# Van Zyl 1992 decomposition

## Description

There is currently a great deal of interest in the use of polarimetry for radar remote sensing. In this context, an important objective is to extract physical information from the observed scattering of microwaves by surface and volume structures. The most important observable measured by such radar systems is the 3x3 coherency matrix [T3]. This matrix accounts for local variations in the scattering matrix and is the lowest order operator suitable to extract polarimetric parameters for distributed scatterers in the presence of additive (system) and/or multiplicative (speckle) noise.

Many targets of interest in radar remote sensing require a multivariate statistical description due to the combination of coherent speckle noise and random vector scattering effects from surface and volume. For such targets, it is of interest to generate the concept of an average or dominant scattering mechanism for the purposes of classification or inversion of scattering data. This averaging process leads to the concept of the « distributed target » which has its own structure, in opposition to the stationary target or « pure single target ».

Target Decomposition theorems are aimed at providing such an interpretation based on sensible physical constraints such as the average target being invariant to changes in wave polarization basis.

Target Decomposition theorems were first formalized by J.R. Huynen but have their roots in the work of Chandrasekhar on light scattering by small anisotropic particles. Since this original work, there have been many other proposed decompositions. We classify four main types of theorem:

1. Those employing coherent decomposition of the scattering matrix (Krogager, Cameron).

2. Those based on the dichotomy of the Kennaugh matrix (Huynen, Barnes).

3. Those based on a “model-based” decomposition of the covariance or the coherency matrix (Freeman and Durden, Dong).

4. Those using an eigenvector / eigenvalues analysis of the covariance or coherency matrix (Cloude, VanZyl, Cloude and Pottier).

The van-Zyl 1992 decomposition was first introduced using a general description of the 3x3 covariance matrix for azimuthally symmetrical natural terrain in the monostatic case. The reflection symmetry hypothesis establishes that in the case of a natural media, as soil and forest, the correlation between co- and cross-polarized channels is assumed to be zero. It follows the corresponding averaged covariance matrix is given by:

with:

The parameters α, ρ, η and μ all depend on the size, shape and electrical properties of the scatterers, as well as their statistical angular distribution. In such a case, it is possible to derive the analytical expressions of the corresponding eigenvalues given by:

And the corresponding three eigenvectors are:

It can be easily shown that the 3x3 averaged covariance matrix can be expressed in the following manner:

with:

The van-Zyl 1992 decomposition thus shows that the two first eigenvectors represent equivalent scattering matrices that can be interpreted in terms of odd and even numbers of reflections.

It follows that power scattered by the surface-like component is given by and the power scattered by the double-bounce component is given by

The term corresponds to the contribution of the volume scattering of the final covariance matrix . Hence, the scattered power by this component can be written as

The expression obtained from an eigenvector / eigenvalue analysis 3x3 averaged covariance matrix , corresponds to the starting point of another class of Target Decomposition Theorems called the Model-Based Decompositions.

## References

Books:

● Jong-Sen LEE – Eric POTTIER, Polarimetric Radar Imaging: From basics to applications, CRC Press; 1st ed., February 2009, pp 422, ISBN: 978-1420054972

● Shane R. CLOUDE, Polarisation: Applications in Remote Sensing, Oxford University Press, October 2009, pp 352, ISBN: 978-0199569731

● Charles ELACHI – Jakob J. VAN ZYL, Introduction To The Physics and Techniques of Remote Sensing, Wiley-Interscience; 2nd edition (July 31, 2007), ISBN-10 0-471-47569-6, ISBN-13 978-0471475699

● Harold MOTT, Remote Sensing with Polarimetric Radar, Wiley-IEEE Press; 1st edition (January 2, 2007), ISBN-10 0-470-07476-0, ISBN-13 978-0470074763

● Jakob J. VAN ZYL – Yunjin KIM, Synthetic Aperture Radar Polarimetry, Wiley; 1st edition (October 14, 2011), ISBN-10 1-118-11511-2, ISBN-13 978-1118115114

● Yoshio Yamaguchi, Polarimetric SAR Imaging : Theory and Applications, CRC Press; 1st ed., August 2020, pp 350, ISBN: 978-1003049753

● Irena HAJNSEK – Yves-Louis DESNOS (editors), Polarimetric Synthetic Aperture Radar : Principles and applications, Springer; 1st edition (Marsh 30, 2021), ISBN 978-3-030-56502-2

Journals:

● J.J. Van Zyl "Unsupervised Classification of Scattering Behaviour Using Radar Polarimetry Data", IEEE Transactions on Geoscience and Remote Sensing, Vol. 27, no. 1, pp. 36-45, July 1989.

● J.J. van Zyl, H.A. Zebker, “Imaging Radar Polarimetry,” Chapter 5, PIERS 3 Progress in Electromagnetic Research, J.A. Kong, Editor, Elsevier, March 1990.