1.2 Review of topic: Why and what is new
Synthetic Aperture Radar (SAR) has been used for several remote sensing applications for several
years. The constantly improving techniques in imaging, instruments and scattering modelling, has
brought this kind of science to study different topics. Especially the introduction interferometry,
(POL-InSAR) a technique based on the combination of Polarimetric and Interferometry has been
improved the possibilities in earth observation using SAR technology to study and monitoring a lot
of parameters. In fact combining orientation, or polarization of radar signals (Polarimetry) with the
analysis of the phase differences in the signal to produce differential range and range-change
measurements (interferometry) from two or more images, is possible to see the Earth in three
dimensions. With this skill, the POLInSAR can be used for monitoring of agriculture, forest, urban
areas, glaciers and sea ice. Especially for forests, this technique is useful to gain understanding
about the impact of global climate change on terrestrial ecosystem, or just to monitor a wide range
of applications such as detection of illegal logging, estimation forest height, or for biomass
estimation. Forest height is one important parameter of a forest which can be measured using SAR.
In literature several models are proposed to estimate forest height. Among these, the Random
Volume over Ground model (RVoG) using PolInSAR, is an established technique for forest height
estimation [11,12]. In the last years, forest height inversion has been demonstrated for several types
of forests and for different frequencies [20,33]. Validated results for boreal forests in X and L band
are presented in [24,25]. Also for a tropical forest, an inversion in L and P band is shown by
[26,27,28,29]. For temperate forests, validated results in L band were presented by
[12,30,31,32,33]. Others authors [14,21] propose the usage of the same frequencies, X and P band
for forest height estimation. The technique is based on the calculus of the height as the difference of
the phase centers. The low frequency penetrates down to the ground, on the contrary scattering
centre is closed to the canopy. Also Cloude proposes in [23] the combination of two frequencies for
forest height estimation, which could reduce errors introduced by the underlying assumptions.
Using a low frequency for the estimation of the ground phase and a high frequency to estimate
forest height using RVoG model, in this thesis the RVoG model will be evaluated using E-SAR
data from the “HOMSAR campaign”. The focus of the thesis is the estimation of extinction and
penetration depth in different frequencies and polarizations. In the end of the second part the forest
height estimation will be performed using a single X band polarization under the assumption that
the signal in X band, comes only from the “volume” part of the forest. In general, the quality of
forest height estimation depends on the realization of the underlying assumptions.
2
Chapter 2: Radar theory
2.1 SAR fundamentals
SAR is an acronym for Synthetic Aperture Radar, and it is an imaging technique for the surface of
the Earth, from a spaceborne or airborne platform. To do this, it points a radar beam approximately
perpendicular to the sensor’s motion vector, transmitting phase-encoded pulses, and recording the
radar echoes by the Earth’s surface. An illustration of the SAR side looking geometry can be found
in fig. 2.1. [5]
y=ground range
x=azimuth
S
l
a
n
t
r
a
n
g
e
Ground range
Swath
F
a
r
r
a
n
g
e
N
e
a
r
r
a
n
g
e
s
l
a
n
t
r
a
n
g
e
F
l
i
g
h
t
d
i
r
e
c
t
i
o
n
ϑ
r
A
l
t
i
t
u
d
e
z
A
l
t
i
t
u
d
e
Figure 2.1. SAR geometry
In the side-looking geometry (see fig 2.1) , the dimension, along the radar beam, is called “slant
range”. The beam of the antenna on the ground (antenna footprint) covers a strip (stripmap mode
operation) on the surface. The width of the strip is called image swath. Then, the platform motion
provides the scanning in the direction of the sensors’ trajectory, which is referred to as azimuth or
along-track direction, which is the second dimension (x-axis). The azimuth dimension of a radar
image (in flight direction) is generated by the sequence of radar pulses that the transmitter emits
consecutively to the ground during the flight.
3
2.2 Range processing and resolution
The transmitter emits fast consecutively short radar pulses to the ground. These pulses are reflected
from a target on the ground and, after a certain time delay, they reach again the receiver. The time
delay, which is a function of the distance between the sensor and the target, can be expressed as
[36]:
c
r
t
d
2
=Δ (2.1)
where
d
r is the distance from the sensor to the target and c is the speed of the light.
