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                                S. Allain, C. Lopez-Martinez, L. Ferro-Famil, E. Pottier

               University of Rennes 1, I.E.T.R, UMR CNRS 6164, Image and Remote Sensing Group
         Campus de Beaulieu - Bat 11.C, 263 Avenue Général Leclerc, CS 74205, 35042 Rennes Cedex, France
            Email :   tel : (+33) fax : (+33)

The aim of this paper is to present two novel polarimetric parameters, the Eigenvalue Relative Difference (ERD) and
the Single bounce Eigenvalue Relative Difference (SERD), to characterize natural media. These parameters are derived
from the eigen-decomposition of the coherency matrix considering the reflection symmetry hypothesis. An analysis of
these parameters is performed on multi-frequency polarimetric SAR data acquired on bare soils and forested areas.
In the first part of this study, the analytical expressions of these polarimetric descriptors is presented based on the
Eigenvector/value based decomposition theorem for PolSAR data. The theoretical study of the ERD and SERD
parameters is also addressed from a statistical point of view in order to determine the effects of speckle noise in their
estimation. The ERD parameter is compared to anisotropy.
In a second step, using real SAR data, the ERD and SERD parameters are analyzed on different natural media such as
bare soils and forested areas. In the case of bare soil, this analysis is led by using the Integral Equation Model (IEM).
The SERD is very high due to the single scattering importance. The ERD variations are strictly monotonous with
surface roughness. These variations are validated by using scatterometric data acquired at EMSL, JRC laboratory (Italy)
and polarimetric SAR data acquired over Alling site. The SERD usefulness is shown on forest areas using multi-
frequency and PolSAR data acquired at P-, L- and C-bands over Nezer forest (France), from where a ground truth is
2.1.      Eigenvector/value based Polarimetric decomposition theorem
For a given measurement configuration, a target is described by its coherent (2×2) complex scattering matrix, S . By
transforming this matrix, in the monostatic case, into the complex target vector kp, the coherency matrix T is defined as
                        T = k Pk P
                                              with            kP =         [S hh + S vv   S hh − S vv   2S hv ]T   (1)

where      represents the average operator.
The eigenvector/value based decomposition theorem presented in [1] allows to split the distributed matrix, T, into a
weighted sum of three orthogonal unitary matrices given by:
                                                                 3                3
                                              T = V Σ V † = ∑ λi v i v † = ∑ λi Ti
                                                                       i                                           (2)
                                                                i =1             i =1

where Σ represents a diagonal non-negative real eigenvalue matrix and V is a special unitary eigenvector matrix.
Pseudo-probabilities of the coherency matrix expansion elements are defined, from a set of sorted eigenvalues, as:
                                                     λi        λi
                                          pi =     3
                                                          =            with λ1 ≥ λ2 ≥ λ3                           (3)
                                                 ∑ λj
                                                  j =1

The distribution of the three probabilities can be fully described by two parameters. The entropy, H, indicates the degree
of statistical disorder of the scattering phenomenon and it is defined as:
                                                        H = −∑ p i log 3 p i with 0 ≤ H ≤ 1                                                       (4)
                                                                i =1

For intermediate entropy values, i.e. different from 0 and 1, a complementary parameter is necessary to fully
characterize the set of probabilities. The anisotropy, A, is defined as the relative importance of the secondary scattering
                                                               λ 2 − λ3
                                                         A=                   with       0 ≤ A ≤1                                                 (5)
                                                               λ 2 + λ3
A value of A close to 0 corresponds to secondary mechanisms with equal importance, while A = 1 indicates that the
power associated to the third matrix, T3 , is null.

