IEEE Geoscience and Remote Sensing Magazine - June 2020 - 34

of infinite bandwidth), whereas the latter spreads throughout a larger range of elevations with more polarimetric
randomness. However, the exact features of each scattering
mechanism and their relative weights depend on many factors, including the tree species, moisture, and wavelength.
Hence, to capture the physical phenomenon, a large set of
parameters is needed, which makes the model difficult to
invert. Because of these issues, model-free approaches are
usually preferred when dealing with forests; two widely
used algorithms are detailed throughout the remainder of
this section: the beamforming algorithm and the Capon
spectral estimator [12].
The beamforming and the Capon methods aim to find
a filter vector h to project the MB data y such that the reflectivity from a specific scatterer situated at elevation z l is
recovered, while the echoes of the scatterers placed at different elevations are rejected. The beamforming algorithm
achieves this goal by constraining the minimum norm of h
mathematically
h BF = min h @ h s.t. h @ a ^ z l h = 1.
h

(7)

The solution to this problem is given by h BF = a (z l)/N,
where the filtered signal's retrieved power is given by
PBF ^ z l h =

a @ ^ z l h Ra ^ z l h
.
N2

(8)

The resolution along the elevation of the beamforming
algorithm is limited and connected to the baseline distribution; also, sidelobes in the vertical impulse-response
function (IRF) inject backscattered power from far-away
elevations. The vertical IRF can be computed in a closed
form through ; h BF ; 2. In the ideal case of regularly spaced
baselines, ; h BF ; 2 is a squared periodic-sinc function, its first
zero being placed at 2r/k z max; the period of ; h BF ; 2 is given
by 2r/k z min. The latter is the tomographic equivalent of the
height of ambiguity for the interferometric case.
The large main lobe of ; h BF ; 2 and slow decay of its response make beamforming unsuitable when the resolution
along the elevation has priority. In this case, a different
nonparametric model may be used: the Capon spectral estimator. The measured MB vector y is correlated to a signal
that has a nonidentity covariance matrix, as shown in (6).
Hence, the filter-design problem in (7) is revised as:
h CP = min h @ Rh s.t. h @ a ^ z l h = 1.
h

(9)

The resulting Capon filter and corresponding intensity are
computed according to the following equations:

and

h CP = 6a @ ^ z l h R -1 a ^ z lh@-1 R -1 a ^ z lh,
PCP ^ z l h = 6a @ (z l) R -1 a ^ z lh@-1.

(10)
(11)

The expression in (11) is also known as the Capon verticalpower spectrum. The inversion in (11) is responsible for sharp
34

peaks in the correspondence of the elevations where targets are present. However, the MB covariance matrix R is
not known and must be estimated from the data itself. This
goal is usually achieved by hypothesizing the ergodicity
and spatial stationarity, but the limited number of looks
can lead to instable inversions; this may be regularized by
considering large estimation windows or introducing an
extra noise term to the estimated R to avoid very small eigenvalues. The same considerations about the periodicity
of the beamforming spectrum apply to h CP as well; in fact,
specifying the resolution is more difficult, as it depends on
the signal-to-noise ratio (SNR). Despite this uncertainty,
the Capon estimator may still be preferred. The limit case is
represented by the European Space Agency's BioSAR 2008
data set [34], where the resolution of the beamforming algorithm was often larger than the forest layer itself. Even
in this extreme scenario, the Capon technique provided
excellent 3D reconstruction [30]. Whenever radiometric
accuracy is the main concern, the beamforming algorithm
should be used instead. Many works relating tomographic
backscattering intensity to forest biomass were based on
beamforming, with the baseline distribution enabling
proper imaging (an approximately 15-m vertical resolution
for the TropiSAR data set [35]).
The two methods mentioned here rely on the linearization of the interferometric phase with respect to the
elevation. This approximation is good for short baselines and as long as the vertical displacement from the
linearization point is small compared to the distance of
the target. However, the airborne-survey geometry of acquisition often makes this approximation unsuitable. In
such cases, the back-projection tomographic algorithm
[13] may be used in place of the beamforming technique.
The back-projection method estimates the reflectivity of
any point in the 3D space by compensating for the signal's delay (the time of arrival and phase value) for all
available sensors. The use of the exact distances does not
imply any linearization and produces clearer reconstructions. A further advantage of this framework involves the
possibility of taking different propagation velocities into
account, such as properly imaging targets that are buried
in snow and ice [36].
The vertical power spectra of the vegetation layer that
are returned by beamforming or the Capon technique reveal the 3D structure of the forest and underlying topography. Figure 2 provides an example of 3D reconstructions
using the AfriSAR P band and UAVSAR L band above La
Lopé, Gabon. In particular, the height-range tomograms
are reported in HH polarization for the specified range line.
Details about polarizations will be provided in the next section. Theoretically, the ground reflectivity in the tomogram
is particularly visible when using low-frequency waves and
in the case of sparse forests. Otherwise, the vegetation's
attenuation can extinguish the signal beforehand; in this
case, the peak associated with the ground in the tomogram
loses power and may fade away. This is particularly evident
IEEE GEOSCIENCE AND REMOTE SENSING MAGAZINE

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IEEE Geoscience and Remote Sensing Magazine - June 2020

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