Theoretical Covariance Matrix by Integration
 Covariance Matrix in the Linear HV Basis
 Covariance Matrix in the Circular LR Basis
 Coherency Matrix by Integration
 Theoretical Kennaugh Matrix
 Polarization Matrices of Canonical Targets
 Mutual Transformation of Polarization Matrices and Summary
 Relation Between Covariance Matrix and Coherency Matrix
 Appendix
 A 4.1: ROTATION OF COHERENCY MATRIX WITH MINIMIZATION OF T33 AND MAXIMIZATION OF T22
 A 4.2: UNITARY TRANSFORMATION OF THE COHERENCY MATRIX WITH MINIMIZATION OF T33 AND MAXIMIZATION OF T22
 A 4.3: DUAL POL DATA MATRIX
 References
In this section, the ensembleaveraged covariance matrix is derived for the purpose of creating physical scattering models. Ensemble averaging is carried out by integration over rotation angles as shown in Figure 4.5. At first, a scattering matrix in the HV polarization basis is rotated around the radar line of sight, then the corresponding covariance matrix is calculated. The elements of the covariance matrix are weighted by the probability density function and integrated over angles. The integral result is used as a theoretical covariance matrix for the scattering model.
Covariance Matrix in the Linear HV Basis
To simplify expressions, we put the scattering matrix in the HV basis as
where S_{HH}=a, S_{vv}= b, and S_{HV} = c. Rotation 0 around the radar line of sight yields, referring to (4.2.9), We can create the covariance matrix [C(0)] based on (4.5.2).
The ensembleaveraged covariance matrix is derived from integration over angles weighted by the probability density function p(6).
FIGURE 4.5 Theoretical averaging by integration.
where the superscript HV in {[C(0)]}^{wl} indicates the polarization basis. After the integration of equation (4.5.4), the elements of ([C(0)])^{HV} become as follows [5]:
where
where /7(0) is the probability density function (PDF) or angular distribution function, satisfying
The final covariance matrix form is dependent on p(6). The PDF p(6) is directly related to physical distributions of the object under observation. It is desirable to take an appropriate function considering actual target distributions. For example, tree branches are randomly oriented if seen from the zenith; however, they are rather oriented in the vertical direction if seen from the horizontal direction. Therefore, we choose three kinds (Figure 4.6) of distribution functions as follows [6]:
Assuming constant PDF, the integrals (4.5.5) yields The elements of equation (4.5.4) become
FIGURE 4.6 Probability density function.
Note that
and commonly appeared terms are, ri + fe", c^{2}, Imjc'(a/>)}.
For example, if we take the scattering matrix of a flat plate, we substitute a = b = l, c = 0 into equation (4.5.10), yielding
Then the ensembleaveraged covariance matrix becomes,
This <[C(0)])™ becomes a scattering model of a flat plate in the covariance matrix formulation.
Assuming the above PDF, the integrals (4.5.5) yields
The elements of equation (4.5.4) become,
Assuming the above р(в), the integrals (4.5.5) yield The elements of equation (4.5.4) become,
Equations (4.5.10), (4.5.12), and (4.5.13) generate the general theoretical covariance matrices for any scattering object. Table 4.2 lists covariance matrices of canonical targets based on the preceding equations. These matrices can be used as a theoretical reference in the modeling.
The important feature of the ensemble average matrix is its invariance with respect to target orientations. The scattering matrix is sensitive to the orientation of target. The horizontal dipole has a scattering matrix different from that of the vertical dipole. However, the covariance matrix has the same form for both dipoles. The form of the covariance matrix remains the same regardless of the dipole orientation. This property is important and convenient for detecting and classifying an object in polarimet ric observation, especially for airborne PolSAR or spaceborne PolSAR observations. This important property comes from the secondorder statistics of polarimetric information contained in the covariance matrix. If the distribution function is different, the elements may change. However, the value itself is close to each other such as 2/8 and 4/15 for C_{22} of dipoles and 2/2 and 16/15 for C_{22} of dihedrals.
Covariance Matrix in the Circular LR Basis
It is anticipated that the circular polarization (LR) basis is invariant with respect to rotation of the target, that is, rollinvariant. It is worth investigating to see the form of the covariance matrix in the circular polarization basis.
The scattering vector in the circular polarization basis after rotation can be written as, Therefore, the covariance matrix becomes,
Theoretical Covariance Matrices of Canonical Targets
Target Type 
Scattering Matrix 
Averaged Covariance Matrix 

