Theoretical Covariance Matrix by Integration

In this section, the ensemble-averaged 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 ensemble-averaged 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 ensemble-averaged 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 second-order 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 C₂₂ of dipoles and 2/2 and 16/15 for C₂₂ 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, roll-invariant. 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
Я-dipole
V-dipole
Я-dihedral
V-dihedral
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, |a-fe|²,|c| and Im|c’(«-/;)|, 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
V-dipole
Я-dihedral
V-dihedral
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 T₂₃, 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
tL-Helix
tf-Helix

Mutual Transformation of Polarization Matrices and Summary

Relation Between Covariance Matrix and Coherency Matrix

These two matrices are 3 x 3 complex valued and semi-definite 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 T₂₂.

Therefore, the rotation angle can be obtained as

By this rotation, T_n is minimized, and T₂₂ is maximized. T₂₂ becomes pure imaginary. This situation is a perfect fit for modeling of helix scattering.

After the rotation, the element of coherency matrix becomes,

T₂₂ increases by the amount Re{r₂₃} sin40, whereas T_n decreases by the same amount, which contributes to the reduction of the volume-scattering power and the increase of the double-bounce scattering power.

If the rotation angle is в = 45°, the positions of T_i} and T₂₂ 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 T₂₂ can be searched by the following equations: 'herefore, the angle can be obtained as

Under this angle, T₂3 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 complex-valued off-diagonal 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 left-handed 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

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