# Multichannel Along-Track Interferometric SAR Systems: Moving Targets Detection and Velocity Estimation

Authors: Alessandra Budillon; Vito Pascazio; Gilda Schirinzi
Journal: International Journal of Navigation and Observation
 (2008)

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## Abstract

Along-track interferometric synthetic aperture radar (AT-InSAR) systems are used to estimate the radial velocity of targets moving on the ground, starting from the interferometric phases, obtained by the combinations of two complex SAR images acquired by two antennas spatially separated along the platform moving direction. Since the radial velocity estimation obtained from a single-phase interferogram (single-channel) suffers from ambiguities, multichannel AT-InSAR systems using more than one interferogram can be used. In this paper, we first analyze the moving target detection problem, evaluating the systems performance in terms of probability of detection and probability of false alarm obtained with different values of target radial velocity, signal-to-clutter ratio, and clutter-to-thermal noise ratio. Then, we analyze the radial velocity estimation accuracy in terms of Cramer-Rao
lower bounds and of mean square error values, obtained by using a maximum likelihood estimation technique. We consider the cases of single-baseline and dual-baseline satellite systems, and we evaluate the detection and estimation performance improvement obtained in the dual-baseline case with respect to the single-baseline one. Sensitivity of the presented method with respect to the involved target and system parameters is also discussed.


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## Body

## 1. Introduction

In this paper,  we review the problem of detecting
the presence of a ground moving
target and estimating its radial velocity by means of along-track
inteferometric synthetic aperture radar (AT-InSAR) systems, mounted on moving
platforms. This kind
of systems can be used, for example, for
continuous (day and night and with any
weather condition) traffic monitoring [1, 2].The detection of  moving
targets on the ground by means of radar systems is addressed in literature as
ground moving target indication (GMTI). GMTI is a very
difficult problem due to the difficulty of separating the signal returned from
a moving target from the stationary background (clutter) [3, 4]. Several
methods, based on very different approaches, have been proposed in literature.
In some of them, radar detection of moving targets on the ground is
accomplished by enhancing the target Doppler signature against the competing
ground clutter returns. Recent clutter suppression techniques use space-time
adaptive processing (STAP) [5–7], requiring more than two channels, and
time-frequency processing [8], requiring high pulse repetition frequency (PRF)
values. While these techniques are effective in improving the detection of fast
targets, for slowly moving targets the signal from clutter separation is more
critical. In particular, clutter reduction becomes more critical when the
Doppler frequency shift due to the target radial velocity falls inside the
clutter azimuth bandwidth. Since such bandwidth increases with the ratio
between the platform velocity and the azimuth antenna dimension, which in
satellite-borne case is high, the range of target velocities values which are
critical to be estimated can be wide in the case of satellite systems.
Moreover, spectral separation requires increased PRF values, which are not
desirable for the very high
data rates and PRF ambiguity problems [9].Other systems that
can be used to detect ground moving target are the AT-InSAR systems, initially
introduced to study ocean currents [10, 11], and then used to detect slow moving
objects (ships, ground vehicles) [12–15] and to estimate their radial velocity.
AT-InSAR systems use more than one SAR antenna (typically two), mounted on the
same platform and displaced along the platform moving direction. The information about the radial
velocity of the moving target is estimated from the interferometric phase of
the images.The accuracy obtained for the estimation of ocean
currents velocity using airborne AT-InSAR sensors can be of the order of few
centimeters per second [10]. These very satisfactory results can be obtained
since the two images acquired with a negligible time delay are very highly
correlated and all the scatterers within a resolution cell move with the same
velocity. In this case, the moving target is an extended one (the sea surface),
while the stationary clutter is absent. The only disturbing signal to be
considered is the additive thermal noise.Differently from the case of ocean currents
estimation, when AT-InSAR techniques are applied to the detection and tracking
of small targets, such as vehicles, the presence of stationary clutter has to
be considered. This heavily affects the interferometric phase values and their
statistical distribution, thus degrading the performance of the moving target
detection and of the radial velocity estimation.For the application of statistical
techniques to the detection and estimation steps, it is necessary to compute the
statistical distribution of the measured interferometric phases. In the
stationary image pixels, the interferometric phase reduces to only phase noise,
whose statistical distribution is well known and depends on the interferometric
signals correlation (the coherence) [16]. In the image pixels where a moving
target is present, the phase statistical distribution diverges from that of
stationary pixels, and strongly depends on the target velocity and on the
statistical model assumed for the target radar cross-section (RCS). The higher
the velocity, the larger the deviation, with
respect to the stationary case. Different
models can be assumed for the radar response of the target. In the following we
will adopt two different RCS models: a deterministic model [13], and a zero
mean Gaussian model [12], underlying differences and analogies.Another problem to be taken into account is that the
interferometric phase is measured in the interval(−π,π], then a phase unwrapping (PhU)
operation is required to retrieve the target radial velocity. The PhU
operation presents
solution ambiguities when only one phase interferogram (single-channel) is
used. It has already been shown in [13, 15] that the joint use of multichannel
configurations (derived from
the use of more than two interferometric images acquired with different
baselines or at different working frequencies) and of classical statistical
estimation techniques allows to obtain very accurate solutions and to overcome
the limitations due to the presence of ambiguous solutions, intrinsic in the
single-channel configurations.In this paper, we show that AT-InSAR systems based on
the use of more interferograms (multichannel) acquired with frequency or
baseline diversity outperform conventional AT-InSAR systems using a single
interferogram. In particular, we show that even a dual-baseline system allows achieving
detection and estimation performance much better than the one obtained by a
single-baseline system. The results obtained in the estimation process are
partially a review of what we have presented in [15]. The analysis, in terms of
moving target detection and radial velocity estimation accuracy, is carried out
varying the main AT-InSAR system and target parameters, such as velocity
values, signal-to-clutter ratio (SCR), defined as the ratio between the power received
from the moving target and the background clutter power, and clutter-to-noise
ratio (CNR), defined as the ratio between the power received from the
background clutter and the additive thermal noise power in the receiver. The
performances presented in the following sections have been obtained considering
two RCS models: deterministic and Gaussian. Finally, a discussion about
robustness of the proposed model with respect to uncertainty on system parameters
has been also included.