It is defined “resolution length” the minimum distance between two objects that are detected as
separate entities and therefore resolved. In the two dimensions there are two different resolutions:
the azimuth resolution δ
az
, and the range resolution δ
sr
.To achieve a high resolution in the
perpendicular flight direction only a short pulse of duration τ is necessary. In fact, to distinguish
two different objects at distances R1 and R2 from the sensor, using the formula (2.1) hence [36]:
τ+=
c
R
c
R 1222
(2.2.1)
where τ is the length of the signal. The difference between R1 and R2 is the resolution:
ΔR=δ
sr
=R2-R1=
2
τc
(2.2.2)
The formula can be expressed as a function of the band width of the signal:
f
c
sr
Δ
≅
2
δ (2.2.3)
Where τ/1≅Δf equals approximately the band width of the pulse. Improvement of the resolution
requires a reduction of the pulse length τ and high peak power for a prescribed mean power
operation. It can be problematic to generate a very short and high power pulse, as the resulting
energy densities are hard to handle. In the spectral domain, short pulse durations are expressed as a
higher signal bandwidth. A high resolution is therefore tantamount with a high signal bandwidth. A
second possibility to generate a high signal bandwidth is to use a long, but frequency modulated
pulse. A way to avoid this limitation is to substitute the short pulse with modulated long ones,
provided that they are followed by a processing step called pulse compression. It is common to use
a linear frequency modulation called “chirp”, linear frequency modulated pulse. Considering a
4
chirp, with a modulated frequency of duration T
p
in the band interval between
2
fΔ
and
2
fΔ
− , after
subtracting the frequency centre f
0
:
22
)(
pp
p
r
T
t
T
witht
T
f
tf ≤≤−
Δ
−= (2.2.4)
the chirp can be expressed, as:
22
2
2
2
)(2
pp
t
T
f
i
r
dttfi
rr
T
t
T
eAeAS with
p
r
≤≤−=
∫
=
Δ
π
π
(2.2.5)
where
r
A is the amplitude of the echo of a point scatterer and
∫
dttf2i
r
e
)(π
describes a quadratic
phase behaviour. The echo of a single point target is contained in many range samples and appears
therefore defocused. The aim of range processing, also called compression, is to focus all the
received energy of a target, distributed over the illumination time, on one point at t=0. In order to
get the resolution of a scatterer, a matched filter has to be correlated with the echo S
r
. The matched
filter is made to compensate the quadratic phase of the echo. The matched filter can be expressed as
[36]:
22
)(
2
pp
t
T
f
i
pr
T
t
T
withefTtR
p
≤≤−Δ=
Δπ
( 2.2.6)
where the energy of the signal is T
p
A
r
2
.
The correlation between the echo and the matched filter can be expressed as [36]:
() () ()
() ( )
()
ft
ft
eAfT
dtRS
T
tRtStV
t
T
f
i
rp
rr
p
rrr
p
Δ
Δ
Δ=
+=
⊗=
Δ
∞+
∞−
∫
π
π
ξξξ
π
sin
*
1
2
( 2.2.7)
The amplitude of ()tV
r
is:
()
()
ft
ft
AfTtV
rpr
Δ
Δ
Δ=
π
πsin
( 2.2.8)
5
The result of this correlation is the range compressed image. The principal shape of the resulting
impulse response corresponds thereby to the impulse response shaped as a sinus cardinal function
(sinc). In figure 2.2 and figure 2.3, examples of echo and the matched filter are illustrated.
Figure 2.2: Real part of the Chirp
Figure 2.3: Graphic of a matched filter shifted
6
The received signal has a constant amplitude and a quadratic phase behaviour (shown is only the
real part of the complex signal). The matched filter has the amplitude of one and exactly the
opposite phase of the signal itself. After the correlation with R
r
(t) the signal appears well located at
the true range position. In fig.2.4 the result of the compressed range signal (sinc) is illustrated:
Figure 2.4: Image resulting from one point scatterer
The resolution in ( 2.2.7 ) corresponds to the 4dB angle of the main lobe of the compressed signal,
(half amplitude of the sinc main lobe). Typically, various functions are applied to suppress side-
lobes, e.g. Hamming weighting.
7
2.3 Azimuth processing and resolution
As seen for the range, in the next step the separation of two points scatterers located along the
azimuth direction is, at the same distance from the acquisition system, investigated. To do this,
consider the geometry of Fig 2.5. For a RAR (Real Aperture Radar), two targets at a given range
cannot be resolved if they are within the antenna beam at the same time.
L
x
L
λ
LrX λ=
r
Fig 2.5: Azimuth geometry (RAR)
Therefore, the antenna beamwidth at the ground represents in the angular azimuth resolution
az
α
for a RAR and is equal to [36]:
L
az
λ
α = (2.3.1)
and the azimuth spatial resolution is equal to :
L
r
X
λ
= (2.31.)
where, r is the slant range and L is the physical antenna dimension along the azimuth direction.
