# H / A / Alpha 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).

In 1997, S.R. Cloude and E. Pottier proposed a method for extracting average parameters from experimental data using a smoothing algorithm based on second order statistics. This method does not rely on the assumption of a particular underlying statistical distribution and so is free of the physical constraints imposed by such multivariate models. An eigenvector analysis of the 3x3 coherency matrix [T] is used since it provides a basis invariant description of the scatterer and also provides a decomposition into types of scattering process (the eigenvectors) and their relative magnitudes (the eigenvalues). This original method, based on an eigenvalue analysis of the coherency matrix, employs a 3-level Bernoulli statistical model to generate estimates of the average target scattering matrix parameters. This alternative statistical model sets out with the assumption that there is always a dominant 'average' scattering mechanism in each cell and then undertakes the task of finding the parameters of this average component.

According to the eigen decomposition theorem, the 3×3 Hermitian coherency matrix can be decomposed as follows where is a 3x3 diagonal matrix with nonnegative real elements (where ), and is a 3x3 unitary matrix of the SU(3) group, where u1, u2, and u3 are the three unit orthogonal eigenvectors.

A parameterisation of the eigenvectors of the 3x3 coherency matrix has been introduced for the case of scattering medium which does not have azimuthal symmetry and which takes the following form .

It follows a revised parameterisation of the coherency matrix as :

with :

In order to simplify the analysis of the physical information provided by this eigen decomposition, three secondary parameters are defined as a function of the eigenvalues and the eigenvectors of the 3×3 coherency matrix

● Entropy (H)

The entropy parameter (H) is defined in the Von Neumann sense from the logarithmic sum of eigenvalues of as where pi is called the probability of the eigenvalue λi with . It represents the relative importance of this eigenvalue respect to the total scattered power, since .

The entropy H determines the degree of randomness of the scattering process, which can be also interpreted as the degree of statistical disorder. In this way:

● If the entropy H is low (H→0), then .

Consequently, the system may be considered as weakly depolarizing and the dominant target scattering matrix component can be extracted as the eigenvector corresponding to the largest eigenvalue and ignore the other eigenvector components. The corresponding coherency matrix presents rank 1 and the scattering process corresponds to a pure target.

● If the entropy H is high (H→1), then

In this situation, the target is depolarizing and we can no longer consider it as having a single equivalent scattering matrix. The full eigenvalue spectrum must be considered. The corresponding coherency matrix presents rank 3, that is, the scattering process is due to the combination of three pure targets. Consequently, corresponds to the response of a distributed target or random target.

● If the entropy H is increasing (0 < H < 1) the number of distinguishable classes identifiable from polarimetric observations is reduced. In this case, the corresponding coherency matrix results from the combination of the three pure targets given by ui for i=1,2,3 weighted by the corresponding eigenvalues.

● Anisotropy (A)

While the entropy is a useful scalar descriptor of the randomness of the scattering problem, it is not a unique function of the eigenvalue ratios. Hence, another eigenvalue parameter defined as the anisotropy A can be introduced, with .

When A→0 the second and third eigenvalues are equal. The anisotropy may reach such a value for a dominant scattering mechanism, where the second and third eigenvalues are close to zero, or for the case of a random scattering type where the three eigenvalues are equal.

The anisotropy A is a parameter complementary to the entropy. The anisotropy measures the relative importance of the second and the third eigenvalues of the eigen decomposition. From a practical point of view, the anisotropy can be employed as a source of discrimination only when H>0.7 where it can clearly discriminate different configurations presenting the same value of entropy.

● Mean alpha angle (Alpha)

The parameterisation of a 3x3 unitary matrix [U3] in terms of column vectors with different parameters α1 β1 etc... is made so as to enable a probabilistic interpretation of the scattering process. In general, the columns of the 3x3 unitary matrix are not only unitary but mutually orthogonal. This means that in practice α1 α2 and α3 are not independent.