## 2. Along-Track Interferometric Sar Systems

Consider an AT-InSAR system constituted by two
antennas moving along the directionx (azimuth) (see Figure 1), and suppose that the two antennas are separated by the
baseline b along the azimuth
direction x, such that b≪H, where H is the platform height. Assume a target on the ground moving with
a constant velocity vT=vTxx+vTrr, where vTx and vTr are the velocity components along the
azimuth and the line of sight direction (range)
r, respectively.Figure 1
Along-track
interferometry system geometry.Both azimuth and range velocity components change the
Doppler history of the moving target (but in different ways) in comparison to
the stationary background. In order to show this behavior, we have simulated an
SAR image of a stationary target shown in Figure2(a) (the horizontal axis
represents azimuth and the vertical axis represents range). In
Figures 2(b) and
2(c), the images of a target moving with only radial velocity and with only azimuth
velocity are shown, respectively. Finally, the image in
Figure 2(d) is related
to a target moving with both radial and azimuth velocity components. We can
observe that the radial velocity component vTr produces an azimuth displacement of the target, due to a Doppler offset (see
Figures 2(b) and
2(d)). The azimuth velocity component vTx, instead, produces a Doppler slope change [3, 5]
causing a defocusing in the moving target image (see
Figures 2(c) and
2(d))
that can be compensated by using autofocusing techniques [17]. These effects
can be exploited to separate moving target from stationary background by means
of Doppler filtering [4]. However, Doppler filtering is effective for fast
moving targets and requires PRF values much higher than the Doppler bandwidth
for making available a certain visibility region in the frequency domain and to
achieve a consistent clutter reduction.(a) Stationary target, (b) target moving with radial
velocity (vTr=27.4Km/h), (c) target moving with
azimuth velocity (vTx=−109.6Km/h), and (d) target moving with both radial and azimuth velocities (vT=vTxx+vTrr).
(a)(b)(c)(d)To avoid excessive data rates and the PRF ambiguity
problem [9], it is desirable to work with low PRF values. In this case, along-track
interferometric systems allow the radial velocity estimation exploiting phase
information.Suppose that|vTx|,|vTr|≪|vP|, where vP=vPx is the velocity of the flying platform and H≫X and H≫W, where X and W are the antenna footprint dimensions. Let Z1 the complex SAR image be acquired
by the first antenna and let Z2 the complex SAR image of the
same ground region be acquired by the second antenna (we
understand the dependence on the pixel coordinates).The two SAR complex images can be modeled as follows:(1)Z1={Zc1+N1+ZT1inpresenceofmovingtarget,Zc1+N1inabsenceofmovingtarget,Z2={Zc2+N2+ZT2inpresenceofmovingtarget,Zc2+N2inabsenceofmovingtarget, where Zc1 and Zc2 are the
clutter signals acquired by the two antennas, N1 and N2 represent the thermal noise at the receivers, and ZT1 and ZT2 denote the SAR images of the moving target
produced by the two interferometric antennas. The SAR target images will
exhibit a phase factor related to the radial velocity [10]: (2)ZT1=A1,ZT2=A2e−jϕv, where A1 and A2 are the target complex images and ϕv is the nominal ATI phase which in the above-mentioned
assumptions is given by (3)ϕv=〈4πbλvr|vP|〉2π=〈4πbλur〉2π, where 〈⋅〉2π is the modulo 2π operation, λ is the wavelength corresponding to the working
frequency f=c/λ of the SAR system, and ur=vr/|vP| is the normalized radial velocity, where the moving
target is present vr≠0 and ϕv≠0.
From (3), it follows  that there
are several velocity values which produce the
same nominal ATI phase. The difference between two velocity values that produce the same nominal ATI phase is  2kur,amb,
where k is an integer and ur,amb=±λ/(4b) is the ambiguity velocity value corresponding to a nominal
ATI phase
equal to ±π.The SAR
interferometric phase signalΦ is (4)Φ=∠Z1Z2*,  where ∠ denotes the principal phase value and * denotes the conjugate.Note that the measured interferometric
phaseϕ differs from the nominal ATI phase ϕv due to the presence of clutter and noise
signals.It is well known that the SAR clutter signalsZc1 and Zc2 can be assumed as random processes, whose real and
imaginary parts are mutually uncorrelated Gaussian signals, with zero mean and
same variance, since they are resulting from the superposition of the signals
backscattered from many scattering centers lying in the resolution cell. N1 and N2 can be modeled as two additive (to the clutter) zero
mean Gaussian complex processes independent of each other, and independent on
the clutter.Then, when the moving target is absent, the two
processesZ1 and Z2 are Gaussian with zero
mean and correlation coefficient γ  given by [18] (5)γ=E[(Z1−E[Z1])(Z2−E[Z2])*]var[Z1]var[Z2]=E[(Zc1+N1)(Zc2+N2)*]E[|Zc1+N1|2]E[|Zc2+N2|2]=γc(1+σn2/σc2)=γc(1+1/CNR), where E[] denotes the expectation operation, γc  is the clutter coherence, representing the correlation between images Zc1 and Zc2, and CNR is given by (6)CNR=σc2σn2, where
2σc2 and 2σn2 are the clutter and thermal noise powers (the factor 2 is due to the sum of the
powers of the real and imaginary parts).In ATI-InSAR space applications,γc is usually
assumed to be equal to
one [12], since the two images are acquired from the same
antenna position with a time lag lower than 1 millisecond. In the case of
bistatic systems, as next generation satellite clusters, clutter
coherence can, instead, assume values smaller than 1. Moreover, a parasitic
cross-track baseline may introduce a height-induced interferometric phase that
needs to be taken into account. Anyway, it can be partly compensated by
exploiting a priori DEM knowledge.In absence of targets, the pdf of the interferometric
phase can be expressed in closed form as [16] (7)fΦ(ϕ;ϕ0,γ)=12π1−|γ|21−|γ|2cos2(ϕ−ϕ0)×{1+|γ|cos(ϕ−ϕ0)cos−1[−|γ|cos(ϕ−ϕ0)][1−|γ|2cos2(ϕ−ϕ0)]1/2},ϕ∈(−π,π], where ϕ0  is the phase of γ, that in this case is equal to zero,
being the coherence given by
(5) real-valued
since the real and imaginary parts
of the clutter signal are uncorrelated.When the moving target is present, two different statistical models
forZT1 and ZT2 can
be considered as follows:(1)
deterministic model: the target
RCS is assumed to be
deterministic;(2)
Gaussian model:  the target RCS is
assumed to be Gaussian
distributed with zero mean.


### 2.1. Statistical Distribution of at-Insar Phase for Deterministic Rcs

A deterministic model is applicable to the
case of a target whose RCS can be expressed by a deterministic function of the
incidence angle. This model applies to canonical scattering objects (such as
corner reflectors, spheres), and to complex or extended targets whose RCS does
not rapidly change between the interferometric acquisitions. Since the RCS
value of a given target mainly depends on incidence angle and target aspect
angle, which does not change in the small time required to the SAR antenna to cover
the baseline length, the target RCS can be usually assumed to be constant in
the interferometric images. Such value influences the signal-to-clutter ratio,
and is not a priori known. This is the model to be used in the interferograms
simulation. It can be adopted also in the velocity estimation procedure if a
precise knowledge of the RCS value is available. An accurate knowledge of the
average RCS values can be available only for accurately characterized targets
[19]. Moreover, this case provides the reference pdfs of the interferometric
phases produced by a given complex target of known RCS. In this assumption, we
can put in (2)A1=A2=A, with A a
deterministic constant. Then, the two processes Z1 and Z2 are Gaussian, with nonzero mean (due to the presence of the target) and the
target RCS can be described in terms of |A|2.The first-order
probability density function (pdf) of the interferometric phaseΦ  can be, in this case, numerically computed via
Monte Carlo techniques, since no closed form, so far, has been found. The pdf depends on the clutter coherence coefficient γc, on the target radial velocity
(as shown by
(2) and
(3))
and on CNR and SCR, where CNR is given by
(6) and SCR is given by (8)SCR=|A|22σc2, where |A|2 is the
signal power. In this case, the SCR values do not affect the coherence γ between the signals Z1 and Z2,
which is still expressed by
(5), but affect the shape of the pdf, which cannot
be expressed by (7). Figure 3 shows the dependence of the pdf fΦ(ϕ) on SCR, CNR, and ur in the case of a deterministic target, evaluated for γc=1  and using the TerraSAR-X parameters [20] of Table 1. In
Figure 3(a), the pdf shape is reported for the values CNR=10dB and
SCR=0,10, and 20 dB and a radial normalized velocity ur=3.25×10−3, corresponding to the nominal ATI
phase value ϕv=π/2.
Figure 3(b) is related to the values
SCR=10dB and CNR=0,10, and 20 dB and to the
same value of ur. In
Figure 3(c), the pdfs are plotted for
SCR=10dB and CNR=10dB and by
varying the normalized
velocity ur.Table 1
Main parameters of TerraSAR-X system.
TerraSAR-XHeight514.8
KmPlatform
velocity7.6
Km/sAlong-track
antenna dimension4.8 mAcross-track
antenna dimension0.8 mAlong-track
baseline1.2 mWorking
frequencyX band—9.65 GHzWavelength3.12 cmRange bandwidth150 MHz(a) Pdf of
the interferometric phase in presence of a deterministic target moving with a
radial velocity such thatϕv=π/2   (the dot on the ϕ axis), for γc=1,
CNR=10dB, and SCR=0,10,20dB, (b)
SCR=10dB and
CNR=0,10,20dB, and (c) SCR=10dB,
CNR=10dB, for seven radial
velocities (urk=(k−1)/6ur,amb,k=1,…,7).
(a)(b)(c)Figure3 shows that the measured phase
pdfs are strongly dependent on SCR values and assume a peak value in a position
which, for low SCR, is different from the nominal ATI phase value
(3), that in
this case is given by π/2 and is highlighted with a dot. The
dependence on CNR is, instead, less pronounced. The sensitivity
of the pdfs
shape with respect to ur
is significant, and the corresponding variances gradually increase by
increasing ur.


