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.