IEEE Geoscience and Remote Sensing Magazine - June 2020 - 33

TOMOSAR SIGNAL MODEL
Let us refer to the geometry of the MB SAR imaging system, where N SAR images along different orbits, not
necessarily coplanar and uniformly spaced, are acquired
across the same target area. The classical focusing algorithm is applied to obtain 2D, full-resolution, single-look
complex (SLC) SAR images from raw data in each track.
Let's assume that the SLC images are co-registered with
subpixel accuracy to the reference master image. After the
co-registration step, the information associated to a single
target appears in the same pixel in all of the images. The
co-registration step usually follows the selection of an image (the master) of the stack to set the reference geometry.
Hence, for a fixed pixel (with x and r coordinates), the generic model of the nth element in the data-stack vector,
y ! C N # 1, is given by [6], [10]
y n ^ x, r h =

#- Tg c^x, r, ghe
Tg

4r
6t ^ x, r, g h - t n ^ x, r, 0 h@
-i
m n

dg + w n ,

L

/ c^ z lha^ z lh + w,

(2)

l=1

where the dependence of the data-stack vector y on (x,r)
has been neglected for the sake of notation simplicity. The
steering vector a (z l) = [e ik z1 (zl) e ik z2 (zl) .... e ik zN (zl)] T , with T as
transpose operator, is constructed by the two-way vertical
wavenumber (k zn) between the master and nth acquisition
tracks, defined as
k zn =

bn
4r
,
m r sin (i)

(3)

where c and b n denote the system wavelength and perpendicular baseline of the nth image with respect to the master.
Let c = [c (z 1) c (z 2) f c (z L)] T ! C L # 1 be the unknown discrete function that contains L uniformly spaced, unknown
samples; (2) can be written in the linear form as
y = A ^ z h c + w,

(4)

where A ! C N # L is a steering matrix that contains L steering vectors. Several inversion techniques exist to retrieve
the unknown reflectivity function c in (4); a detailed overview of these approaches is presented in the next section.
We note that the complex additive-noise vector w in (4)
JUNE 2020

IEEE GEOSCIENCE AND REMOTE SENSING MAGAZINE

mr
dg = 2B .
n

(5)

(1)

where t n (x, r, g) represents the distance between the scatterer located at position (x, r, g) and the nth sensor. The exponential factor highlights the so-called de-ramping compensation, 2Tg expresses the extension of the cross range
in the imaged scene, c represents the scene's scattering
properties, and w n represents the complex additive noise.
After expanding the difference t k (g) - t k (0), associating the g-dependent term in c, and moving from the cross
range to the elevation in accordance with z = g sin (i), the
discretized version of the TomoSAR equation in correspondence to any (x, r, z) coordinate can be derived by sampling
in L steps as
y=

follows a Gaussian distribution with zero mean and a standard deviation of v w.
Some remarks are now in order. The factor 2Tg in (1)
and consequently in (4) plays a key role in the inversion process. To identify the reconstructed signal, the solution space
should be restricted to the range of the height of ambiguity
given by the Nyquist interval, that is, 2Tg = mr/2br n, where
br n is the mean baseline separation between the acquisitions.
It is worth highlighting that the ideal discrete version of (1) is
obtained by uniformly sampling g through the illuminated
scene's cross-range extension, with the sampling step being
lower than the Nyquist-resolution value. In particular, the
cross-range Nyquist resolution dg can be derived based on
the overall length of the orthogonal baseline (B n), given by

RECONSTRUCTION APPROACHES
The purpose of the tomographic-reconstruction techniques
presented in this section is to estimate c from the measurements y or, rather, the vertical backscatter intensity c 2.
Under the hypothesis of uncorrelation along z of the vertical2
reflectivity profile (that is, E {c (z) c (s) @} = d (z - s) c (z) ),
the second-order statistics of the MB measurements y can
be expressed as
R = E " yy @ , = A ^ z h R c A ^ z h@ + v 2w I ! C N # N,

(6)

where R and R c = diag {| c |2} represent the covariance matrices of y and c, respectively, and I is the identity matrix.
In the equations presented so far, E, d, @, and diag {.} denote the expectation, delta function, Hermitian operators,
and diagonal operators, respectively.
Equation (6) states that each element of the covariance
matrix of the random vector y represents an estimation
of the Fourier transform of the vertical reflectivity profile
| c |2. This feature implies that the problem of estimating the
scene's 3D structure amounts to a spectral-estimation problem. During the past decades, many algorithms have been
developed for estimating the spectrum of a signal; some of
them are described in this section, with considerations of
the physical assumptions they rely on.
Spectral-estimation algorithms can be divided into
two main groups: model-free (nonparametric) and modelbased (parametric). Model-based algorithms have significant features that can describe the forest layer. Conversely,
model-free approaches do not rely on any a priori knowledge of the vegetation layer and can result in a more flexible reconstruction. The vertical structure of a vegetation
layer is often described in terms of the vertical distribution
of the ensembles' backscattering, which can be classified
based on their polarimetric signature. Most models include
a mixture of contributions from the ground and canopy
[33]. Generally speaking, the former is highly concentrated
along the vertical direction (a Dirac delta, in the limit case
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IEEE Geoscience and Remote Sensing Magazine - June 2020

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