### 2.2. Statistical Distribution of at-Insar Phase for Gaussian Rcs

A Gaussian
model is applicable when
the RCS of the targetsA1  and A2 and then the signals ZT1 and ZT2  are
assumed to be zero mean (complex) Gaussian processes, as in [12]. This model
allows to take into account the lack of knowledge of the target RCS values
(that can be described in terms of variance σT2) and then of
the SCR. It applies to complex or extended targets which can be considered to
consist of a large number of isotropic scattering elements, randomly
distributed in a region whose dimensions are large compared to the wavelength
of the illuminating radiation, and all contributing to the overall signal with
the same weight [21].  When the
number of the elementary scatterers in which the target can be decomposed is
small and/or some of them are dominant
with respect to the others, the pdf of the
backscattered signal is not zero mean Gaussian and is difficult to derive [22].
However, even if this model could not be always appropriate for the description
of the signal intensity distribution, it has the advantage of providing an
analytical form for the interferometric phase pdf, which in many cases well
approximates the true distribution. Moreover, as it will be shown in the next
section, the adoption of a Gaussian model for the moving target RCS instead of
the actual deterministic model will not impair significantly the GMTI
performance.Since the signalsZ1 and
Z2 acquired
by the two interferometric antennas are still zero mean
Gaussian signals, as
happens when the target is absent, the pdf of the
interferometric phase can be
expressed in the closed form given by (7),
where γ is the coherence
coefficient between the images Z1 and
Z2,
and ϕ0
is the phase of
γ.
Now,the expression of
γ
is changed with respect to
(5)
and is given by [12] (9)γ=E[(Zc1+N1+ZT1)(Zc2+N2+ZT2)*]E[|Zc1+N1+ZT1|2]E[|Zc2+N2+ZT2|2]=γcσc2+σT2γTσc2+σn2+σT2=γc+γTSCR1+1/CNR+SCR,  where SCR=σT2/σc2, and γT is the target (complex) coherence and depends on the
target velocity through the nominal phase (1): (10)γT=E[ZT1ZT2*]E[|ZT1|2]E[|ZT2|2]=E[A1A2*]E[|A1|2]E[|A2|2]ejϕv=γT0ejϕv, where γT0 is the target coherence for zero radial velocity, equal to one. It has
to be noted that ϕ0, the phase of γ, is different from ϕv.Figure4 shows
the dependence of the pdf fΦ(ϕ) on SCR, CNR, and ur in the case of a Gaussian distributed
target, evaluated for γc=γT0=1 and using the TerraSAR-X parameters [20] of Table 1. In
Figure 4(a), the pdf shape is reported for the values CNR=10dB and
SCR=0,10, and 20 dB and a radial normalized velocity ur=3.25×10−3, corresponding to the nominal
ATI phase value ϕv=π/2.
Figure 4(b) is related to the values
SCR=10dB and CNR=0,10, and 20 dB and to the same value of ur. In
Figure 4(c), the pdfs are plotted for
SCR=10dB and CNR=10dB and by varying the
normalized velocity ur.(a) Pdf of
the interferometric phase in presence of Gaussian target moving with a radial
velocity such thatϕv=π/2   (the dot on the ϕ axis), for γc=1, CNR=10dB and SCR=0,10,20dB,
(b) SCR=10dB and
CNR=0,10,20dB, and (c) SCR=10dB,
CNR=10dB, for seven radial
velocities (urk=(k−1)/6ur,amb,k=1,…,7).
(a)(b)(c)Figure4 shows
that also in this case the measured phase pdfs are not centered on the noise-free
value highlighted with a dot and given by (3), and that their shape strongly
depends on SCR and weakly on CNR. Moreover, we note that now the pdfs have
always a behavior that is symmetrical around the phase ϕ0, contrarily to what happened in the deterministic case (see
Figure 3).
The sensitivity of pdfs shape with respect to ur
is similar to
the deterministic case and also in this case the corresponding variances gradually
increase by increasing ur.However, for high SCR (strongly reflective target),
the pdfs derived in the two cases are quite similar,
as shown in Figure5,
where we have reported the pdfs related to the two models,
(a) with SCR=0dB,
(b) SCR=10dB, and (c) SCR=20dB.Pdfs of the
interferometric phase in presence of a deterministic and a Gaussian target
moving with a radial velocity such thatϕv=π/2  (the dot on the ϕ axis), for γc=1CNR=10dB and (a) SCR=0dB, (b) SCR=10dB, and (c)
SCR=20dB.
(a)(b)(c)Moreover, the availability of an
analytical expression for the pdf and for the coherence
(see (7)
and (10))
allows to easily discuss the effects that CNR, SCR, and radial velocity
variations have on the interferometric measured phase distribution. First of
all, we note that the pdf behavior changes according to the changes of |γ| and ϕ0. In particular, when |γ| decreases, the
spreading of the measured phase values around ϕ0 increases, while the changes in the
values of ϕ0 determine a simple circular shift of
the curves. The first effect affects the velocity estimation accuracy, while
the latter introduces a phase polarization ϕ0-ϕv, which, if known, can be implicitly taken into
account and compensated in the velocity estimation procedure.We can easily analyze the
pdf dependence on the system and target parameters in three limit cases.(i) Strong Targets
SCR≫1   and
CNR≫1; then, from (9)
and  (10), |γ|≅|γT0|    is independent on the velocity value and ϕ0≅ϕv,    that is, the pdf maximum position does not depend on
SCR and CNR. For high SCR values we expect that the velocity estimation
accuracy does not depend on the velocity value.(ii) Camouflaged Targets
SCR≅1, CNR≫SCR,  and assume γc≅γT0; then,
from (9)  and
(10), γ≅γc(1+exp(jϕv))/2=γcexp(jϕv/2)cos(ϕv/2).
Consequently, |γ|≅|γccos(ϕv/2)|  and ϕ0≅ϕv/2. In this case, the pdf spreading and the phase
polarization are strongly dependent on the velocity value.(iii) Weak Targets
SCR≪1;   then, from
(9)
and (10), γ≅γc/(1+CNR).
In this case ϕ0≅0,  the clutter term is dominant, and the pdf of the phase is in practice
independent on the velocity value. As expected, the velocity estimation problem
cannot be solved for very low SCR values.The
behavior of|γ| and ϕ0
versus SCR using the TerraSAR-X
parameters, with ϕv=π/2  and CNR values of
10, 20, and 30 dB, is
shown in Figure 6. Note that ϕ0
is independent on CNR for the considered
values of SCR, while |γ| is practically independent on CNR
for SCR values larger than 0 dB. Then,
we expect that for SCR values greater
than 0 dB, measurement errors on CNR
will not influence significantly the
velocity estimation accuracy.TerraSAR-X
parameters and Gaussian target model:
(a)|γ| versus SCR, and (b) ϕ0  versus SCR, with
CNR = 10 dB (dashed line), CNR = 20 dB (solid
line), CNR = 30 dB (dotted line),
and γc=γT0=1. Note that the ϕ0  curves obtained for different values
of CNR are coincident.
(a)(b)Note further that, for
small changes of SCR around the value SCR = 0 dB (SCR = 1), the curves exhibit
strong variations. This implies that small errors in the knowledge of the SCR
can affect the velocity estimation accuracy.For the deterministic
model the pdf is not given by
(6), and there is not a direct relation between |γ| and the pdf spreading, and between ϕ0 and the position of the pdf maximum. Even so, the
general qualitative behavior of |γ| and ϕ0  with
respect to changes of SCR, CNR, γc, and γT0 is similar to
the one obtained in the Gaussian case.


## 2.1. Statistical Distribution of at-Insar Phase for Deterministic Rcs

A deterministic model is applicable to the
case of a target whose RCS can be expressed by a deterministic function of the
incidence angle. This model applies to canonical scattering objects (such as
corner reflectors, spheres), and to complex or extended targets whose RCS does
not rapidly change between the interferometric acquisitions. Since the RCS
value of a given target mainly depends on incidence angle and target aspect
angle, which does not change in the small time required to the SAR antenna to cover
the baseline length, the target RCS can be usually assumed to be constant in
the interferometric images. Such value influences the signal-to-clutter ratio,
and is not a priori known. This is the model to be used in the interferograms
simulation. It can be adopted also in the velocity estimation procedure if a
precise knowledge of the RCS value is available. An accurate knowledge of the
average RCS values can be available only for accurately characterized targets
[19]. Moreover, this case provides the reference pdfs of the interferometric
phases produced by a given complex target of known RCS. In this assumption, we
can put in (2)A1=A2=A, with A a
deterministic constant. Then, the two processes Z1 and Z2 are Gaussian, with nonzero mean (due to the presence of the target) and the
target RCS can be described in terms of |A|2.The first-order
probability density function (pdf) of the interferometric phaseΦ  can be, in this case, numerically computed via
Monte Carlo techniques, since no closed form, so far, has been found. The pdf depends on the clutter coherence coefficient γc, on the target radial velocity
(as shown by
(2) and
(3))
and on CNR and SCR, where CNR is given by
(6) and SCR is given by (8)SCR=|A|22σc2, where |A|2 is the
signal power. In this case, the SCR values do not affect the coherence γ between the signals Z1 and Z2,
which is still expressed by
(5), but affect the shape of the pdf, which cannot
be expressed by (7). Figure 3 shows the dependence of the pdf fΦ(ϕ) on SCR, CNR, and ur in the case of a deterministic target, evaluated for γc=1  and using the TerraSAR-X parameters [20] of Table 1. In
Figure 3(a), the pdf shape is reported for the values CNR=10dB and
SCR=0,10, and 20 dB and a radial normalized velocity ur=3.25×10−3, corresponding to the nominal ATI
phase value ϕv=π/2.
Figure 3(b) is related to the values
SCR=10dB and CNR=0,10, and 20 dB and to the
same value of ur. In
Figure 3(c), the pdfs are plotted for
SCR=10dB and CNR=10dB and by
varying the normalized
velocity ur.Table 1
Main parameters of TerraSAR-X system.
TerraSAR-XHeight514.8
KmPlatform
velocity7.6
Km/sAlong-track
antenna dimension4.8 mAcross-track
antenna dimension0.8 mAlong-track
baseline1.2 mWorking
frequencyX band—9.65 GHzWavelength3.12 cmRange bandwidth150 MHz(a) Pdf of
the interferometric phase in presence of a deterministic target moving with a
radial velocity such thatϕv=π/2   (the dot on the ϕ axis), for γc=1,
CNR=10dB, and SCR=0,10,20dB, (b)
SCR=10dB and
CNR=0,10,20dB, and (c) SCR=10dB,
CNR=10dB, for seven radial
velocities (urk=(k−1)/6ur,amb,k=1,…,7).
(a)(b)(c)Figure3 shows that the measured phase
pdfs are strongly dependent on SCR values and assume a peak value in a position
which, for low SCR, is different from the nominal ATI phase value
(3), that in
this case is given by π/2 and is highlighted with a dot. The
dependence on CNR is, instead, less pronounced. The sensitivity
of the pdfs
shape with respect to ur
is significant, and the corresponding variances gradually increase by
increasing ur.