Sphere. Plate 

Яdipole 

Vdipole 

Яdihedral 

Vdihedral 

Left helix 

Right helix 
where
Assuming constant distribution of Рв), we obtain the following result:
Since the final equation (4.5.18) is of diagonal form, we can see the diagonal elements corresponding to eigenvalues. Therefore, it is convenient for eigenvalue analysis.
On the other hand, the number of independent parameters is three and is less than four in other polarization matrices. Therefore, this form is not so convenient for classifying or identifying a target. For reference, the form of the canonical object becomes as follows:
Coherency Matrix by Integration
The coherency matrix has the advantage of mathematical orthogonality and at the same time representing physical scattering mechanisms. It is useful for the interpretation of the PolSAR image. Here we derive the coherency matrix by integration in the same way as the covariance matrix using the scattering vector.
A scattering vector k,, and its rotation vector k,, (0) can be expressed as Based on equation (4.2.9), the rotation relation can be expressed in a simple form as
so that where
Therefore, the coherency matrix [r(0)] after rotation is given by The elements are:
In order to derive ensemble averaging, we carry out the following integration with three kinds of probability density functions.
The integration yields
Note that Trace ([т(в )]^ = «“ + 2 c" +1/?' = Span [S] applies to all coherency matrices (4.6.7—4.6.9).
It is understood that four terms. j« + /?^{2}, afe^{2},c and Imc’(«/;), appear as independent parameters in the coherency matrix in the same way as in the covariance matrix. These four terms are important polarimetric indexes.
Canonical targets are listed in Table 4.3. Plate, sphere, and helix have the same form regardless of probability functions. Dipoles and dihedrals have slightly different forms; however, the element values are close to each other as regards to PDF. Table 4.3 shows the basic scattering models in the scattering power decomposition (Chapters 7 and 8).
The terminology is given to coherency matrix whose constitutes are as follows:
Coherency Matrix of Canonical Objects by Integration
Target 
Vector 
Coherency Matrix 
Normalized Coherency Matrix by Integration 

Me) 

Sphere. Plate 

//dipole 

Vdipole 

Яdihedral 

Vdihedral 

Left helix 

Right helix 
Theoretical Kennaugh Matrix
Using a rotated scattering matrix (4.5.2), the corresponding Kennaugh matrix [/f (0)] becomes,
Assuming a uniform PDF, the Kennaugh matrix by integration is given by
By using equation (4.5.9), it is written as
Since Re {a//} = 4(a + b“ we can consider + , a/b^{2}, cj", and Im{c*(a/;)} as four
independent parameters again.
Polarization Matrices of Canonical Targets
Canonical targets expressed by the covariance matrix, coherency matrix, and Kennaugh matrix are listed and compared in Table 4.4. They are normalized so that the trace becomes 1.
It is interesting to compare the forms and check which matrix is suitable for the classification of objects. For example, coherency formulation has the simplest form for a plate or sphere. For a helix target, the coherency matrix formulation gives pure imaginary for T_{23}, and its sign indicates the sense of rotation. For dihedrals, the C_{l3} component of a covariance matrix yields negative values, which are easily found. The dipole expression is a sum of a plate and dihedral, etc. These matrix forms are important references for target classification and identification.
TABLE 4.4
Canonical Target Expressed in Various Polarization Matrices
Target 
Covariance 

