IEEE Geoscience and Remote Sensing Magazine - June 2020 - 95

are critical to the final forest parameter inversion accuracy. The basic idea of system parameter optimization is to
quantify the influence of system parameters on the final
inversion error using theoretical or simulation-based studies of the coherence optimization-based forest-height inversion. Through theoretical derivation, we have developed an
analytic model between forest-height estimation error and
PolInSAR system parameters. With this error-dependence
model, it is easy to understand the reverse requirements of
PolInSAR system parameters from the end application of
forest-height mapping.
In this part, we first study the impact of PolSAR system
parameters. A fully polarimetric radar system can measure
the 2 # 2 complex scattering matrix for each resolution cell
in the image. Due to the limitations of the radar hardware,
the measured scattering matrix is inevitably distorted.
Assuming that R is the receiving system, T is the transmitting system, and S is the scattering matrix of objects,
then the measured scattering matrix Z can be expressed
as [111], [112]
1 d h S hh S hv 1 d v
n hh n hv
Z hh Z hv
E=<
E$<
E
F$;
F+;
Z =;
dv f
Z vh Z vv
S vh S vv d h f
n vh n vv
= R·S·T + N,
(46)
where d v represents the crosstalk from the vertically polarized
channel to the horizontally polarized channel, d h represents
the crosstalk from the horizontally polarized channel to the
vertically polarized channel, and f represents the complexvalued channel imbalance (amplitude and phase) between
the H and V channels.
If we assume that the crosstalk for vertical and horizontal polarization are identical (i.e., d v = d h = d) and, in addition, that the reciprocity theorem constrains the scattering
matrix to be symmetric (i.e., S hv = S vh, Z hv = Z vh,), then the
measured scattering matrix Z in (46) can be vectorized, and
the distorted 3D Pauli scattering vector k z is given as
Z hh + Z vv
S hh + S vv
n hh + n vv
1
kv z =
>Z vv - Z hhH = Q(d, f ) $ 1 >S vv - S hhH + 1 >n vv - n hhH
2
2
2
2Z hv
2S hv
2n hv
(47)
= Q(d, f ) $ kv s + nv ,

Now, we analyze how the polarimetric distortions affect the forest RVoG scattering model in (22). According
to the expression of the scattering vector kv z, we can obtain the expression of the distorted coherency matrices
T11 ^Q 1, nv 1h, T22 ^Q 2, nv 2 h, and Ω 12 ^Q 1, Q 2 h as follows:
T11(Q 1, nv 1) = kv z, 1 $ kv Hz, 1
= Q 1 $ T11 $ Q H1 + nv 1 $ nv H1

(49)

T22 ^Q 2, nv 2h = kv z, 2 $ kv Hz, 2
= Q 2 $ T22 $ Q H2 + nv 2 $ nv H2

(50)

Ω 12 ^Q 1, Q 2 h = kv z, 1 $ kv Hz, 2 = G Q 1 $ Ω 12 $ Q H2 }.

(51)

We assume that the radar systems at opposite ends of the
baseline have the same polarimetric distortion, that is,
Q 1 = Q 2 = Q, and that the system noise is not taken into
consideration, that is, nv 1 = nv 2 = 0. Then, affected by the polarimetric distortion, the observed coherence for the forest
scattering model becomes
v , d, f ) = e^iz 0h
cu Vol(w

v , d, f )
cu V + m(w
v , d, f ) ,
1 + m(w

(52)

where cu V still represents the coherence for the volume layer. Compared with the original ground-to-volume scatterv , d, f ) rev ) in (22), the new component m(w
ing ratio m(w
v but also to the
lates not only to the polarization vector w
polarimetric distortions. Equation (52) still represents a
line on the coherence plane, the slope of which remains
unchanged. In Figure 9, we briefly plot the shift diagram
of the coherence values under the impact of polarimetric distortions. The blue points indicate the complex coherence values without distortions, and the red points
indicate the complex coherence values with distortions.
In this example, the polarimetric distortions lead to
higher ground-to-volume ratios, which causes these
points to move toward the ground phase. Conversely, if
the distortions lead to lower ground-to-volume ratios,
Im (γ )

where the distortion matrix Q(d, f ) is given by
R
V
2
2
f2 - 1
S 1 + 2d + f
W
+
f
d
d
S
W
2
2
S f2 - 1
W
1 - 2d 2 + f 2
Q (d, f ) = S
df - d W .
2
2
SS
W
d + df
df - d
f + d2 W
T
X

(48)

The distortion matrix Q(d, f ) is a function of polarization
crosstalk d and channel imbalance f, which indicates how
the distortion contaminates the true scattering vector. In
the case of perfect radar polarization, the crosstalk is d = 0,
and the channel imbalance is f = 1; then, the distortion
matrix Q (0, 1) becomes an identity matrix.
JUNE 2020

IEEE GEOSCIENCE AND REMOTE SENSING MAGAZINE

O

Re (γ )

FIGURE 9. The impact of polarization distortions on coherence

values.
95
IEEE Geoscience and Remote Sensing Magazine - June 2020

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