## 2.2. Statistical Distribution of at-Insar Phase for Gaussian Rcs

A Gaussian
model is applicable when
the RCS of the targetsA1  and A2 and then the signals ZT1 and ZT2  are
assumed to be zero mean (complex) Gaussian processes, as in [12]. This model
allows to take into account the lack of knowledge of the target RCS values
(that can be described in terms of variance σT2) and then of
the SCR. It applies to complex or extended targets which can be considered to
consist of a large number of isotropic scattering elements, randomly
distributed in a region whose dimensions are large compared to the wavelength
of the illuminating radiation, and all contributing to the overall signal with
the same weight [21].  When the
number of the elementary scatterers in which the target can be decomposed is
small and/or some of them are dominant
with respect to the others, the pdf of the
backscattered signal is not zero mean Gaussian and is difficult to derive [22].
However, even if this model could not be always appropriate for the description
of the signal intensity distribution, it has the advantage of providing an
analytical form for the interferometric phase pdf, which in many cases well
approximates the true distribution. Moreover, as it will be shown in the next
section, the adoption of a Gaussian model for the moving target RCS instead of
the actual deterministic model will not impair significantly the GMTI
performance.Since the signalsZ1 and
Z2 acquired
by the two interferometric antennas are still zero mean
Gaussian signals, as
happens when the target is absent, the pdf of the
interferometric phase can be
expressed in the closed form given by (7),
where γ is the coherence
coefficient between the images Z1 and
Z2,
and ϕ0
is the phase of
γ.
Now,the expression of
γ
is changed with respect to
(5)
and is given by [12] (9)γ=E[(Zc1+N1+ZT1)(Zc2+N2+ZT2)*]E[|Zc1+N1+ZT1|2]E[|Zc2+N2+ZT2|2]=γcσc2+σT2γTσc2+σn2+σT2=γc+γTSCR1+1/CNR+SCR,  where SCR=σT2/σc2, and γT is the target (complex) coherence and depends on the
target velocity through the nominal phase (1): (10)γT=E[ZT1ZT2*]E[|ZT1|2]E[|ZT2|2]=E[A1A2*]E[|A1|2]E[|A2|2]ejϕv=γT0ejϕv, where γT0 is the target coherence for zero radial velocity, equal to one. It has
to be noted that ϕ0, the phase of γ, is different from ϕv.Figure4 shows
the dependence of the pdf fΦ(ϕ) on SCR, CNR, and ur in the case of a Gaussian distributed
target, evaluated for γc=γT0=1 and using the TerraSAR-X parameters [20] of Table 1. In
Figure 4(a), the pdf shape is reported for the values CNR=10dB and
SCR=0,10, and 20 dB and a radial normalized velocity ur=3.25×10−3, corresponding to the nominal
ATI phase value ϕv=π/2.
Figure 4(b) is related to the values
SCR=10dB and CNR=0,10, and 20 dB and to the same value of ur. In
Figure 4(c), the pdfs are plotted for
SCR=10dB and CNR=10dB and by varying the
normalized velocity ur.(a) Pdf of
the interferometric phase in presence of Gaussian target moving with a radial
velocity such thatϕv=π/2   (the dot on the ϕ axis), for γc=1, CNR=10dB and SCR=0,10,20dB,
(b) SCR=10dB and
CNR=0,10,20dB, and (c) SCR=10dB,
CNR=10dB, for seven radial
velocities (urk=(k−1)/6ur,amb,k=1,…,7).
(a)(b)(c)Figure4 shows
that also in this case the measured phase pdfs are not centered on the noise-free
value highlighted with a dot and given by (3), and that their shape strongly
depends on SCR and weakly on CNR. Moreover, we note that now the pdfs have
always a behavior that is symmetrical around the phase ϕ0, contrarily to what happened in the deterministic case (see
Figure 3).
The sensitivity of pdfs shape with respect to ur
is similar to
the deterministic case and also in this case the corresponding variances gradually
increase by increasing ur.However, for high SCR (strongly reflective target),
the pdfs derived in the two cases are quite similar,
as shown in Figure5,
where we have reported the pdfs related to the two models,
(a) with SCR=0dB,
(b) SCR=10dB, and (c) SCR=20dB.Pdfs of the
interferometric phase in presence of a deterministic and a Gaussian target
moving with a radial velocity such thatϕv=π/2  (the dot on the ϕ axis), for γc=1CNR=10dB and (a) SCR=0dB, (b) SCR=10dB, and (c)
SCR=20dB.
(a)(b)(c)Moreover, the availability of an
analytical expression for the pdf and for the coherence
(see (7)
and (10))
allows to easily discuss the effects that CNR, SCR, and radial velocity
variations have on the interferometric measured phase distribution. First of
all, we note that the pdf behavior changes according to the changes of |γ| and ϕ0. In particular, when |γ| decreases, the
spreading of the measured phase values around ϕ0 increases, while the changes in the
values of ϕ0 determine a simple circular shift of
the curves. The first effect affects the velocity estimation accuracy, while
the latter introduces a phase polarization ϕ0-ϕv, which, if known, can be implicitly taken into
account and compensated in the velocity estimation procedure.We can easily analyze the
pdf dependence on the system and target parameters in three limit cases.(i) Strong Targets
SCR≫1   and
CNR≫1; then, from (9)
and  (10), |γ|≅|γT0|    is independent on the velocity value and ϕ0≅ϕv,    that is, the pdf maximum position does not depend on
SCR and CNR. For high SCR values we expect that the velocity estimation
accuracy does not depend on the velocity value.(ii) Camouflaged Targets
SCR≅1, CNR≫SCR,  and assume γc≅γT0; then,
from (9)  and
(10), γ≅γc(1+exp(jϕv))/2=γcexp(jϕv/2)cos(ϕv/2).
Consequently, |γ|≅|γccos(ϕv/2)|  and ϕ0≅ϕv/2. In this case, the pdf spreading and the phase
polarization are strongly dependent on the velocity value.(iii) Weak Targets
SCR≪1;   then, from
(9)
and (10), γ≅γc/(1+CNR).
In this case ϕ0≅0,  the clutter term is dominant, and the pdf of the phase is in practice
independent on the velocity value. As expected, the velocity estimation problem
cannot be solved for very low SCR values.The
behavior of|γ| and ϕ0
versus SCR using the TerraSAR-X
parameters, with ϕv=π/2  and CNR values of
10, 20, and 30 dB, is
shown in Figure 6. Note that ϕ0
is independent on CNR for the considered
values of SCR, while |γ| is practically independent on CNR
for SCR values larger than 0 dB. Then,
we expect that for SCR values greater
than 0 dB, measurement errors on CNR
will not influence significantly the
velocity estimation accuracy.TerraSAR-X
parameters and Gaussian target model:
(a)|γ| versus SCR, and (b) ϕ0  versus SCR, with
CNR = 10 dB (dashed line), CNR = 20 dB (solid
line), CNR = 30 dB (dotted line),
and γc=γT0=1. Note that the ϕ0  curves obtained for different values
of CNR are coincident.
(a)(b)Note further that, for
small changes of SCR around the value SCR = 0 dB (SCR = 1), the curves exhibit
strong variations. This implies that small errors in the knowledge of the SCR
can affect the velocity estimation accuracy.For the deterministic
model the pdf is not given by
(6), and there is not a direct relation between |γ| and the pdf spreading, and between ϕ0 and the position of the pdf maximum. Even so, the
general qualitative behavior of |γ| and ϕ0  with
respect to changes of SCR, CNR, γc, and γT0 is similar to
the one obtained in the Gaussian case.


