IEEE Geoscience and Remote Sensing Magazine - June 2020 - 38

approach is the sum of Kronecker products (SKP) [17],
which provides the basis for the decomposition of the MB
data from a forested area along the elevation direction.
The method models the polarimetric radar backscattering
from the ground and canopy layers in forested areas. Suppose that the polarimetric MB covariance matrix (R p) is
estimated from the generated lexicographic target vector,
y p; the method then expresses R p by the KP between two
matrices, which is obtained by singular-value decomposition; that is,
Rp =

/ Ru j 7 Cu j,

(22)

j=1

u j ! C 3 # 3 and Ru j ! C N # N account for the correlation
where C
among different polarizations and baselines, respectively.
u j and Ru j are related to
It can be shown that the matrices C
the polarimetric covariance matrix, (C g, C v), and interferometric structure matrix, (R g, R v), of the ground and aboveground volumetric scattering from the canopy via a linear
invertible transformation as
R g = aRu 1 + ^1 - a h Ru 2
R v = bRu 1 + (1 - b) Ru 2
1
u 1 - bC
u 2h
C g = a - b ^^1 - b h C
1
u 1 + aC
u 2 h,
C v = a - b ^- ^1 - ahC

(23)

where a and b are two real numbers obtained by requiring the semipositive definitiveness condition in the previous linear equations. The polarimetric matrices of C g and
C v represent information from the backscattering mechanisms about the ground and volume, respectively, while
matrices R g and R v contain interferometric information
that can be employed for height mapping and the reconstruction of the vertical structure. The results of SKP decomposition using BioSAR 2007 data [34] are shown in
Figure 5. The tomograms are reconstructed using the
decomposed R g and R v.

INHERENT DIFFICULTIES OF TOMOSAR
The principle of TomoSAR is similar to that of X-ray computed tomography (CT). However, compared with CT, the
application of TomoSAR is limited by some additional
problems. First, while a sufficient number of uniformly
distributed sensors is available in CT, the number of MB
SAR data sources is generally limited in TomoSAR, with
only a relatively low number of available passes, which
do not necessarily provide regular sampling. Moreover,
MB TomoSAR data acquisitions may span a long interval,
which could cause an unavoidable temporal decorrelation
and the appearance of disturbances due to atmospheric
variations. Furthermore, the focusing procedure given
with the TomoSAR equation in (4) might be affected by a
phase error because of the real tracks' deviation from the
nominal positions. These issues can impair focusing and
3D-information extraction. They and their possible effects
must be considered when MB SAR data are processed.
More details of these issues and the possible remedies are
discussed next.
NONOPTIMAL MB CONFIGURATION
In CT applications, a narrow X-ray beam is transmitted
for a set of angle positions that are 360º around the object; in TomoSAR, under realistic conditions, acquiring the
equally spaced baselines may not be feasible. Moreover, due
to some practical limitations, only a relatively small number of passes and trajectories is available. In practice, and
more specifically in the spaceborne case, the MB data are
obtained through irregular sampling and a small number
of baselines. This problem introduces a degree of ill-conditioning into the inversion process, and it may lead to ambiguities in the reconstruction [10].
Generally, the MB data-sampling distance must be sufficiently small to fulfill the Nyquist criterion for the spatial
bandwidth of y; otherwise, as in the case of azimuth focusing, the occurrence of sidelobes and ambiguities in the
reconstruction is inevitable. The optimal sampling distance
(d opt) between tracks depends on the slant range and target
height (H) and can be represented as [6]
mr
d opt # H .

(a)

(b)
FIGURE 5. The Capon-based tomograms of (a) the ground and
(b) the volume obtained by SKP decomposition. The horizontal
and vertical axes of the tomograms are related to the slant range
and elevation directions.

(24)

In such a case, the total number of required baselines might
be obtained by dividing the overall baseline, B n, by the optimal sampling distance. The TomoSAR-achievable vertical
resolution is highly dependent on the extension of the tomographic aperture (B n). Although it may not always be possible
for TomoSAR users to design the optimal MB configuration
for the sampling, resolution, and number of images, some
feasible strategies might be useful to mitigate a nonoptimal
configuration's blurring and sidelobe effects. The baseline
interpolation [21], in which a set of uniform baseline data
is recovered from the available nonuniform baseline, may
enhance the reconstruction quality for typical approaches,
such as beamforming and the Capon spectral estimator.
IEEE GEOSCIENCE AND REMOTE SENSING MAGAZINE

JUNE 2020

IEEE Geoscience and Remote Sensing Magazine - June 2020

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