794 IEEE GEOSCIENCE AND REMOTE SENSING LETTERS, VOL. 8, NO.

4, JULY 2011
Efcient Stripmap SAR Raw Data Generation Taking
Into Account Sensor Trajectory Deviations
A. S. Khwaja, L. Ferro-Famil, Member, IEEE, and E. Pottier, Senior Member, IEEE
AbstractIn this letter, a simulation procedure for airborne
stripmap synthetic aperture radar with zero squint angle is pro-
posed. The procedure is based on the idea of inverse processing
and 1-D summation. It utilizes narrow bandwidth approxima-
tion and is more efcient compared to a time-domain simulator.
Moreover, it allows for the simulation of raw data for higher
aperture angle congurations compared to existing approaches.
The effectiveness of the approach is demonstrated by means of
examples for point scatterers.
Index TermsAirborne stripmap synthetic aperture radar
(SAR), broadside geometry, inverse processing, SAR raw data
simulation.
I. INTRODUCTION
S
YNTHETIC aperture radar (SAR) raw data simulation can
be useful for evaluating different processing algorithms,
testing and validating different system design parameters, plan-
ning and preparing for future SAR missions [1], studying
scattering effects [2], studying the effects of motion errors, etc.
For a large realistic scene, the most straightforward approach
of time-domain simulation is very time consuming [3]. In
order to design a simulator that may execute efciently on
ordinary computing platforms, it is necessary to reduce the
computational complexity. To achieve this, frequency-domain
simulation schemes have been proposed [4][6]. However,
these techniques assume an ideal trajectory and cannot be used
directly for simulating raw data in the presence of motion er-
rors. Different approaches have been proposed in the literature
to take this case into account. Vandewal et al. [7] consider
spotlight, while Franceschetti et al. [8] consider stripmap SAR
conguration in broadside geometry, i.e., zero squint angle.
The methods presented in [7] and [8] consist of simulation in
the frequency domain along with a summation, making use of
approximations. Franceschetti et al. [8] also present an error
analysis for the proposed simulation methods that provides an
upper limit of the motion errors for which the methods are valid,
assuming motion errors to be constant. In this letter, we describe
a new approach for raw data simulation based on narrow band-
width (BW) approximations for stripmap SAR conguration
Manuscript received June 24, 2010; revised November 5, 2010; accepted
November 25, 2010. Date of publication March 16, 2011; date of current
version June 24, 2011.
A. S. Khwaja was with the Institute of Electronics and Telecommunications
of Rennes, University of Rennes 1, 35042 Rennes, France (e-mail: shahar-
yar.khwaja@yahoo.com).
L. Ferro-Famil and E. Pottier are with the Institute of Electronics and
Telecommunications of Rennes, University of Rennes 1, 35042 Rennes, France
(e-mail: Laurent.ferro-famil@univ-rennes1.fr; eric.pottier@univ-rennes1.fr).
Color versions of one or more of the gures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identier 10.1109/LGRS.2011.2111411
with zero squint angle. This approach can simulate raw data for
high azimuth aperture angle. This letter is organized as follows.
Section II presents a SAR raw data model in the presence
of motion errors. Section III presents the proposed simulation
approach based on narrow BW approximation. Section IV sum-
marizes the errors arising due to approximations used in this
method. Section V gives illustrative examples where the quality
of motion-compensated images obtained from simulated data is
compared with that of the images obtained from reference raw
data generated by a time-domain simulator.
II. SAR RAW DATA MODEL
SAR raw data acquired by a sensor emitting a pulse p(t) at
a given ight position for a at scene consisting of n
r
n
y
scatterers in the range (across ight) and azimuth (along ight)
directions, respectively, can be described as a sum of individ-
ual contributions weighed by the backscattering coefcients
(r
m
, y
n
)
s(t, y)=

m,n
(r
m
, y
n
)p
_
t
2d
mn
(y)
c
_
exp(j2f
c
t). (1)
The size of s(t, y) is denoted as N
r
N
y
pixels, and values
of (r
m
, y
n
) (m denotes the range position, whereas n denotes
the azimuth position) are given by a discrete reectivity map
(r, y) dened as follows:
(r, y) =