## 3. Multichannel Along-Track Sar Interferometry

The along-track
interferometric phase depends on radial velocity, baseline, and wavelength as
shown in (3). Phase values outside interval (−π,π) wrap mod(2π), so that such values are
indistinguishable from the ones differing for 2π multiples. The same holds for the
corresponding radial velocity values. The radial velocity ambiguity value ur,amb=±λ/(4b), corresponding to the interferometric phase ±π, is then the maximum velocity value that
can be unambiguously detected. Moreover, in the realistic case of noisy data, this
ambiguity problem can be present also for normalized radial velocities smaller
than ur,amb. Such effect can be particularly critical either
for detection applications or for velocity estimation ones.A
method for overcoming these limitations, restoring the solution uniqueness,
consists in exploiting different datasets acquired with different baselines, or
with frequency diversity [13, 15].Different
baseline datasets (at least two) can be
generated when the AT-InSAR system is
constituted by more than two antennas (at least three). Different frequency
datasets can  be generated in two ways. In the first, we can suppose that the
SAR sensors can operate at different working frequencies, for instance in X and
C bands simultaneously. In the second, the multifrequency interferograms can be
obtained by subband filtering of the interferometric images splitting the overall bandwidth as
shown in Figure7. Azimuth band partition produces the conventional azimuth
looks, while range band partition produces different range looks. Note that
this second partition generates looks with a small frequency diversity. Their
generation is finalized to phase noise suppression as in conventional multilook
procedure.Figure 7
Partition of the dual-band spectrum of a hypothetical SAR interferometric
system. The two bandsBI  and BII are subband
filtered into Nr=8 range
subband (central frequencies v1,v2,…,v8), and the Doppler band Bd into Na=4 azimuth looks (L1,L2,L3,L4). Each (n=1,…,N=Na×Nr=32) identifies a portion of the 2D
frequency domain not overlapping with the others. Absence of overlapping
guarantees the statistical independence of interferograms.We
will refer in the following to multifrequency and/or multibaseline
configuration as multichannel configuration. The moving target detection and
the radial velocity estimation
are performed from the knowledge of such multiple wrapped interferometric phase
(statistically independent) signals obtained with different baselines or with
different working frequencies. It has to be noted that when the channels
originate by band partition, the multichannel approach has a drawback, as the SCR
is reduced, as we will show in the following.The CNR is given, in the case of a single-look SAR image, by(11)CNRSLook=PClutterPNoise=(KaKr)2(PTxGAλ2/(4π)3R4)(δxδrσ0)KaKrPNoise, where Ka  and Kr are the integratio samples along the azimuth
and the range direction, respectively, PTx is the power transmitted by the SAR antenna, GA is the radar antenna gain, R  is the distance between the SAR antenna and the ground region
where this ratio is evaluated, σ0 is the normalized radar cross-section relative to the background (the clutter), δx and δr are the spatial resolutions of the images, and PNoise  is the thermal power at the receiver. Note that
the product (δxδrσ0)
represents the RCS of a clutter resolution cell.The SCR is given, in the case of a single-look radar image, by(12)SCRSLook=PTargetPClutter=(KaKr)2(PTxGAλ2/(4π)3R4)σTarget(KaKr)2(PTxGAλ2/(4π)3R4)(δxδrσ0)=σTarget(δxδrσ0), where σTarget is the moving target RCS.In the case of a multichannel system, two cases have to be
distinguished: the case where more antennas (multibaseline) or more working
frequencies are used, and the case where the overall system bandwidth is
partitioned into different subbands (multilook). In the first case, the spatial
resolution does not change, and supposing that the integration samples are the same
for each channel, the CNR and the SCR values do not change. In the latter case,
as the band partition reduces the spatial resolution, and supposing thatNa and Nr  are the number of azimuth looks and range looks,
respectively, the CNR and the SCR values
change in (13)CNRMLook=(KaKr/NaNr)2(PTxGAλ2/(4π)3R4)(NaδxNrδrσ0)(KaKr/NaNr)PNoise=CNRSLook,(14)SCRMLook=σTarget(NaδxNrδrσ0)=1(NaNr)SCRSLook. The multichannel (derived by band partition) along-track SAR interferometry system
scheme is depicted in
Figure 8.Figure 8
Multichannel
along-track SAR interferometry system scheme.


## 4. Multichannel at-Insar Moving Target Detection

The
interferometric phaseΦ is distributed
according to a pdf depending on several
parameters: (15)fΦ(ϕ∣ur;λ,b,γc,SCR,CNR). Of course, in
the absence of a moving target (ur=0,andSCR=0), Φ reduces to pure phase noise.A moving
target can be detected by comparing the interferometric phaseΦ  with a threshold ϕT in the interval (−π,π]. We can evaluate the detection
probability (PD) and false
alarm probability (PFA) in
the following way: (16)PD=∫−π−ϕTfΦ(ϕ∣ur;λ,b,γc,SCR,CNR)dϕ+∫ϕTπfΦ(ϕ∣ur;λ,b,γc,SCR,CNR)dϕ,PFA=2∫ϕTπfΦ(ϕ∣0;λ,b,γc,CNR)dϕ.The
performance of the detection process is, as expected, better for high values of
SCR, that is, when the moving targets power is significantly larger than the
clutter power. For moving targets mingling with the background clutter, the
detection capability worsens,
so that if one wants low values ofPFA, the PD can decrease to very low values, not consistent with the applications [23]. This
approach, based on a single interferogram value, does not provide the desired
results in terms of simultaneous low values of  PFA  and high
values of PD. An alternative improved detection strategy is
based on the use of multichannel interferograms. After the application of the
threshold to each channel, a binary integration procedure can be adopted to
combine single-channel decisions.We use a
hypothetical dual-baseline system working at the frequencyf=9.65GHz, with three antennas separated by the two baselines b1=1.2m and b2=1.8b1m. Both interferometric signals are partitioned into
4 azimuth looks, for a total of N=8 channels. Note that the same effect could be obtained with a dual frequency
system, with a first working frequency f1 and a second working frequency equal to f2=1.8f1 (e.g., C band and X band).Suppose that
the detection probability of one of the channels corresponding to the first
baseline (or first frequency) is equal toPD1,
and that the detection probability of one of the channels corresponding to the
second baseline (or second frequency) is equal to PD2; we can evaluate the probability that the
target  is detected from (N/2+j) channels (j=1,…,N/2) on a total of N channels: (17)Pj=∑k=jN/2[(N2k)PD1k(1−PD1)N/2−k]×[(N2N2−k+j)PD2N/2−k+j(1−PD2)k−j].We have
developed two possible strategies and compared them with the one based on a
single interferogram. Strategy 1 consists in   considering present the moving
target when the majority of the interferogram values are above prefixed
thresholds. We obtain the following estimated detection probability:(18)PD>N/2=∑j=1N/2Pj.For the
estimation of the false alarm probability, we have used the same reasoning,
using the single-channel false alarm probabilitiesPFA1  and PFA2 in place of PD1 and  PD2 in
(17).Strategy 2
consists in considering present the moving target when more than 3/4 of the
total interferogram values are above prefixed thresholds. We obtain(19)PD>3N/4=∑j=n/4+1N/2Pj.In Figure9,
we have reported all the PD’s corresponding to different
operating conditions and in the case of the deterministic model (the Gaussian
model provides similar results). We have reported also the estimation of  PD  adopting Strategy 1 (dashed line) and Strategy 2 (solid line). The single-channel
detection probabilities are depicted with dash-dotted and dotted lines. All the
results refer to a moving target with ur=1×10−3,
SCR = 10 dB, CNR = 10 dB, and γc=1. The PD’s have
been plotted versus the velocity values ur,T corresponding to the thresholds ϕT.Figure 9
PD’s
adopting Strategy 1 (dashed line) and Strategy 2 (solid line), in presence of a
moving target with ur=0.001,
SCR = CNR = 10 dB, and γc=1.
Single-channel PD’s are depicted with dash-dotted line (first
baseline) and dotted line (second baseline).In Figure10,
we have reported all the PFA’s corresponding to the same operating
conditions, adopting Strategy 1 (dashed line) and Strategy 2 (solid line). All
the results are evaluated in absence of a
moving target and for CNR = 10 dB and
γc=1.Figure 10
PFA’s
adopting Strategy 1 (dashed line) and Strategy 2 (solid line), in absence of
moving targets, CNR = 10 dB, γc=1. Single-channel PFA’s are depicted with dash-dotted line (first baseline) and
dotted line (second baseline).In Figure11,
Strategies 1 and 2 are compared in terms of PFA and PD.
Following Strategy 2, it can be found
that a threshold exists where performances are
quite good with PD approaching 1 and PFA approaching 0.Figure 11
PFA and PD adopting Strategy 1 (dashed line) and Strategy 2 (solid
line). ur=1×10−3,
SCR = CNR = 10 dB, and γc=1.Figure11
has been plotted for a fixed velocity value
(ur=1×10−3). When the velocity value changes, the detection probability changes, while the false alarm probability remains unchanged. In particular, considering Strategy
2, in Figures 12(a) and
12(b) it can be appreciated that, as expected, by increasing
the velocity the detection probability increases.PFA
and PD adopting Strategy
2 for SCR = CNR = 10 dB, γc=1,0;
(a) ur=0.5×10−3 and (b) ur=1.5×10−3.
(a)(b)To better show this effect, the detection probability versus velocity at
fixed values of false alarm probability is shown in
Figures13(a) and 13(b) for
CNR = 10, γc=1, with SCR = 10
(Figure 13(a))
and SCR = 20 (Figure 13(b)) for two different
fixed values of PFA(10−2and10−3).PD versus
velocity for fixed values of PFA(10−2and10−3) for CNR = 10, γc=1; (a) SCR = 10
and (b) SCR = 20.
(a)(b)It can be observed that in these examples a
target can be detected with probability approaching 1 starting from normalized
velocity values approximately equal tour=10−3,
also for PFA=10−3.
In this case, the phase thresholds corresponding to ur=10−3 and that guarantee PFA=10−3 and PD≅1  are ϕT1=4πb110−3/λ=0.49  and ϕT2=4πb210−3/λ=0.88.
In general, the phase thresholds depend on the minimum detectable velocity
according to (3). The velocities values that can be detected reduce
significantly when PFA increases, or
when the target RCS increases.