2.2.          Reflection symmetry
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 [2] [3]. In this case, it is then possible to derive, from
the coherency matrix T, the analytical expressions of its corresponding eigenvalues. The different backscattering
coefficients are gathered into a coherency matrix, T, defined as follows:

                                             A + 2ℜB A + 2ℑB 0                                  A = σ hhhh + σ vvvv
                                        T =  A − 2ℑB A − 2ℜB 0                       with      B = σ hhvv                                      (6)
                                                0       0    4C 
                                                                                                 C = σ hvhv

The literal expressions of the Non-Ordered in Size (“nos”) eigenvalues are [4]:
λ1nos =      (σ hhhh + σ vvvv ) + f (σ hhhh , σ vvvv )
           2                                                                f (σ hhhh , σ vvvv ) = (σ hhhh + σ vvvv ) 2 + 4 ρ hhvvσ hhhh σ vvvv
           1                                                                                  σ hhvv
λ2 nos   = (σ hhhh + σ vvvv ) − f (σ hhhh , σ vvvv )                   with ρ                                                                     (7)
           2                                                                  hhvv =

λ3nos    = 4σ hvhv                                                                       σ hhhh 2σ vvvv 2

The first and second eigenvalues depend on the co-polarized backscattering coefficients and on the correlation between
the vertical and horizontal channels, ρ hhvv . In this case λ1nos > λ 2 nos always hold. The third eigenvalue corresponds to
cross-polarized channel and it is related to multiple scattering for rough surfaces.In order to determine the scattering
mechanism, an analysis is led on the αi angle parameter. This parameter is calculated from the eigenvector :

                                                                               S hh − S vv 
                                                             α i = arctan
                                                                               S hh + S vv 
The nature of the scattering mechanism is determine by the following method:
                                    π                                                                                 π
     S hh − S vv             αi <                                                      S hh − S vv             αi >
if                 <1                4      !Single Reflection                    if                   >1             4        !Double Reflection
     S hh + S vv                  *                                                    S hh + S vv
                          ℜ(S hh S vv ) > 0                                                                 ℜ(S hh S* )
                                                                                                                    vv    <0

Moreover, the orthogonality condition is assumed between the different eigenvectors and we obtain:
(S hh1 + S vv1 )(S hh 2 + S vv 2 ) * + (S hh1 − S vv1 )(S hh 2 − S vv 2 ) * = 0                         α1 + α 2 =                                (9)

The two first eigenvectors correspond to complementary mechanisms.
         π            π                                                                  π              π
α1 <         ⇒ α2 >        then λs = λ1nos and λd = λ 2 nos and                   α1 >        ⇒ α2 <         then λs = λ 2 nos and λd = λ1nos
         4            4                                                                   4             4

From the “nos” eigenvalues presented above, two new parameters called the Eigenvalue Relative Difference [5] and the
Single bounce Eigenvalue Relative Difference, denoted respectively by ERD and SERD, are built up to compare the
relative importance of the different scattering mechanisms. There are defined as:
                                                    λs − λ3nos
                                          ERD =                      with   −1 < ERD < 1                                  (10)
                                                    λs + λ3nos

                                                   λd − λ3nos
                                         SERD =                      with   −1 < SERD < 1                                 (11)
                                                   λd + λ3nos

These two parameters permit to cover all the “nos” eigenvalues spectrum and to compare the importance of the various
scattering mechanisms. The first parameter shall be compared, in what it follows, with the anisotropy derived from the
second and the third eigenvalues of the coherency matrix. An important difference between these two parameters is that
the dynamic range of the ERD parameter is larger [–1;1] than the anisotropy range [0;1].
The SERD usefulness becomes important for media with large entropy values, in order to determine the nature and the
importance of the different scattering mechanisms.
2.3.      Statistical Analysis
As observed in (1), the coherency matrix T needs to be estimated from data due to the presence of speckle noise.
Consequently, the estimation of the eigenvalues of T, as well as the parameters H, A, ERD and SERD shall be also
affected by this noise component.
In the case of the matrix T, it is complex to determine the effects of speckle noise in every one of its entries since they
present a different mathematical nature. Nevertheless, this task is less difficult in the case of the equivalent covariance
matrix C, as all its elements correspond to the hermitian product of the different complex SAR images. The estimation
of C, via multilooking, is statistically characterized by the Wishart distribution. Additionally, in [6] it was demonstrated
that speckle noise was due to the mixture of unity-mean multiplicative and zero-mean additive noise sources, which
combination is determined by the data’s correlation structure. In the case of a particular element of C, i.e., the hermitian
product of a pair of SAR images, where the indices {p,q,r,s} refer to any pair of orthogonal polarization states, the
speckle noise model establishes:

