# Freeman-Durden 3 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 Freeman-Durden 3 components decomposition is a technique for fitting a physically based, three-component scattering mechanism model to the polarimetric SAR observations without utilizing any ground truth measurements. The mechanisms are canopy scatter from a cloud of randomly oriented dipoles, even- or double-bounce scatter from a pair of orthogonal surfaces with different dielectric constants, and Bragg scatter from a moderately rough surface. This composite scattering model is used to describe the polarimetric backscatter from naturally occurring scatterers.

To summarize, this decomposition models the 3x3 covariance matrix as the contribution of three scattering mechanisms

● Volume scattering where a canopy scatterer is modeled as a set of randomly oriented dipoles.

● Double-bounce scattering modeled by a dihedral corner reflector.

● Surface or single-bounce scattering modeled by a first-order Bragg surface scatterer.

The volume scattering from a forest canopy is modeled as the contribution from an ensemble of randomly oriented thin dipoles. The scattering matrix of an elementary dipole, expressed in the orthogonal linear (h,v) basis, when horizontally oriented, has the expression that reduces to in the case of thin dipole.

Considering a set of randomly oriented dipoles, characterized by the previous scattering matrix and oriented according to a uniform phase distribution, the covariance matrix of the ensemble of thin dipoles can be modeled by where fv corresponds to the contribution of the volume scattering component.

The covariance matrix presents rank 3, thus, the volume scattering cannot be characterized by a single scattering matrix of a pure target.

The second component of the Freeman-Durden 3 components decomposition corresponds to the double-bounce scattering. In this case, a generalized corner reflector is employed to model the scattering process. The diplane itself is not considered metallic. Hence, we consider that the vertical surface has reflection coefficients Rth and Rtv for the horizontal and the vertical polarizations, whereas the horizontal one presents the coefficients Rgh and Rgv for the same polarizations. Additionally, two phase components for the horizontal and the vertical polarizations are considered, i.e., and , respectively. The complex phase terms and account for any attenuation or phase change effect. Hence, the scattering matrix of the generalized dihedral is which gives rise to the covariance matrix of the double-bounce scattering component. After normalization respect to the Svv component, this covariance matrix can be written following where and where corresponds to the contribution of the double-bounce scattering to the |Svv|2 component.

The third component of the Freeman-Durden 3 components decomposition consists of a first-order Brag surface scatterer modeling surface scattering. The scattering mechanism is represented by the scattering matrix . Consequently, the covariance matrix corresponding to this scattering component is where corresponds to the contribution of the double-bounce scattering to the |Svv|2 component and where .

Assuming that the volume, double-bounce, and surface 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 four equations in five unknowns. However, the volume contribution can then be subtracted off the |SHH|2, |SVV|2 and SHHSVV* terms, leaving three equations in four unknowns:

In general, a solution can be found if one of the unknowns is fixed. According to van Zyl, based on the sign of the real part of , double-bounce or surface scatter is considered as the dominant contribution in the residual.

● If , the surface scatter is considered as dominant and the parameter α is fixed with: α = -1.

● If , the double bounce scatter is considered as dominant and the parameter β is fixed with: β = +1.

Then the contribution fS and fD and the parameters α or β can be estimated from the residual radar measurements. Finally, 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 . Hence, the scattered power by this component can be written as follows

It can be concluded that 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

The following figure presents the scheme employed to invert the Freeman-Durden 3 components decomposition.

## 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.