## 5. Multichannel at-Insar Moving Target Radial Velocity Estimation

As discussed
in Section2, the accuracy of the velocity estimation, obtainable with a given
AT-InSAR system configuration, depends on the statistical model assumed for the
target image, on the target radial velocity, and on the following parameters:
SCR, CNR, γc, and γT0.We have
already presented the multichannel system (multifrequency and/or multibaseline)
configuration, providing the different phase measurements which are required to
find a reliable solution for the detection and estimation problems [13, 15].The ML
estimation of   the normalized radial velocity from
multichannel data is given by(20)u^r=argmaxurL(ur),L(ur)=∏n=1,…,Nm=1,…,MfΦ(ϕn,m∣ur;λn,bm,γc,γT0,SCR,CNR)︸singlechannellikelihoodfunction,   where L(⋅) is the multichannel likelihood
function, obtained by multiplying the likelihood
functions corresponding to the
central frequency of the subbands and/or to the different baselines, and ϕn,m represent the wrapped phase values
relative to the N frequencies c/λn and to the M baselines bm. The factorization
(20) comes from the
assumed statistical independence of the multichannel interferograms.We evaluate
numerically the Cramer-Rao lower bound (CRLB) of the estimated (normalized)
velocity for the two different target statistical models considered above, and
we estimate the target radial velocity using
(20). We compare the CRLB with the root mean square error (RMSE) values.
It has to be reminded that the CRLB represents the lower bound for the variance
of the estimated parameter (the normalized radial velocity, in this case),
whatever unbiased estimator working on the available set of data (the wrapped
phases) may be considered [24].We use the
TerraSAR-X parameters introduced in Table1 for the
numerical simulation, and we
consider a single-baseline system and a dual-baseline system (b1=1.2mandb2=1.8b1m), as in Section 4. For each baseline we
considered 2 subbands and 2 azimuth looks, in total 4 different channels [23].
For baseline b1 the maximum
radial velocity value that can be unambiguously detected is |ur,amb1|=λ/(4b1)=6.5×10−3 and for baseline  b2 is
|ur,amb2|=λ/(4b2)=3.6×10−3.  A correspondence table between
normalized and true radial velocity values in the TerraSAR-X case is reported
in Table 2.Table 2
Correspondence between normalized
and true velocity values in the TerraSAR-X case.
ur1×10−44×10−47×10−41×10−32×10−33×10−34×10−35×10−36×10−3vr[m/s]0.763.045.327.6015.222.830.438.045.6vr[km/h]2.7410.913.727.454.782.1109.4137.0164.4TheCRLB1/2 and RMSE values (in
logarithmic scale) for the deterministic case and for the Gaussian case, with γc=1,
SCR = 0, 10, and 20 dB, CNR = 10 dB,
and b1=1.2m, are reported in Figure 14. Of course, the SCR
values are the ones obtained after the subband filtering
(14).CRLB1/2 (solid line) and RMSE (dashed line) values (in log scale) of the estimated
target radial normalized velocity, with CNR = 10 dB, SCR=0,10,20dB,
γc=1,
for a single-baseline system (b1=1.2m) for (a) a deterministic modeled moving target and (b)
a Gaussian modeled moving target.
(a)(b)The CRLB
values vary with the normalized velocityur  to be estimated
since the pdfs of the interferometric phase change with it (see
Figures 3(c)
and 4(c)), as shown in Section
2. In particular, in all cases considered, the
CRLB values increase with the increasing of velocity. Estimation of velocity
values is more accurate for small values of velocity (the CRLB are lower), and
less accurate when velocity increases.We observe
also that, with the same SCR, CNR, andγc values, the deterministic target model exhibits lower CRLB values than
the Gaussian model. This effect is due to the different variance values
corresponding to the deterministic and Gaussian cases for the same SCR, CNR,
and γc values (see
Figures 3(c) and
4(c)). The larger the
variance (the larger the phase noise), the larger the corresponding CRLB.We observe
further that in all considered cases the CRLB values decrease by increasing SCR
under the same CNR andγc values. Then, as expected, velocity estimation is more
accurate when signal-to-clutter ratio is larger. It can be noted that the RMSE values for velocities
far from the ambiguity value are only “slightly” larger than the CRLB1/2 ones and the RMSE values obtained
using the deterministic model, likewise the CRLB1/2, are lower than the RMSE in the
Gaussian case, as expected. This means that 4 channels are sufficient, for the
chosen values of CNR, SCR, and γc, to obtain quality performance very close to the best theoretical
results when  there are no ambiguity problems. Instead, it can be noted that the
RMSE for velocities approaching the ambiguity value (from 4×10−3to6×10−3) and for low SCR tends to
deviate from the CRLB values. This
behavior is more pronounced in the Gaussian
case.In case we
consider two baselines, we can obtain CRLB and RMSE values less variable with
velocity and less sensitive to ambiguity problems. In particular, consideringb1=1.2m and b2=1.8b1m, γc=1, SCR=0,10, and
20 dB, CNR = 10 dB, we get the CRLB1/2 and RMSE
curves (in logarithmic scale) shown in Figure
15, for the deterministic and
Gaussian cases.CRLB1/2 (solid line) and RMSE (dashed line) values (in log scale) of the estimated
target radial normalized velocity, with CNR = 10 dB,
SCR = 0, 10, 20 dB, γc=1,
for a dual-baseline system (b1=1.2m, b2=1.8b1m) for (a) a
deterministic modeled moving target and (b) a
Gaussian modeled moving target.
(a)(b)It can be appreciated that the CRLBs are
reduced with respect to the single-baseline case, and that they are less variable
with the velocity. Differently from the single-baseline system, in the dual-baseline
system the RMSE values for all the velocities considered are very close to the
CRLBs at least for high SCR values. Only for SCR = 0 dB it can be observed that a
degradation of the estimation performances exists. The performance analysis shows that the
considered AT-InSAR system allows to estimate normalized radial velocity with RMSE
of the order of10−4 (using two baselines and 4 channels for baseline)
even for SCR = 10 dB. In traffic monitoring applications, when the moving targets
are cars and trucks, the RCS can be of the order of, or larger than, 100m2.
In this case, the SCR can be significantly larger than 10 dB, allowing the
application of this kind of sensors to practical situations.


### 5.1. Multichannel Ml Velocity Estimation Algorithm Robustness

#### 5.1.1. Robustness with Respect to The Target Model

As discussed in Section2, the
deterministic target model is less tractable because, so far, no analytical
statistical description has been obtained, but it is more realistic (targets of
interest—cars, trucks—exhibit RCSs that
do not vary significantly with the radar observation angles [19]). The Gaussian
target model is less realistic (it is rare that cars and trucks response is a
zero mean Gaussian signal which varies significantly inside the synthetic
aperture), but it allows to perform the velocity estimation using analytical
likelihood functions, since an analytical expression for the interferometric
phase pdf is available with great advantages in terms of the computational
efficiency. For this
reason, we present the performance results in terms of RMSE values obtained
using data generated with deterministic RCS, processed with the likelihood functions derived from the Gaussian model.
In other words, we process the AT-InSAR data with a model different from the
actual one used to simulate them. We compare again the single-baseline system
with the dual-baseline one. The RMSE values for
CNR = 10 dB, SCR=0,10,20dB, and
γc=1 for a deterministic modeled moving target obtained
using a likelihood function derived from the deterministic and the Gaussian
models are reported in Figure 16, for (a) the single-baseline system (b1=1.2m) and (b) the dual-baseline system (b1=1.2m, b2=1.8b1m).RMSE values of the estimated target radial normalized velocity, for a
deterministic target likelihood (solid line) and RMSE values for a Gaussian
target likelihood (dashed line), in log scale,
with CNR = 10 dB,SCR=0,10,20dB,  γc=1
and for (a) a single-baseline system
(b1=1.2m)
and (b) for a dual-baseline system
(b1=1.2m, b2=1.8b1m).
(a)(b)It can be noted, quite surprisingly,
that the RMSE values processed with the likelihood functions derived from the
Gaussian model are very close to the ones relative to the deterministic case.
In such a way it is possible to process efficiently actual data with a
tractable estimation algorithm without impairing significantly the estimation
performance.


