IEEE Geoscience and Remote Sensing Magazine - June 2020 - 20

with their very small wavelengths (approximately 3.1 cm),
allow measurement of small surface displacements, even
those caused by thermal dilation of the imaged targets [17],
[92]. The Sentinel-1 A/B constellation, characterized by a
high temporal resolution (a revisiting cycle of six days) and
wide-area coverage, is well suited for wide-area monitoring.
However, when InSAR techniques are applied to urban
areas, the interpretation of phenomena occurring at the
local scale or the level of a single structure can be difficult
due to the layover effect. The presence of multiple scatterers interfering in the same resolution cell may impair the
method's performance, and misdetections of persistent
scatterers and inaccuracies in height-velocity and timeseries measurements may occur [93], [94]. This potential
limitation can be overcome by using differential TomoSAR
techniques, which integrate the TomoSAR and differential
interferometry concepts [24]. Differential tomography allows estimation of, in addition to the elevation, the average
deformation velocity, producing a 4D reconstruction (3D
plus velocity) [95], or even the
MONITORING
thermal dilation of the scatAPPLICATIONS REQUIRE
terers, which leads to a 5D reTHE IDENTIFICATION OF A
construction (3D plus velocity
FEW RELIABLE CONTROL
plus temperature) [25], [96].
POINTS PROVIDING
Monitoring applications
require
the identification of a
ACCURATE ESTIMATIONS
few
reliable
control points proOF THE STRUCTURE
viding
accurate
estimations of
DEFORMATIONS.
the structure deformations.
Then, the development of appropriate techniques capable
of detecting reliable scatterers, unmixing their signals, and
separating them from noise and clutter is of fundamental
importance for these applications. Standard PSI techniques
detect single scatterers by applying a threshold on a decision variable, obtained by matching the observed phase
values to an extended phase model. TomoSAR techniques
extended to the 4D and 5D cases rely on the same extended phase model used in PSI, but scatterer detection is
performed by comparing fully focused scattering distribution in the 4D and 5D spaces with a proper threshold. The
detection process, then, benefits the gain of the aperture
synthesis in the elevation direction, which increases the
SNR and signal-to-clutter ratio of the coherent scatterers
matching the adopted phase model.
The extended phase model, which takes into account
temporal deformation and thermal dilation for a scatterer
located on the structure at elevation h, at acquisition time
t n and temperature Tn, is typically assumed to be composed
of linear and nonlinear contributions [96]:
d (h, t n, Tn) = n defo t n + k th Tn + d NL (h, t n, Tn),

(20)

where n defo is the velocity of the slow linear temporal deformation, having the dimension of m/s, and k th is a thermal
20

coefficient, having the dimension of m/°C, representing the
phase-to temperature sensitivity and depending on the material and/or physical structure. The term d NL (h, t n, Tn) is the
nonlinear contribution and, in general, is not considered.
The N # M sensing matrix A now depends on (h, n defo, k th),
where n defo ! R Nv collects the N v samples along the velocity dimension, k th ! R Nk collects the N k samples along
the thermal dilation dimension, and M = N h # N v # N k .
Introducing the velocity and thermal angular frequencies,
h n = 4rt n m, and g n = 4rTn m, respectively, and the corresponding vectors h, g ! R N, the associated steering vector a
of size N # 1, for a fixed triplet of parameters (h k, n defo j, k th l),
can be defined as
a (h k, n defo j, k th l) = exp [- j (ph k + hn defo j + gk th l)].

(21)

The extended TomoSAR signal model can be written as
v = A (h, n defo, k th) u + e,

(22)

where matrix A (h, n defo, k th) is the concatenation of the extended steering vectors defined by (21).
The Fast-Sup-GLRT presented in the "Scatterer Extraction" section can be applied by following the same procedure; the only difference is that, in the generic hypothesis
t , k th)
H m, with m = 1, ..., K M, m triplets of parameters (ht , n defo
must be estimated by properly selecting m columns of the
matrix A (h, n defo, k th) . Each position of the selected columns corresponds to a parameter triplet value.
To show the thermal dilation effect on a tall building
in an urban area, a Paris skyscraper, 100 m tall, is considered. The results obtained with Fast-Sup-GLRT are shown
in "A 4D/5D Analysis of Scatterers" for two cases: when
no thermal dilation is considered (3D case) and when
it is taken into account (4D case). Figure S16(a)-(c) in
"A 4D/5D Analysis of Scatterers" shows the point clouds
obtained with, respectively, 3D Fast-Sup-GLRT and colorized by height, 4D Fast-Sup-GLRT (height plus thermal dilation) and colorized by height, and 4-D FastSup-GLRT and colorized by thermal dilation. The thermal map shows values that roughly range from 0 to
0.5 mm/°C, compatible with a coefficient of thermal expansion of 9.8 # 10-6 /°C of a concrete structure of 100 m.
In the 4D reconstruction, a better structure description
can be observed with an increment of approximately 5%
detected points.
The Barcelona data set has been considered to illustrate
the results of a 5D reconstruction on a wide area. (The
intensity SAR image is shown in Figure S17 in "A 4D/5D
Analysis of Scatterers.") In the area, the subsidence caused
by water abstraction and an uplift probably related to the
cessation of water pumping are expected, as reported in
[97]. Figure S17(b)-(d) in "A 4D/5D Analysis of Scatterers"
shows the height, deformation, and thermal dilation maps,
respectively. As expected, in agreement with what was
found on the same data set in [97], there is an area, near the
IEEE GEOSCIENCE AND REMOTE SENSING MAGAZINE

JUNE 2020

IEEE Geoscience and Remote Sensing Magazine - June 2020

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