IEEE Geoscience and Remote Sensing Magazine - June 2020 - 94

v ) is set to 3]. Point
phase point exp^iz 0 h [when m(w
V represents the volume coherence value cu V $ exp^iz 0 h
v ) is set to 0]. A line can be fitted using the es[when m(w
timated coherence points, and it intersects the unit circle
at two points, z 1 and z 2.
◗ Stage 2-Vegetation bias removal: The ground phase zt
of each pixel can be judged from the two intersection
points by the polarization order. The coherence point
nearer the ground has a lower ground-to-volume amplitude ratio, and generally, the HV channel has a lower
ground-to-volume amplitude ratio than HH + VV or HH
- VV channels. Therefore, it is reasonable to expect that
the HV coherence will be ranked the farthest distance
away from the ground phase.
◗ Stage 3: Height and extinction estimation: After obtaining the
ground phase, the scattering model in (23) can be used to
v V) measured under volmatch the coherence value cu Vol(w
v V to obtain the forest height.
ume polarization channel w
The specific inversion process is given as [81], [82]
v V h - cu V $ exp(izt ) .
min cu Vol ^w
h V, v

(44)

This inversion algorithm considers the wave attenuation
and the vertical structure of the canopy, which has high
accuracy. Equation (44) represents minimizing the difference between the model prediction and the observation in
a least-squares sense. To solve (44), we can search the forest
height and attenuation coefficient v from a lookup table.
COHERENCE OPTIMIZATION
In the "Inversion Algorithm of PolInSAR" section, we introduced two inversion methods: DEM differencing and

Design of PolInSAR
System Parameters

PolSAR System
Parameters

InSAR System
Parameters

δ, f, n

γ SNR, γ Quant,
γ Coreg . . .

~
γ Vol(w, δ, f ) = eiφ0
~
γ + m (w, δ, f )
. V
1 + m (w, δ, f )

~
γ Vol(w, γSNR) = γSNR
~
γ + m (w)
. eiφ0 V
1 + m (w)

RVoG

RVoG

Coherence Optimization
Forest-Height
Estimation Error
FIGURE 8. A framework for the analysis of PolInSAR system

parameters.
94

three-stage inversion. For three-stage inversion, the true
volume-scattering point V usually cannot be observed
directly from the data, and the visible line is only a fraction of VG. To obtain the best height estimates, the optimum volume coherence can be solved by the coherence
optimization method. The complex coherence varies with
polarization, as shown in (16), and coherence optimization
is achieved by choosing the desired linear combination of
polarization states that yields the highest coherence. This optimization problem can be solved by maximizing the complex
Lagrangian function; it can further be transformed into
two 3 # 3 complex eigenvalue problems as [1], [108]
v 1 = vw
v 1, K 1 = T -111 Ω 12 T -221 Ω H12
K1 w
v 2 = vw
v 2, K 2 = T -221 Ω H12 T -111 Ω 12 .
K2 w

(45)

The matrices K 1 and K 2 have the same real nonnegative eigenvalues. Each eigenvalue is related to a pair of eigenvectors:
v 1,opt1, w
v 2,opt1 ,, " w
v 1,opt2, w
v 2,opt2 ,, and " w
v 1,opt3, w
v 2,opt3 , . The
"w
v
w
polarization vector opt1 represents the optimum groundv opt3 represents the
dominated scattering mechanism, and w
optimum volume-dominated scattering mechanism [7].
Finally, the obtained optimum complex coherences lead to
best height estimates when the inversion algorithm of (44)
is used. In addition to the Lagrange multiplier technique,
many other approaches can also be used to find the optimum polarization states [89], [109], [110].
The technique described is three-stage inversion combined
with coherence optimization for forest-height inversion. Actually, by substituting the optimum complex coherence into
(41), the DEM differencing can also obtain its best height
estimates. This is the DEM differencing method with optimized coherences, which also has a high inversion accuracy. Comparing (41) and (44), we can easily find that DEM
differencing has an advantage in error analysis. Therefore,
in the next forest-height inversion and PolInSAR system
parameter analysis, we use the DEM differencing method
with optimized coherence.
PolInSAR SYSTEM PARAMETER ANALYSIS
This section shows how to analyze the error sources of PolInSAR so that one can understand how PolInSAR system
parameters should be determined with the target inversion
accuracy in mind. The polarimetric system and platform
parameters are analyzed; then, the forest-height estimation
error model is established and discussed.
POLARIMETRIC SYSTEM PARAMETERS
As shown in Figure 8, polarimetric and conventional interferometric system parameters are analyzed separately.
PolSAR system parameters mainly include polarimetric
distortions-that is, the polarization crosstalk and channel imbalance (amplitude and phase)-whereas the InSAR
system parameters mainly include various kinds of decorrelation factors. All of these system parameters have impacts on the performance of radar observation and, thus,
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