Since X is dependent on r, it would be possible to improve the azimuth resolution either by reducing
the wavelength λ or by increasing the antenna length L. Either case are difficult to be achieved.
Indeed, short wavelengths can cause electromagnetic field dispersion through the atmosphere (i.e.
space born remote sensing) while large distances between the sensor and the ground would require
unfeasible antenna dimensions.
8
A SAR, as above mentioned, provides a solution to these limiting problems, as it synthesizes a
larger antenna aperture along the azimuth direction by coherently combining the backscattered
echoes received along the flight path.
So, considering the SAR geometry in figure 2.6 below:
y
=
r
a
n
g
e
x
=
a
z
i
m
u
t
h
S
a
t
e
l
l
i
t
e
f
l
i
g
h
t
p
a
t
h
r
d
z=altitude
r
g
n
g
z
1
L
s
a
r
0
α a
z
Figure 2.6: Geometry of a SAR image. Lsa, length synthetic aperture antenna,
az
α , angular azimuth
resolution,
0
r , slant range coordinate,
1
z altitude to nadir point
The angular resolution of a synthetic aperture (SAR),
azSAR
α , of the length L
sa
is two times higher
than the one of a real aperture of the same length:
==
azazSAR
αα 2
sa
L2
λ
(2.3.2)
A SAR system takes advantage of the fact that the response of a scatterer on the ground is contained
in more than a single radar echo, and shows a typical phase history over the illumination time. An
appropriate coherent combination of several pulses leads to the formation of a synthetically
enlarged antenna called synthetic aperture. A graphic of the imaging geometry of SAR is shown in
fig. 2.6.
9
The phase differences between elements of the synthetic aperture result from a two-way path
difference and are therefore, two times larger than in the case of a real antenna. Like for the Chirp
in range, also in azimuth the phase history is quadratic, because the distance changes quadratically
with the flight motion x [36]:
2
)(
2
kx
i
r
xi
ra
eAeAS
−
==
ϕ
(2.3.3)
and like for the chirp, a point scatterer can be reconstructed by the convolution of the signal with a
matched filter function:
222
)(
2
2
sasa
kx
i
saa
L
x
L
fore
k
LxWR ≤≤−
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
π
(2.3.4)
where L
sa
is the length of the synthetic antenna, W(x) is a weight function or a rect function defined
as:
⎪
⎩
⎪
⎨
⎧
≤≤−
=
Elsewhere
T
t
T
for
tW
PP
0
22
1
)( (2.3.5)
The correlation is in this case expressed as:
() ( )
∫
+∞
∞−
+=⊗= ξξξ dxRS
L
RSxV
aa
sa
aaa
1
)( (2.3.6)
Substituting S
a
and R
a
with equation (2.3.3) and (2.3.4) the correlation is expressed as:
()
∫
−
+=
2
2
2
2
2
)(
sa
sa
L
L
ikx
kx
i
ra
dexWe
k
AxV ξξ
π
ξ
(2.3.7)
considering the approximation ()( ) ( )ξξξ −==+ WWxW is possible to express the correlation as:
()
(){}ξπ
π
ξξ
π
ξ
WFe
k
A
deWe
k
AxV
kx
kx
i
r
L
L
ikx
kx
i
ra
sa
sa
2
2
2
)(
2
2
2
2
2
2
≅
≅
∫
−
(2.3.8)
Where with F
kx
has been denoted the Fourier-Transform of the weight function (2.3.4) can be
expressed in module as:
10
(){}ξ
π
WFA
k
xV
kxra
2
)( = (2.3.9)
The last expression can be expressed also as :
2
2
sin
2
)(
sa
sa
sarr
L
kx
L
kx
LA
k
xV
⎟
⎠
⎞
⎜
⎝
⎛
=
π
(2.3.10)
Interestingly, the achieved resolution is in SAR, completely independent of the range distance and
is determined only by the size of the real antenna. The resolution can be expressed as:
2
L
az
=δ (2.3.11)
In summary, the radar response takes the waveform of sin(x)/x in both range and azimuth direction,
it is a result of compressing a signal with a quadratic phase history. In Figures (2.7.1 and 2.7.2) the
raw data and the compressed version of a single point scatterer in range and azimuth direction is
shown:
Fig. 2.7.1: Raw data of a single scatterer point. Amplitude.
11
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