                                                                 (              )
                        S pq S * = ψN c z n n m exp( jφ x ) + ψ ρ pqrs − N c z n exp( jφ x ) +ψ (n ar + jn ai )
                               rs                                                                                         (12)

where nm refers to the unity mean multiplicative speckle component, nar+jnar is the complex, zero-mean additive term
and |ρpqrs| is the coherence which characterizes S pq S* . This model is presented in details within [6]. On the other hand,
it has been demonstrated that speckle noise has the important effect of introducing a bias into the estimated eigenvalues
respect to their true values [7]. Consequently, this bias is also transmitted to H and A, in such a way that H is
underestimated, whereas A is basically underestimated. In what it follows we analyze the effects of speckle over the
estimation of the new parameters and which are the consequences of the reflection symmetry assumption.
The speckle noise model given at (10) is now employed to determine the effects of the different noise components over
the estimated eigenvalues. For clarity reasons, the following analysis is considered in the case of 2x2 hermitian
covariance matrices C, since the extension for 3x3 matrices is straightforward. Let C be a single-look 2x2 hermitian
covariance matrices. Given the speckle noise model, it is possible to derive its average characteristic polynomial for the
calculation of the corresponding eigenvalues as:

                                  ψ                  2        2         22
                             λ2 + ( pq + ψ rs )λ + ψ pqrs − ψ pqrs ρ − ψ pqrs E n ar + jn ai
                                                                                                        }= 0              (13)

                                     { }
where ψ pq = E  S pq  , ψ rs = E S rs
                                             and ψ pqrs = ψ pqψ rs . The previous polynomial has been obtained by

neglecting the correlation between the multiplicative and the additive noise due to its small value [6]. From (11), it is
possible to derive the analytical expression of the corresponding eigenvalues

                                                                   2          2       2
                           (ψ pq + ψ rs ) ± (ψ pq + ψ rs ) 2 − 4ψ pqrs (1 − ρ ) + 4ψ pqrs E n ar + jn ai
                                                                                                                   }   (14)
                                                                                                                   
As it can be observed in the previous equation, only the additive speckle noise term is able to explain the bias of the
sample eigenvalues, which present maximum variance for low coherences [6]. Consequently, the reflection symmetry
assumption considered to obtain ERD and SERD has two important consequences on its estimation. On the one hand, as
a zero value is introduced in the cross-polarized channels of C, the additive noise component is completely eliminated.
It has to be mentioned at this point that up to date, there is not a speckle filter considering this additive noise term. On
the other hand, since the additive noise term is eliminated, the biases over the sample eigenvalues shall be reduced.
Consequently, the ERD parameter will present a lower bias than the A parameter.
In the case of rough surfaces, single scattering dominates the mean scattering mechanism, even, on very rough surfaces,
whereas the probabilities of double bounce and multiple scattering phenomena are smaller. Thus, the SERD parameter
values are very high and closed to onewhereas the ERD variations are very sensitive to surface roughness.
3.1.              Polarimetric scattering model IEM
Soil is mainly characterized by its roughness and its dielectric constant. A stochastic surface is defined by its correlation
function, ϕxx(x,y) and correlation length, Lc, its height probability density function and standard deviation, σ. The
surface spectrum shape is considered gaussian. In order to characterize natural surfaces, the IEM is employed to derive
the backscattering coefficients [8]. This model is widely used due to its large validity domain and since it has been
validated on large sets of experimental data. The cross-polarized information is derived from the multiple scattering
formulation taking into account the coherent SAR integration process inside each resolution cell. This model satisfies
the reflection symmetry assumption. Using this model, the ERD obtained from the “nos” eigenvalues is compared to
anisotropy, usually employed as a surface roughness descriptor [9]. Figs. 1-a and -b show, respectively, the anisotropy
and ERD variations versus the roughness relative to the number wave, kσ obtained using the IEM model for various
dielectric constants, ε. In this illustration case, the considered surface spectrum is gaussian, the incident angle is 40° and
the radar frequency is 1.3 GHz.
        1                                                                                      1
                                      ε=5                                                                                                                 ε=5
                                      ε = 10                                                                                                              ε = 10
       0.8                            ε = 15                                                 0.6                                                          ε = 15
                                      ε = 25                                                                                                              ε = 25
                                      ε = 35                                                                                                              ε = 35
       0.6                                                                                   0.2