In this case a statistical model of the scatterer is considered as a 3 symbol Bernoulli process i.e. the target is modeled as the sum of three [S] matrices, represented by the columns of [U3] occuring with probabilities pi ,given by the normalized eigenvalues so that p1 + p2 + p3 = 1. In this way for example, the parameter α follows a random sequence as:

and the best estimate of this parameter is given by the mean of this sequence, easily evaluated as

The study of the mean mechanism which can occur inside a pixel, is mainly performed through the interpretation of the mean alpha angle, since its values can be easily related with the physics behind the scattering process. The next list reports the interpretation of α:

● α→0: The scattering corresponds to single-bounce scattering produced by a rough surface.

● α→π/4: The scattering mechanism corresponds to volume scattering.

● α→π/2: The scattering mechanism is due to double-bounce scattering.

The eigen decomposition of the coherency matrix is also referred as the H/A/α decomposition, but, in fact, this decomposition provides only three output parameters: the entropy (H), the anisotropy (A) and the mean alpha angle (α).

However, it is always possible to introduce and define a 3x3 coherency matrix decomposition based on the eigenvalues and eigenvectors decomposition that is the following.

Indeed, a mean unitary eigenvector can be defined following:

where the remaining average angles are defined in the same way as α, with:

The mean magnitude of the mechanism is obtained as

It follows the decomposed coherency matrix that can be expressed as the outer product of a single target vector k0 with

## 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:

● S.R. Cloude « Uniqueness of target decomposition theorems in radar polarimetry », Direct and Inverse Methods in Radar Polarimetry, Part 1, NATO-ARW, W.M. Boerner et al., Eds. New York : Kluwer Academic, 1992, pp 267-296.

● S.R. Cloude « Symmetry, Zero Correlations and Target Decomposition Theorems », Proceedings of 3rd International Workshop on Radar Polarimetry (JIPR '95), IRESTE, University of Nantes, March 1995, pp 58-68.

● S.R. Cloude, E. Pottier, "A Review of Target Decomposition Theorems in Radar Polarimetry", IEEE Transactions on Geoscience and Remote Sensing, Vol. 34 No. 2, pp 498-518, March 1996.

● S.R. Cloude, E. Pottier, "An Entropy Based Classification Scheme for Land Applications of Polarimetric SAR", IEEE Transactions on Geoscience and Remote Sensing, Vol. 35, No. 1, pp 68-78, January 1997.

● J.R. Huynen « Phenomenological theory of radar targets », Ph. D. dissertation, Drukkerij Bronder-offset, N.V. Rotterdam, 1970.

● Ngheim, S.H. Yueh, R. Kwok, F.K. Li « Symmetry properties in Polarimetric Remote Sensing », Radio Science, Vol. 27. No. 5, pp 693-711 Oct 1992.

● E. Pottier « On Dr J.R. Huynen’s main contributions in the development of polarimetric radar techniques, and how the « radar targets phenomenological concept » becomes a theory », SPIE Vol 1748, Radar Polarimetry, pp 72-85, 1992.

● E. Pottier, “Unsupervised Classification Scheme and Topography Derivation from POLSAR data based on the H/A polarimetric decomposition” Proceedings of 4th International Workshop on Radar Polarimetry (JIPR ‘98), IRESTE, University of Nantes, France, pp 535-548, July 1998.

● E. Pottier, D.L. Schuler, J.S. Lee, T. Ainsworth, "Estimation of Terrain Surface Azimuthal / Range Slopes using Polarimetric Decomposition of POLSAR Data", IEEE International Geoscience and Remote Sensing Symposium. Hamburg, 28 June - 2 July 1999.

● E.Pottier, J.S. Lee "Unsupervised Classification Scheme of POLSAR Images Based on the Complex Wishart Distribution and the H/A/ Polarimetric Decomposition Theorem" 3th European Conference on Synthetic Aperture Radar, EUSAR 2000, Munich, 23-25 May 2000.

● D.L. Schuler, J.S. Lee, T.L. Ainsworth, E. Pottier, « Terrain Slope Measurement Accuracy Using Polarimetric SAR Data », Proceedings of IGARSS’99, Hamburg, July 1999.