one. This choice implicates the decomposition of the Kennaugh matrix into three matrices. The definition of the nine Huynen parameters, which depend on the symmetric part of the scattering matrix, is kept. Seven new parameters have been introduced to determine the Kennaugh matrix. Because the polarimetric dimension of the target equals 7 for a bistatic configuration, nine inde© 2001 CRC Press LLC
pendent bistatic target equations link together the 16 parameters. The target diagram is also extended to the bistatic case. Moreover, the 14 characteristic polarization states of the scattering matrix that form the bistatic polarization fork are calculated in the characteristic basis. Six angular parameters allow the location of these different polarization states at the surface of the Poincaré sphere. They are called with the maximum copolarized power, bounded to the radius of the Poincaré sphere, the bistatic Euler parameters. We can extend the basic principles of monostatic radar polarimetry theory to bistatic radar polarimetry. However, our analytical treatment assumes a pure target with no noise and extra clutter. Realistic applications will not be pure target cases; therefore, the clutter due to the environment and the speckle must be modeled to extract the target. Different approaches used for target decomposition theory in radar polarimetry exist in the literature: those based on the Kennaugh matrix and Stokes vector using an eigenvector analysis of the covariance or coherency matrix, and those employing coherent decomposition of the scattering matrix. Several monostatic decomposition theorems can solve the problem and allow the separation between the stationary target and the noise. A second study object was to extend the existing decomposition theorem from the monostatic configuration to the bistatic configuration.
REFERENCES 1. A.B. Kostinski and W.M. Boerner, “On foundations of radar polarimetry,” IEEE Trans. Antennas and Propagation, Vol. AP-34, No. 12, pp. 1395–1404, December 1986. 2. W.M. Boerner, A.K. Jordan and I.W. Kay, guest editors, special issue on “Inverse methods in electromagnetics,” IEEE Trans. Antennas and Propagation, AP-29, March 1989. 3. S.K. Chaudhuri and W.M. Boerner, “A polarimetric model for the recovery of high frequency scattering centers from bistatic-monostatic scattering data,” IEEE Trans. Antennas and Propagation, Vol.AP-35, No. 1, pp. 87–93, January 1987. 4. E.M. Kennaugh, “Polarization dependence of RCS—A geometrical interpretation,” IEEE Trans. Antennas and Propagation, vol. AP-29, N.02, march 1981. 5. G.A. Deschamps, “Geometrical representation of the polarization of a plane electromagnetic wave,” Proc. IRE., pp. 540–544, 1951. 6. J.R. Huynen, “Phenomenological theory of radar targets,” (PhD dissertation), PQR press, Drukkerij Bronder-offset N.V. Rotterdam, 1970. 7. E. Pottier, “Contribution de la polarimétrie dans la discrimination de cibles radar. Application à l’imagerie électromagnétique haute résolution,” Thèse de Doctorat, Université de Rennes I, 1990. 8. Z.H. Czyz, “An alternative approach to foundations of radar polarimetry,” Direct and Inverse methods in radar polarimetry, W.M. Boerner et al., NATO-ARW-DIMPRP, pp. 247–266, 1988. 9. Z.H. Czyz, “Bistatic radar target classification by polarization properties,” ICAP ’87, IEE Conf., York, UK, No. 274, Pt. 1, pp. 545–548, 1987. 10. Z.H. Czyz, “Characteristic polarization states for bistatic nonreciprocal coherent scattering case,” ICAP ’91, IEE Conf., Publ. 333, York, UK, pp. 253–256, 1991. 11. M. Davidovitz and W.M. Boerner, “Extension of Kennaugh optimal polarization concept to the asymmetric scattering matrix case,” IEEE Trans. Antennas and Propagation, AP-34, (4), pp. 569–574, 1986. 12. W.M. Boerner and Z.H. Czyz, “A rigorous formulation of the characteristic polarization state concept and its solution for the bistatic coherent case,” ETC, vol. 1, November 1991. 13. W-M. Boerner, Direct and inverse methods in radar polarimetry, Vol. 1, Dorecht, Boston, London, Kluwer Academic Publishers, 1992. 14. A-L. Germond, E. Pottier, J. Saillard, “Nine polarimetric bistatic target equations,” Electronics Letters, Vol. 33, No. 17, pp. 1494–1495, 1997. 15. A-L. Germond, E. Pottier, J. Saillard, “Two bistatic target signatures: The bistatic equations and the bistatic polarization fork,” MIKON, pp. 123-127, Cracow, 1998. 16. J.I. Glaser, “Some results in the bistatic radar cross section of complex objects,” Proc. IEEE, Vol. 77, No. 5, pp. 639–648, May 1989. © 2001 CRC Press LLC
17. V.W. Pidgeon, “Bistatic cross section of the sea,” IEEE. Trans. Antennas and Propagation, Vol.1 4, No. 3, pp. 405–406, May 1966. 18. G.W. Ewell, “Bistatic radar cross section measurements,” Radar Reflectivity Measurement: Techniques and Applications, Chap. 5, pp. 139–176, Ed. Currie, 1989. 19. R.E. Kell, “On the derivation of bistatic RCS from monostatic measurements,” Proc. IEEE, Vol. No. 8, pp. 983–987, August 1965. 20. R.W. Larson et al., “Bistatic clutter measurements,” IEEE Trans. Antennas and Propagation, Vol. 26, No. 6, pp. 801–804, November 1978. 21. N.J. Willis, Bistatic Radar, Artech House, 1991. 22. M.R.B. Dunsmore, “The principles and applications of bistatic radars,” PIERS ’98, Ispra, 1998. 23. A. Guissard, “Mueller and Kennaugh matrices in radar polarimetry,” IEEE Transactions on geoscience and remote sensing, Vol. 32, No. 3, May, pp. 590–597, 1994. 24. S.R. Cloude and E. Pottier, “A review of target decomposition theorems in radar polarimetry,” IEEE Trans on geoscience and remote sensing, Vol. 34, pp. 498–518, 1996. 25. S.R. Cloude, “Group theory and polarization algebra,” Optik, Vol. 75, No. 1, pp. 26–36, January 1986.
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