A                                                                                     ERD
       0.4                                                                                   -0.2

       0.2                                                                                   -0.6

        0                                                                                     -1
                 0.5      1          1.5               2     2.5                                        0.5         1           1.5               2          2.5
                                     kσ                                                                                         kσ
                                                 -a-                                                                                        -b-

                                                 Fig. 1. -a- Anisotropy, -b- ERD simulated with the IEM model
The ERD parameter is similar to the anisotropy for small roughness values, but presents a different behaviour for high
frequencies. As it can be observed, these parameters are very sensitive to surface roughness relative to frequency. For
each ε value, one anisotropy value corresponds to two different values of kσ introducing an ambiguity for surface
roughness extraction, whereas the ERD is strictly monotonic with kσ. An important difference between these two
parameters is that the dynamic range of the ERD parameter is larger [–1;1] than the anisotropy range [0;1].
3.2.              Application on scatterometric data
        1                                                                                       1
                                                             θ = 10°
       0.9                                                                                    0.8

       0.8                                                   θ = 20°                          0.6

       0.7                                                                                    0.4

       0.6                                                                                    0.2
                          1                        2         θ = 30°
                                                                                       ERD      0
A      0.5
                                                             θ = 40°                                                                                        θ = 50°
                                                                                             -0.2                                                           θ = 40°
                                                             θ = 50°                                                                                        θ = 30°
                                                                                                                                                            θ = 20°
       0.1                                                                                                                                                  θ = 10°
        0                                                                                           0         0.2   0.4   0.6         0.8             1    1.2        1.4
             0     0.2   0.4   0.6         0.8         1   1.2         1.4                                                      kσ
                                                 -a-                                                                                        -b-

                                                                 Fig. 2. -a- Anisotropy, -b- ERD versus kσ
A comparative analysis is now led using monostatic indoor scatterometric data acquired in the European Microwave
Signature Laboratory (EMSL) anechoic chamber at JRC laboratory for frequencies in the range [1-19 GHz] and for five
incident angles from 10° to 50° [10]. The correlation length and the rms height of the surface are respectively 6 cm and
0.4 cm. It is important to notice that the correlation between the co- and the cross-polarized channels is non-null. To
build our coherency matrix based on the reflection symmetry hypothesis, these correlations are considered equal to
zero. The anisotropy and the ERD are plotted versus kσ on the Figs. 2-a and -b. As it has been obtained theoritically
with the IEM model, the anisotropy decreases for small kσ, corresponding to smaller frequencies, and increases as kσ
increases, whereas, the ERD decreases always with kσ. These data permit to validate the polarimetric parameter
variations with kσ derived with the IEM model.
3.3.              Analysis on real PolSAR data
The polarimetric SAR datasets under analysis were acquired by the German Aerospace Center (DLR) E-SAR sensor, at
L- band, over the Alling test site in Germany. The considered scene is mainly composed by agricultural fields and
forested areas. As we can observe on fig. 3, the ERD permits to separate various field areas, whereas there is not visible
difference with the anisotropy. This is mainly due to the high correlation between the co- and cross-polarized channels
on these particular dataset, which correspond to “noise information”. Moreover, the ERD presents a larger dynamic
range than anisotropy and a larger roughness sensitivity that permits to separate more easily different areas.
 0 .4

