IEEE Geoscience and Remote Sensing Magazine - June 2020 - 98

distortion, the larger the height estimation error becomes.
Only in the absence of system noise, that is, Th V ^nv h = 0,
does the height estimation error become 0 and the polarimetric distortion have no impact on it.
Now, we analyze the height estimation error Th V ^nv h
caused by system noise in the absence of polarimetric distortions. Here, the system noise contains the various types
of nonvolume decorrelation sources mentioned in (61),
which impacts the coherence amplitude in the same way.
To study the sensitivity of forest height to small changes in
coherence amplitude, for the estimated height in (41), we
take the first-order derivative of the coherence amplitude as
follows [7]:
2zt
2h V
2F o
1
= k e u + 2f· u
,
z 2 c Vol
2 cu Vol
2 c Vol

(66)

where the function F represents F = sin c -1 ^ cu Vol h . We
use a third-order polynomial to fit for F and take f = 0.4
for the L band. According to [7], 2zt /2 | cu Vol | can be ignored
compared to 2F/2 | cu Vol | . Through the derivative analysis
of the two terms in (66), we can derive the height estimation error Th V (nv ) caused by system noise as follows [7]:
Th V (nv ) = g ^k z, ; cu vol ;, SNR h
1
1
. A $ k $ ` 1+ SNR j $ ^8.2 $ ; cu vol ;3 - 6.9 $ ; cu vol ;2 + 3 $ ; cu vol ; h,
z
(67)
where A is an unknown constant. We can see from (67) that
the height estimation error caused by system noise relates
not only to SNR but also to the volume coherence coefficient cu vol and the vertical wavenumber k z .
Finally, considering both the impact of polarimetric
distortions and system noise together, we directly give
the theoretical error model between the forest-height estimation error and the PolInSAR system parameters as
follows [7]:
1
1
1
1
Th V ^k z, ;cu vol ;, SNR, d, ; f ; h . A $ k $ ` 1 + SNR j $
$
z
(1 - d)2 | f |2
$ ^8.2 $ ;cu vol ;3 - 6.9 $ ;cu vol ;2 + 3 $ ;cu vol ;h.
(68)

Equation (68) predicts that the height estimation error Dh V
caused by system imperfectness might be dependent on
the system parameters, including polarization crosstalk d,
polarization channel imbalance f, system noise SNR, and
the volume coherence cu vol and vertical wavenumber k z .
Among the five system parameters, the unit k z is given in
meters-1, and the other four parameters are unitless. The unknown constant A is empirically determined. Details can be
found in [7], and a brief description is given here. We denote
the observed estimation error as y i, and the corresponding
theoretical height errors in (67) are denoted by Ax i . The solution of A can be regarded as a linear fitting problem of y i and
x i . The residual norm is given by the l 2 norm [7]:
98

l 2 = Ax - y

2

= / ^ Ax i - y ih2.

(69)

i

By minimizing the residual norm, we get
2l 2
2 ^ Ax i - y ih x i = 0.
2A = /
i

(70)

The optimal value A yields
A=

/ i xi yi
/ i x 2i .

(71)

After experimenting with real SAR data from temperate
forests in southern Germany, we determined the unknown
constant to be A = 3.67. This airborne SAR data over the
Traunstein forest area was acquired by DLR's E-SAR system
[121], [122]. The theoretical relationship has been validated
on both simulated and real airborne SAR data. The numerical results match well with the theoretical errors, which
verifies the proposed theoretical model. With this errordependence model, we can determine suitable PolInSAR
system parameters according to the requirements of forestheight mapping.
Equation (68) shows the error-dependence model of
forest-height estimation, which can be used to analyze
PolInSAR system parameters. In [7], we provide some error maps of forest-height estimation as a function of system
parameters. For the system parameters in (68), the vertical
wavenumber k z is the key parameter for forest-height inversion, determining the available height range possible to invert and scaling the height inversion sensitivity. Therefore,
choosing an appropriate k z can avoid overestimating or
underestimating tree height. From [123], we know that the
appropriate range for k z is 0.05-0.15. It is reasonable to set
the wavenumber to k z = 0.1 rad/m. The volume coherence
coefficient cu vol depends on the forest structures. For forest scatterers, the SNR can usually reach above 15 dB. For a
typical SNR and coherence coefficient, we plot the contour
maps of the height estimation error, varying with polarization crosstalk d and amplitude imbalance ; f ; in Figures 11
and 12, respectively.
The contour maps in Figures 11 and 12 indicate how the
expected forest-height estimation error varies with PolInSAR
system parameters. We can see that larger volume coherence values cause larger height estimation error, which corresponds to lower tree height. By comparing the error maps
under different SNRs, we can see that increasing the SNR
can effectively reduce the forest-height inversion error. For
a specific SNR value, reducing the polarization crosstalk
and channel imbalance can also reduce the forest-height
estimation error. If the SNR is sufficiently large, the height
estimation error approaches zero, and the polarimetric distortions have no impact on the inversion result.
The error budget shown in Figures 11 and 12 is just
one part of the final estimation error. For PolInSAR forestheight mapping, the estimation error mainly comes from
two sources: RVoG model error and radar measurement
IEEE GEOSCIENCE AND REMOTE SENSING MAGAZINE

JUNE 2020
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