#### 5.1.2. Robustness with Respect to The System Parameters

A further experiment directed to test
robustness consists in varying the parameters of the model that are not
perfectly known. In particular, while CNR andγc can be easily obtained from the system characteristics
(noise power of the SAR sensor) and from the processed SAR
images (clutter
power and coherence), the SCR value is unknown until the target is unknown (we
do not a priori know its position or its RCS). Note also that it is not
necessary to test robustness against CNR since, as highlighted in the comments
to Figures 3 and
4, errors on it do not influence significantly the velocity
estimation accuracy. Instead the role of SCR is quite determinant in the target
velocity estimation procedure with respect to the other parameters, as shown by
the pdf shapes and in the CRLB and RMSE behaviors. For this reason, we test the
performance algorithm when the target RCS value is not a priori known.First of all, we note that the SCR
can be estimated  from the data and the final estimation can be casted as a
joint estimation of velocity and SCR:(21)[u^r,SC^R]=argmaxur,SCRL(ur,SCR),L(ur,SCR)=∏n=1,…,Nm=1,…,MfΦ(ϕn,m∣ur,SCR;ϑ)︸singlechannellikelihoodfunction,   where ϑ denotes  the following parameters λn,bm,γc=1,γT0=1,  CNR0 is computed from the data. We assume a Gaussian target model to derive the likelihood since it allows
to perform the velocity estimation using analytical likelihood function and it
does not degrade the estimation performance, as previously seen, even if the
data are generated using a deterministic target model.As far as the
accuracy of the estimation of SCR is concerned, we show in Figure17(a) some
cuts (relative to different SCR values) of the likelihood function (21) versus
SCR for a fixed radial velocity value. Note that the curves exhibit a similar
shape since the axis of abscissas is in log scale, while in linear scale they
would appear very different from each other, and their spreading increases by
increasing SCR. Consequently, as the best theoretical obtainable accuracy (the
Cramer-Rao lower bound (CRLB)) is directly connected to the spreading of the
likelihood function, the accuracy worsens increasing the SCR. The complete
behavior of the square root of the CRLB for SCR is presented in
Figure 17(b)
and confirms the mentioned trend.(a) Likelihood functions (21) versus SCR
for a fixed radial velocity value; (b) square
root of the CRLB for
SCR.
(a)(b)We analyze also the system performance in
terms of RMSE in the three following cases: (1) the estimation is performed by
adopting for SCR the nominal valueSCRn; (2) SCR is estimated in
conjunction with the normalized radial velocity; (3) the estimation is
performed by adopting for SCR a fixed value SCRf different from the true one. In Table 3,
we report the RMSE values
of the estimated radial velocity using AT-InSAR images simulated with the
system parameters reported in Table 1, with
CNR = 10 dB, SCR = 10 dB, γc=1, and with normalized target velocities ur=1×10−3, ur=2×10−3, ur=3×10−3. The multichannel MLE algorithm
(21)
adopts a single baseline (b1=1.2m)  and 4 different
channels (the 2 subbands and 2
azimuth looks used for the CRLB evaluation).Table 3
RMSE values of the estimated radial
velocity relative to data generated with different values of SCR using the
target deterministic RCS model processed with Gaussian RCS-derived likelihood
functions, using, respectively, the nominal value of SCR(SCRn), the
estimated (by (9)) value of SCR (SCRe), and a fixed value of
SCR = 30 dB (SCRf). All other parameters
(CNR = 10 dB, γc=1,
etc.) are fixed.
SCR = 5 dBSCR = 10 dBSCR = 15 dBSCR = 20 dBurSCRnSCReSCRfSCRnSCReSCRfSCRnSCReSCRfSCRnSCReSCRf1e−33.77e−44.24e−44.14e−41.73e−41.82e−42.26e−49.45e−59.68e−51.23e−45.07e−55.23e−56.33e−52e−35.37e−45.55e−46.48e−42.97e−42.99e−43.54e−41.46e−41.51e−41.93e−48.62e−58.98e−51.08e−43e−31.30e−31.15e−31.21e−33.58e−43.73e−44.75e−41.96e−41.97e−42.71e−41.05e−41.08e−41.42e−4The best performance has been
(of course) obtained using the nominal SCR values (SCRn column) but
the adoption of the estimated SCR values through (21) (SCRe column)
or the fixed one (SCRf column)
do not impair significantly the velocity estimation performance, showing
that there is a weak sensitivity of the velocity estimation from the SCR
values. We have tested the robustness of the ML algorithm with respect to an
unknown SCR in the single-baseline system but we expect a similar behavior in
the dual-baseline system.


## 5.1. Multichannel Ml Velocity Estimation Algorithm Robustness

### 5.1.1. Robustness with Respect to The Target Model

As discussed in Section2, the
deterministic target model is less tractable because, so far, no analytical
statistical description has been obtained, but it is more realistic (targets of
interest—cars, trucks—exhibit RCSs that
do not vary significantly with the radar observation angles [19]). The Gaussian
target model is less realistic (it is rare that cars and trucks response is a
zero mean Gaussian signal which varies significantly inside the synthetic
aperture), but it allows to perform the velocity estimation using analytical
likelihood functions, since an analytical expression for the interferometric
phase pdf is available with great advantages in terms of the computational
efficiency. For this
reason, we present the performance results in terms of RMSE values obtained
using data generated with deterministic RCS, processed with the likelihood functions derived from the Gaussian model.
In other words, we process the AT-InSAR data with a model different from the
actual one used to simulate them. We compare again the single-baseline system
with the dual-baseline one. The RMSE values for
CNR = 10 dB, SCR=0,10,20dB, and
γc=1 for a deterministic modeled moving target obtained
using a likelihood function derived from the deterministic and the Gaussian
models are reported in Figure 16, for (a) the single-baseline system (b1=1.2m) and (b) the dual-baseline system (b1=1.2m, b2=1.8b1m).RMSE values of the estimated target radial normalized velocity, for a
deterministic target likelihood (solid line) and RMSE values for a Gaussian
target likelihood (dashed line), in log scale,
with CNR = 10 dB,SCR=0,10,20dB,  γc=1
and for (a) a single-baseline system
(b1=1.2m)
and (b) for a dual-baseline system
(b1=1.2m, b2=1.8b1m).
(a)(b)It can be noted, quite surprisingly,
that the RMSE values processed with the likelihood functions derived from the
Gaussian model are very close to the ones relative to the deterministic case.
In such a way it is possible to process efficiently actual data with a
tractable estimation algorithm without impairing significantly the estimation
performance.


### 5.1.2. Robustness with Respect to The System Parameters

A further experiment directed to test
robustness consists in varying the parameters of the model that are not
perfectly known. In particular, while CNR andγc can be easily obtained from the system characteristics
(noise power of the SAR sensor) and from the processed SAR
images (clutter
power and coherence), the SCR value is unknown until the target is unknown (we
do not a priori know its position or its RCS). Note also that it is not
necessary to test robustness against CNR since, as highlighted in the comments
to Figures 3 and
4, errors on it do not influence significantly the velocity
estimation accuracy. Instead the role of SCR is quite determinant in the target
velocity estimation procedure with respect to the other parameters, as shown by
the pdf shapes and in the CRLB and RMSE behaviors. For this reason, we test the
performance algorithm when the target RCS value is not a priori known.First of all, we note that the SCR
can be estimated  from the data and the final estimation can be casted as a
joint estimation of velocity and SCR:(21)[u^r,SC^R]=argmaxur,SCRL(ur,SCR),L(ur,SCR)=∏n=1,…,Nm=1,…,MfΦ(ϕn,m∣ur,SCR;ϑ)︸singlechannellikelihoodfunction,   where ϑ denotes  the following parameters λn,bm,γc=1,γT0=1,  CNR0 is computed from the data. We assume a Gaussian target model to derive the likelihood since it allows
to perform the velocity estimation using analytical likelihood function and it
does not degrade the estimation performance, as previously seen, even if the
data are generated using a deterministic target model.As far as the
accuracy of the estimation of SCR is concerned, we show in Figure17(a) some
cuts (relative to different SCR values) of the likelihood function (21) versus
SCR for a fixed radial velocity value. Note that the curves exhibit a similar
shape since the axis of abscissas is in log scale, while in linear scale they
would appear very different from each other, and their spreading increases by
increasing SCR. Consequently, as the best theoretical obtainable accuracy (the
Cramer-Rao lower bound (CRLB)) is directly connected to the spreading of the
likelihood function, the accuracy worsens increasing the SCR. The complete
behavior of the square root of the CRLB for SCR is presented in
Figure 17(b)
and confirms the mentioned trend.(a) Likelihood functions (21) versus SCR
for a fixed radial velocity value; (b) square
root of the CRLB for
SCR.
(a)(b)We analyze also the system performance in
terms of RMSE in the three following cases: (1) the estimation is performed by
adopting for SCR the nominal valueSCRn; (2) SCR is estimated in
conjunction with the normalized radial velocity; (3) the estimation is
performed by adopting for SCR a fixed value SCRf different from the true one. In Table 3,
we report the RMSE values
of the estimated radial velocity using AT-InSAR images simulated with the
system parameters reported in Table 1, with
CNR = 10 dB, SCR = 10 dB, γc=1, and with normalized target velocities ur=1×10−3, ur=2×10−3, ur=3×10−3. The multichannel MLE algorithm
(21)
adopts a single baseline (b1=1.2m)  and 4 different
channels (the 2 subbands and 2
azimuth looks used for the CRLB evaluation).Table 3
RMSE values of the estimated radial
velocity relative to data generated with different values of SCR using the
target deterministic RCS model processed with Gaussian RCS-derived likelihood
functions, using, respectively, the nominal value of SCR(SCRn), the
estimated (by (9)) value of SCR (SCRe), and a fixed value of
SCR = 30 dB (SCRf). All other parameters
(CNR = 10 dB, γc=1,
etc.) are fixed.
SCR = 5 dBSCR = 10 dBSCR = 15 dBSCR = 20 dBurSCRnSCReSCRfSCRnSCReSCRfSCRnSCReSCRfSCRnSCReSCRf1e−33.77e−44.24e−44.14e−41.73e−41.82e−42.26e−49.45e−59.68e−51.23e−45.07e−55.23e−56.33e−52e−35.37e−45.55e−46.48e−42.97e−42.99e−43.54e−41.46e−41.51e−41.93e−48.62e−58.98e−51.08e−43e−31.30e−31.15e−31.21e−33.58e−43.73e−44.75e−41.96e−41.97e−42.71e−41.05e−41.08e−41.42e−4The best performance has been
(of course) obtained using the nominal SCR values (SCRn column) but
the adoption of the estimated SCR values through (21) (SCRe column)
or the fixed one (SCRf column)
do not impair significantly the velocity estimation performance, showing
that there is a weak sensitivity of the velocity estimation from the SCR
values. We have tested the robustness of the ML algorithm with respect to an
unknown SCR in the single-baseline system but we expect a similar behavior in
the dual-baseline system.