 0 .3

 0 .2
                                                                                   Six fields with roughness ground truth measurements are
 0 .1                                             ERD
                                                                                   selected. On Fig. 3, the corresponding anisotropy and ERD
                                                  A n is o t r o p y               parameters are plotted versus kσ.
                                                                                   The ERD decreases with roughness which corresponds to
-0 .1
                                                                                   the results obtained with the IEM model whereas the
-0 .2
                                                                                   anisotropy value is constant over all fields.
-0 .3
     0 .8     1      1 .2   1 .4    1 .6   1 .8            2           2 .2
                              σ en cm

                                                                 Fig. 3. Anisotropy and ERD versus kσ
These parameters are now calculated on the Nezer site situated in the south west of France. Fully polarimetric data have
been acquired by the NASA JPL AIRSAR sensor at P-, L-, and C- bands. The scene contains bare soil areas and a large
amount of homogeneous forested areas of maritime pines. Several tree-age classes are distinguished from 5-8 years of
age to more than 41 years. Backscattering from the tree parcels is highly correlated to the age of the trees.
4.1.              Multi-frequency Polarimetric behavior
The reflection symmetry assumption is largely considered in forested areas and also in theoretical volume scattering
models as radiative transfer model and first order distorted Born approximation [3]. In the case of forested area, the
entropy is quite high and it becomes more difficult to discriminate the different scattering mechanisms. SERD permits
to compare co- and cross-polarized channels.
4.2.              Analysis on real SAR data
The ERD is not shown in this case but it can be observed that it decreases with the age in P-band and is almost constant
with tree species in L- and C- bands. In order to validate the usefulness of the SERD parameter, it is represented for a
range from 0 to 1 on the figs. 4-b, -c and -d. The results can be compared to ground truth measurements represented on
fig 4-a. As it has been observed over the Alling dataset, for rough surfaces, the SERD parameter values are very high on
rough surfaces and close to one due to the large importance of single scattering. The SERD decreases on bare soils with
frequency because the roughness parameter, kσ, increases and the secondary (volume) mechanisms importance
increases. At P-band, the SERD decreases with the age of the trees and we can observe a significant analogy between
the SERD parameter and the ground truth measurements. As frequency increases, the SERD value becomes more
homogenous and it becomes more difficult to rely this parameter with the characteristics of the trees as wave
penetration decreases.


                       Bare soil     15-19 years                                                                                0
                       5-8 years     33-41 years                                                                                SERD
                       8-11 years    > 41 years
                       11-14 years   N/ A
                      -a- ground truth                                             -b- SERD at P band


                    -c- SERD at L band                                             -d- SERD at C band
                                            Fig. 4. SERD for different frequency bands
In this paper, two new polarimetric descriptors based on the reflection symmetry assumption: the Eigenvalue Relative
Difference (ERD) and the Single bounce Eigenvalue Relative Difference (SERD) have been presented.Using a speckle
noise model established in [6], the effects of the different noise components over the estimated eigenvalues are
analyzed. In the reflection symmetry assumption considered to obtain ERD and SERD, the additive noise term is
eliminated and consequently, the biases over the sample eigenvalues are reduced. From measured data, the ERD is
shown relevant for surface characterization. Using the IEM surface scattering model, it has been demonstrated that ERD
has a larger validity domain than Anisotropy and that ERD is a better roughness discriminator. The SERD usefulness is
shown over forest zones, as it discriminates tree species. Both parameters are sensitive to natural media characteristics
and will be employed for quantitative inversion of bio and-geo physical parameters.
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