*Bounty: 50*

*Bounty: 50*

I’ve tried to have a look on some parts of the books:

But I still don’t seem to understand?!

Consider the following material:

- Subsection 6.4.1 of the book Polarimetric Radar Imaging: From Basics to Applications > Freeman-Durden Three Component Decomposition
- A three-component scattering model for polarimetric SAR data
- Three-component scattering model to describe polarimetric SAR data

In order to build the base covariance matrices we have taken the following steps:

**1. For volume scattering:**

Consider the s-matrix for a vertically oriented infinitely thin dipole as:

$$S=begin{bmatrix}0 & 0 \ 0 & 1end{bmatrix}$$

It can be oriented about the radar look direction by the angle $phi$

$$S(phi)=begin{bmatrix}cos phi & sin phi \ -sin phi & cosphiend{bmatrix}begin{bmatrix}0 & 0 \ 0 & 1end{bmatrix}begin{bmatrix}cos phi & -sin phi \ sin phi & cosphiend{bmatrix}=begin{bmatrix}sin^2phi & sinphicosphi \ sinphicosphi & cos^2phiend{bmatrix}$$

Then the `3-D Lexicographic feature vector`

will be:

$$Omega = begin{bmatrix}sin^2phi \ sqrt{2}sinphicosphi \ cos^2phiend{bmatrix}$$

And the covariance matrix is:

$$C(phi)=Omega.Omega^{*T}=

begin{bmatrix}

sin^4phi & sqrt{2}sin^3phicosphi & sin^2phicos^2phi \

sqrt{2}sin^3phicosphi & 2sin^2phicos^2phi & sqrt{2}sinphicos^3phi\

sin^2phicos^2phi & sqrt{2}sinphicos^3phi & cos^4phi

end{bmatrix}$$

The **second-order statistics** of the resulting covariance matrix will be:

$$C_V = textstyleint_{-pi}^pi C(phi)p(phi), dphi =frac{1}{8}begin{bmatrix}3 & 0 & 1 \ 0 & 2 & 0 \ 1 & 0 & 3end{bmatrix}$$

assuming that $p(phi)=frac{1}{2pi}$ is the probability density function and $phi$ is uniformally distributed.

Why does it say the **second-order statistics**? Isn’t it just the **average** or **expected-value**?

**2. For double-bounce scattering:**

This component is modeled by scattering from a dihedral corner reflector, where the reflector surfaces can be made of different dielectric materials. The vertical (trunk) surface has Fresnel reflection coefficients $R_{TH}$ and $R_{TV}$ for vertical and horizontal polarizations, respectively. And the horizontal (ground) surface has Fresnel reflection coefficients $R_{GH}$ and $R_{GV}$ for vertical and horizontal polarizations, respectively. Assuming that the complex coefficients $gamma_H$ and $gamma_V$ represent any propagation attenuation and phase change effects, The S-matrix for double-bounce scatter will be:

$$S=begin{bmatrix}e^{2jgamma_H}R_{TH}R_{GH} & 0 \ 0 & e^{2jgamma_V}R_{TV}R_{GV}end{bmatrix}$$

Multiple the matrix by $frac{e^{-2jgamma_V}}{R_{TV}R_{GV}}$ and assume $alpha=e^{2j(gamma_H-gamma_V)}frac{R_{TH}R_{GH}}{R_{TV}R_{GV}}$, the s-matrix can be written in the form:

$$S=begin{bmatrix}alpha & 0 \ 0 & 1end{bmatrix}$$

Then the `3-D Lexicographic feature vector`

will be:

$$Omega = begin{bmatrix}alpha \ 0 \ 1end{bmatrix}$$

and the covariance matrix will be:

$$C(phi)=Omega.Omega^{*T}=begin{bmatrix}|alpha|^2 & 0 & alpha \ 0 & 0 & 0 \ alpha^* & 0 & 1end{bmatrix}$$

Which is in fact the second-order statistics for double-bounce scattering, after

normalization, with respect to the VV term.

Why does it say the **second-order statistics**? Here we have no **probability distribution function** and so no **average** or **expected value** is computed but we can say the expected value of a fixed quantity is itself. So is second-order statistics the same as expected value?

Then what about the definition in the book `Order Statistics`

:

In statistics, the kth order statistic of a statistical sample is

equal to its kth-smallest value

which suggests that for finding the second-order statistics of random variables, we should write them in a nondescending order and the choose the 2nd in the queue?