## 5.1.1. Robustness with Respect to The Target Model

As discussed in Section2, the
deterministic target model is less tractable because, so far, no analytical
statistical description has been obtained, but it is more realistic (targets of
interest—cars, trucks—exhibit RCSs that
do not vary significantly with the radar observation angles [19]). The Gaussian
target model is less realistic (it is rare that cars and trucks response is a
zero mean Gaussian signal which varies significantly inside the synthetic
aperture), but it allows to perform the velocity estimation using analytical
likelihood functions, since an analytical expression for the interferometric
phase pdf is available with great advantages in terms of the computational
efficiency. For this
reason, we present the performance results in terms of RMSE values obtained
using data generated with deterministic RCS, processed with the likelihood functions derived from the Gaussian model.
In other words, we process the AT-InSAR data with a model different from the
actual one used to simulate them. We compare again the single-baseline system
with the dual-baseline one. The RMSE values for
CNR = 10 dB, SCR=0,10,20dB, and
γc=1 for a deterministic modeled moving target obtained
using a likelihood function derived from the deterministic and the Gaussian
models are reported in Figure 16, for (a) the single-baseline system (b1=1.2m) and (b) the dual-baseline system (b1=1.2m, b2=1.8b1m).RMSE values of the estimated target radial normalized velocity, for a
deterministic target likelihood (solid line) and RMSE values for a Gaussian
target likelihood (dashed line), in log scale,
with CNR = 10 dB,SCR=0,10,20dB,  γc=1
and for (a) a single-baseline system
(b1=1.2m)
and (b) for a dual-baseline system
(b1=1.2m, b2=1.8b1m).
(a)(b)It can be noted, quite surprisingly,
that the RMSE values processed with the likelihood functions derived from the
Gaussian model are very close to the ones relative to the deterministic case.
In such a way it is possible to process efficiently actual data with a
tractable estimation algorithm without impairing significantly the estimation
performance.


## 5.1.2. Robustness with Respect to The System Parameters

A further experiment directed to test
robustness consists in varying the parameters of the model that are not
perfectly known. In particular, while CNR andγc can be easily obtained from the system characteristics
(noise power of the SAR sensor) and from the processed SAR
images (clutter
power and coherence), the SCR value is unknown until the target is unknown (we
do not a priori know its position or its RCS). Note also that it is not
necessary to test robustness against CNR since, as highlighted in the comments
to Figures 3 and
4, errors on it do not influence significantly the velocity
estimation accuracy. Instead the role of SCR is quite determinant in the target
velocity estimation procedure with respect to the other parameters, as shown by
the pdf shapes and in the CRLB and RMSE behaviors. For this reason, we test the
performance algorithm when the target RCS value is not a priori known.First of all, we note that the SCR
can be estimated  from the data and the final estimation can be casted as a
joint estimation of velocity and SCR:(21)[u^r,SC^R]=argmaxur,SCRL(ur,SCR),L(ur,SCR)=∏n=1,…,Nm=1,…,MfΦ(ϕn,m∣ur,SCR;ϑ)︸singlechannellikelihoodfunction,   where ϑ denotes  the following parameters λn,bm,γc=1,γT0=1,  CNR0 is computed from the data. We assume a Gaussian target model to derive the likelihood since it allows
to perform the velocity estimation using analytical likelihood function and it
does not degrade the estimation performance, as previously seen, even if the
data are generated using a deterministic target model.As far as the
accuracy of the estimation of SCR is concerned, we show in Figure17(a) some
cuts (relative to different SCR values) of the likelihood function (21) versus
SCR for a fixed radial velocity value. Note that the curves exhibit a similar
shape since the axis of abscissas is in log scale, while in linear scale they
would appear very different from each other, and their spreading increases by
increasing SCR. Consequently, as the best theoretical obtainable accuracy (the
Cramer-Rao lower bound (CRLB)) is directly connected to the spreading of the
likelihood function, the accuracy worsens increasing the SCR. The complete
behavior of the square root of the CRLB for SCR is presented in
Figure 17(b)
and confirms the mentioned trend.(a) Likelihood functions (21) versus SCR
for a fixed radial velocity value; (b) square
root of the CRLB for
SCR.
(a)(b)We analyze also the system performance in
terms of RMSE in the three following cases: (1) the estimation is performed by
adopting for SCR the nominal valueSCRn; (2) SCR is estimated in
conjunction with the normalized radial velocity; (3) the estimation is
performed by adopting for SCR a fixed value SCRf different from the true one. In Table 3,
we report the RMSE values
of the estimated radial velocity using AT-InSAR images simulated with the
system parameters reported in Table 1, with
CNR = 10 dB, SCR = 10 dB, γc=1, and with normalized target velocities ur=1×10−3, ur=2×10−3, ur=3×10−3. The multichannel MLE algorithm
(21)
adopts a single baseline (b1=1.2m)  and 4 different
channels (the 2 subbands and 2
azimuth looks used for the CRLB evaluation).Table 3
RMSE values of the estimated radial
velocity relative to data generated with different values of SCR using the
target deterministic RCS model processed with Gaussian RCS-derived likelihood
functions, using, respectively, the nominal value of SCR(SCRn), the
estimated (by (9)) value of SCR (SCRe), and a fixed value of
SCR = 30 dB (SCRf). All other parameters
(CNR = 10 dB, γc=1,
etc.) are fixed.
SCR = 5 dBSCR = 10 dBSCR = 15 dBSCR = 20 dBurSCRnSCReSCRfSCRnSCReSCRfSCRnSCReSCRfSCRnSCReSCRf1e−33.77e−44.24e−44.14e−41.73e−41.82e−42.26e−49.45e−59.68e−51.23e−45.07e−55.23e−56.33e−52e−35.37e−45.55e−46.48e−42.97e−42.99e−43.54e−41.46e−41.51e−41.93e−48.62e−58.98e−51.08e−43e−31.30e−31.15e−31.21e−33.58e−43.73e−44.75e−41.96e−41.97e−42.71e−41.05e−41.08e−41.42e−4The best performance has been
(of course) obtained using the nominal SCR values (SCRn column) but
the adoption of the estimated SCR values through (21) (SCRe column)
or the fixed one (SCRf column)
do not impair significantly the velocity estimation performance, showing
that there is a weak sensitivity of the velocity estimation from the SCR
values. We have tested the robustness of the ML algorithm with respect to an
unknown SCR in the single-baseline system but we expect a similar behavior in
the dual-baseline system.


## 6. Conclusions

In this paper, we presented the performance evaluation of multichannel
AT-InSAR systems in terms of moving target detection ability and target radial
velocity estimation accuracy. The analysis has been performed with different
target statistical model and system parameters, such as radial velocity, SCR,
CNR, and number of system channels. In particular, we compared a single-baseline
system with a dual-baseline system. Regarding the detection process, the use of
multichannel interferograms, after the application of a threshold stage to each
channel, allows to adopt a binary integration to combine single-channel
decisions. Such a strategy, compared with the one based on a single
interferogram, provides better results in terms of simultaneous low values ofPFA and high values of PD.With reference to the target radial velocity estimation, two target
models have been considered: deterministic target response and Gaussian target
response. The first model is more realistic and applies to well-characterized
targets, while the latter applies when the target RCS is not accurately known.
The use of the Gaussian model in the velocity estimation procedure has proved
to be particularly appropriate to the case of realistic datasets, since it
allows to take into account the uncertainty on the knowledge of the target RCS.
The mean square errors obtained using the Gaussian model are not very different
from the lower bounds which are obtained using a deterministic model exploiting
a very accurate knowledge of the target RCS. The estimation errors obtained
using the Gaussian model exhibit a low sensitivity to the errors on the
knowledge of the SCR value, which is a parameter difficult to be estimated in the
absence of an accurate target characterization.The analysis of performance which has been presented evidences that a
single-baseline AT-InSAR system allows to obtain accurate radial velocity
estimations approaching the CRLB also in the case of a small number of channels
(only four, in the presented case) for velocities far from the ambiguity value,
while to solve ambiguities at least a dual-baseline AT-InSAR system is
required.The present paper is focused on detection and estimation of interferometric phase
data in order to highlight the performance improvement derived by the use of
more than one channel (baseline). The use of more than two baselines would
allow further performance improvement at the expenses of growing system
complexity and cost.



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*Source: 310656-2008-12-15.xml*