# Yamaguchi 4 components 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 Yamaguchi 4 components decomposition is directly based on the Freeman-Durden 3 components decomposition which models the scattering power in three distinct components and which can be successfully applied to decompose SAR observations under the reflection symmetry condition. However, it can be possible to find some areas in a SAR image for which the reflection symmetry condition does not hold.

Based on the Freeman-Durden 3 components decomposition, Yamaguchi et al. proposed, in 2005, a 4-component scattering model by introducing an additional term corresponding to non-reflection symmetric cases and .

In order to accommodate the decomposition scheme for the more general scattering case encountered in complicated geometric scattering structures, the fourth component introduced is equivalent to a helix scattering power. This helix scattering power term, that corresponds to and , appears in heterogeneous areas (complicated shape targets or man-made structures) whereas disappears for almost all natural distributed scattering.

The concept of helix mechanism has been mainly developed by Krogager in his coherent Target Decomposition Theorem, where it was shown that a helix target generates a left-handed or a right-handed circular polarization for all incident linear polarizations, according to the target helicity. The scattering matrices, corresponding to a left-helix target or to a right-helix target, have the form: and .

These two scattering matrices yield left / right helix covariance matrices given by.

and

where in both case, fC corresponds to the contribution of the helix scattering component.

The second important contribution concerns the modification of the volume scattering matrix in the decomposition according to the relative backscattering magnitudes of versus . In the theoretical modeling of volume scattering, a cloud of randomly oriented dipoles is implemented with a probability function being uniform for the orientation angles. However, for vegetated areas where vertical structure seems to be rather dominant, the scattering from tree trunks and branches displays a non-uniform angle distribution. The proposed new probability distribution is given by

where θ is taken from the horizontal axis seen from the radar.

It follow that in the case of:

● a cloud of randomly oriented, very thin horizontal () cylinder-like scatterers, the volume scattering averaged covariance matrix is thus given by

● a cloud of randomly oriented, very thin vertical () cylinder-like scatterers, the volume scattering averaged covariance matrix is thus given by

In both case, fV corresponds to the contribution of the volume scattering component.

The asymmetric form of the two volume scattering averaged covariance matrices seems to be of considerable use because it can be adjusted to the measured data according to the ratio .

Depending on the scene, the Yamaguchi 4 components decomposition proposes to select the appropriate volume scattering averaged covariance matrices by choosing one of the asymmetric forms if the relative magnitude difference is larger than 2dB, or the symmetric form if the difference is within ±2dB.

As a result, this choice allows making a straightforward best fit to measured data.

Assuming that the volume, double-bounce, surface and helix scatter components are uncorrelated, the total second-order statistics are the sum of the above statistics for the individual mechanisms. Thus, the model for the total backscatter is:

This model gives five equations in six unknowns . The parameters a, b, c and d are fixed according to the chosen volume scattering averaged covariance matrix .

The contribution of each scattering mechanism can be estimated to the span, following with:

The term fv corresponds to the contribution of the volume scattering of the final covariance matrix . The power scattered by the double-bounce component of the final covariance matrix has the expression and the power scattered by the surface-like component is . At last, the term fC corresponds to the contribution of the helix scattering of the final covariance matrix .

Although the Yamaguchi 4 components decomposition is intended to apply to non-reflection symmetry case, the scheme automatically includes the reflection symmetry condition, thus proposing a decomposition scheme for the more general scattering case encountered in complicated geometric scattering structures.

There exist three mode of decomposition which are the following.

o Y4O: original Yamaguchi 4 components decomposition

o Y4R: Yamaguchi 4 components decomposition with rotation transformation

o S4R: original Yamaguchi 4 components decomposition with special unitary transformation from oriented dihedral component.

The following figures present the scheme employed to invert the Yamaguchi 4 components decomposition according to the decomposition used.

Y4O decomposition mode

Y4R decomposition mode

S4R decomposition mode

## 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:

● Freeman A. and Durden S., “A three-component scattering model to describe polarimetric SAR data,” in Proc. SPIE Conf. Radar Polarimetry, vol. SPIE-1748, pp. 213-225, San Diego, CA, July 1992.

● Freeman A. and Durden S., “A Three-Component Scattering Model for Polarimetric SAR Data”, IEEE Trans. Geosci. Remote Sens., vol. 36, no. 3, May 1998.

● Krogager E. and Freeman A., “Three component break-downs of scattering matrices for radar target identification and classification”, in Proc. PIERS '94, Noordwijk, The Netherlands, July 1994.

● Yamaguchi Y., Moriyama T., Ishido M. and Yamada H., “Four-Component Scattering Model for Polarimetric SAR Image Decomposition”, IEEE Trans. Geos. Remote Sens., vol. 43, no. 8, August 2005.

● Yamaguchi Y., Yajima Y. and Yamada H., “A Four-Component Decomposition of POLSAR Images Based on the Coherency Matrix”, IEEE Geos. Rem. Sens. Letters, vol. 3, no. 3, July 2006.

● Y. Yamaguchi, A. Sato, W.M. Boerner, R. Sato, H. Yamada, “4-component scattering power decomposition with rotation of coherency matrix”, IEEE TGRS vol. 49, no. 6, June 2011

● A. Sato, Y. Yamaguchi, G. Singh, and S.-E. Park, “4-component scattering power decomposition with extended volume scattering model”, IEEE GRS Letters, vol. 9, no. 2, pp. 166–170, March 2012

● G. Singh, Y. Yamaguchi, S.E. Park, Y. Cui, H. Kobayashi, « Hybrid Freeman/Eigenvalue Decomposition Method With Extended Volume Scattering Model » IEEE GRS Letters, vol. 10, no. 1, January 2013

● G. Singh, Y. Yamaguchi, S.E. Park, « General Four-Component Scattering Power Decomposition With Unitary Transformation of Coherency Matrix » IEEE TGRS vol. 51, no. 5, May 2013