Covariance ^[C’(tfV)]^ 
([ca/o]) 
Coherency 
Kennaugh 

Plate, Sphere 

Dihedral 

Dipole 

tLHelix 

tfHelix 
Mutual Transformation of Polarization Matrices and Summary
Relation Between Covariance Matrix and Coherency Matrix
These two matrices are 3 x 3 complex valued and semidefinite matrices. Since they are frequently used in the data analysis, the relation is explained again. As shown in Section 4.3, the transformation is carried out as,
Since UЛ is a unitary matrix, the covariance matrix and coherency matrix are equivalent mathematically. Therefore, the information contained inside is the same. This also indicates the eigenvalues of both matrices are the same.
In the same way, covariance in the HV basis can be transformed to that in the circular LR basis.
So far, various polarization matrices are introduced and compared. As a summary, the mutual relations and transformations can be visualized as shown in Figure 4.7. These matrices are connected by unitary transformation. Once the scattering matrix is obtained, all polarization matrices can be derived by unitary transformation as shown in Figure 4.4. The number of independent parameters is nine, even if the matrix form is different. The four key parameters a + b, д  b~, cj~, and Im appear in the theoretically averaged matrices.
FIGURE 4.7 Mutual transformation of polarization matrices. (From Yamaguchi, Y., Radar Polarimetry from
Basics to Applications: Radar Remote Sensing Using Polarimetric Information (in Japanese), IEICE, 2007.)
Appendix
A 4.1: ROTATION OF COHERENCY MATRIX WITH MINIMIZATION OF T33 AND MAXIMIZATION OF T22
Rotation of the coherency matrix around the radar line of sight is carried out by the following equation (Figure A4.1):
More explicitly, it can be written as The element becomes,
To find the minimum value of T33, we search В by its derivative = 0
FIGURE A4.1 Rotation around the radar line of sight.
The same equation can be obtained for maximizing T_{22}.
Therefore, the rotation angle can be obtained as
By this rotation, T_{n} is minimized, and T_{22} is maximized. T_{22} becomes pure imaginary. This situation is a perfect fit for modeling of helix scattering.
After the rotation, the element of coherency matrix becomes,
T_{22} increases by the amount Re{r_{23}} sin40, whereas T_{n} decreases by the same amount, which contributes to the reduction of the volumescattering power and the increase of the doublebounce scattering power.
If the rotation angle is в = 45°, the positions of T_{i}} and T_{22} are mutually interchanged. T_{[2} and T_{l} }also change their positions as in the following equation:
A 4.2: UNITARY TRANSFORMATION OF THE COHERENCY MATRIX WITH MINIMIZATION OF T33 AND MAXIMIZATION OF T22
This transformation is intended to reduce Тзз by mathematical operations [7]. The unitary transformation below is not physically realizable. However, this complex transform also minimizes the T33 element.
then the elements become
The minimum Г33 and the maximum T_{22} can be searched by the following equations: 'herefore, the angle can be obtained as
Under this angle, T_{2}3 becomes a real value, and the imaginary part of 7*23 vanishes. The element of the coherency matrix after the unitary transformation becomes
If this angle is chosen as 45°, the positions of the element change as follows:
A 4.3: DUAL POL DATA MATRIX
Dual pol data has four independent polarimetric parameters as shown in the following equation. From the 3 x 3 covariance matrix expression in the HV polarization basis, we eliminate the VV component. Then the covariance matrix by HH + HV becomes as follows:
There are two real diagonal terms and one complexvalued offdiagonal term in the 2 x 2 covariance matrix. We have four real polarimetric parameters in total, which is much less than nine in the quad pol case.
For compact pol data in which the lefthanded circular wave transmission and H and V channel reception are assumed, the scattering equation and the covariance matrix become,
From this expression, we can see the matrix is the same form as the preceding 2x2 matrix, and that four real parameters are available for the compact pol. Since it is impossible to retrieve S_{HH}, S_{HV}, S_{vv }from this expression, the compact pol cannot be substituted by the quad pol [8].
References
 1. W.M. Boerner et al., eds., Direct and Inverse Methods in Radar Polarimetry, Parts 1 and 2, NATO ASI Series, Mathematical and Physical Sciences, vol. 350, Kluwer Academic Publishers, the Netherlands, 1988.
 2. F. M. Henderson and A. J. Lewis, Principles & Applications of Imaging Radar, Manual of Remote Sensing. 3rd ed„ vol. 2, ch. 5, pp. 271357. John Wiley & Sons, New York. 1998.
 3. S. R. Cloude and E. Pottier, “A review of target decomposition theorems in radar polarimetry,” IEEE Trans. Geosci. Remote Sens., vol. 34, no. 2. pp. 498518, March 1996.
 4. J. S. Lee and E. Pottier, Polarimetric Radar Imaging from Basics to Applications, CRC Press, 2009.
 5. Y. Yamaguchi, M. Ishido, T. Moriyama, and H. Yamada, “Fourcomponent scattering model for polarimetric SAR image decomposition,” IEEE Trans. Geosci. Remote Sens., vol. 43, no. 8, pp. 16991706, 2005.
 6. Y. Yamaguchi, Radar Polarimetry from Basics to Applications: Radar Remote Sensing using Polarimetric Information (in Japanese), IEICE, Tokyo, December 2007. ISBN: 9784885542277.
 7. G. Singh, Y. Yamaguchi, and S.E. Park, “General fourcomponent scattering power decomposition with unitary transformation of coherency matrix,” IEEE Trans. Geosci. Remote Sens., vol. 51, no. 5, pp. 30143022, 2013.
 8. J. S. Lee, M. R. Grunes, and E. Pottier, “Quantitative comparison of classification capability: Fully polarimetric versus dual and singlepolarization SAR,” IEEE Trans. Geosci. Remote Sens., vol. 39, no. 11, pp. 23432351,2001.