m,n
(r
m
, y
n
)(r r
m
, y y
n
). (2)
The sensor target distance is
d
mn
(y) =
_
{r
m
r
em
(y)}
2
+{y y
n
y
e
(y)}
2
(3)
where r
em
(y) and y
e
(y) represent the effective motion errors
in the slant-range and azimuth directions, respectively.
The former term can be rewritten as [9] r
em
(y) =
d
e
(y) sin(
m
+
xz
(y)), where d
e
(y) =
_
x
2
e
(y) + z
2
e
(y),

xz
(y) = arctan{z
e
(y)/x
e
(y)}. x
e
(y) and z
e
(y) represent
the motion errors in the ground range and height directions,
respectively.
m
is dened in Table I. The motion errors in the
azimuth direction are usually removed by azimuth resampling
of SAR raw data [10] and henceforth shall be ignored. The
motion errors in the range direction can be compensated using
motion compensation techniques [9], [11], [12].
Equation (1) describes the time-domain simulation consist-
ing of 2-Dsummations and is computationally intensive, requir-
ing n
r
n
y
N
r
N
y
operations. However, it serves as a reference to
compare the accuracy of the simulation method proposed in the
following section.
1545-598X/$26.00 2011 IEEE
KHWAJA et al.: EFFICIENT STRIPMAP SAR RAW DATA GENERATION 795
TABLE I
LIST OF SYMBOLS
III. EFFICIENT SAR RAW DATA SIMULATION
The expression of raw data in the transmitted-wavenumber
azimuth-distance domain can be calculated using the principle
of stationary phase (POSP) [13], as given hereinafter
S(k, y) = P(k)

m,n
(r
m
, y
n
)S
emn
(k, y)
exp
_
j2k
_
r
2
m
+ (y y
n
)
2
_
(4)
where
S
emn
(k, y) = exp {j2kr
emn
(y)} (5)
r
emn
(y) =
_
{r
m
r
em
(y)}
2
+ (y y
n
)
2
+
_
r
2
m
+ (y y
n
)
2
(6)
represents the motion errors. P(k) is the Fourier transform
(FT) of p(t). Equation (4) consists of 2-D summations, similar
to (1), and its direct use for raw data generation has no advan-
tage over the time-domain simulation: The motion errors vary
for each point scatterer position, and raw data for each point
have to be generated and summed together.
Equation (6) can be expanded to observe that the motion
errors consist of two main parts
r
emn
(y) r
em
(y)
(y
n
y)
2
2 (r
m
r
em
(y))
+
(y
n
y)
2
2r
m
r
em
(y)
(y
n
y)
2
r
em
(y)
2r
2
m
. (7)
The rst part represents the error in the center of the azimuth
beam, known as narrow beamwidth error, and the second part
represents the azimuth-position-dependent motion errors.
An efcient method to simulate the data is considered
in [7] and [8], by means of partial summing along with
frequency-domain simulation. Franceschetti et al. [8] consider
the narrow beamwidth approximation [9][11] for motion er-
rors. This approximation states that motion errors across the
aperture are the same as those in the center of the azimuth
beam, i.e., S
em
(k, y) = exp{j2kr
em
(y)} where r
em
(y)
x
e
(y) sin
m
+ z
e
(y) cos
m
. Making use of this approximation
and the POSP in the azimuth direction, the raw data in the 2-D
wavenumber domain can be written as
S(k, y) =

m
F
1
y
_
exp
_
j
_
4k
2
k
2
y
r
m
_
(r
m
, k
y
)
_
P(k)S
em
(k, y) (8)
where (r
m
, k
y
) is the 1-D FT of the reectivity map in
azimuth direction
(r
m
, k
y
) =

n
(r
m
, y
n
) exp(jk
y
y
n
)(r r
m
). (9)
This is the most accurate of the two simulation techniques
presented in [8] and is described as narrow azimuth beam and
arbitrary deviation approach. However, it ignores the azimuth-
position-dependent motion errors leading to a phase error of

e1
=k(y
n
y)
2
r
em
(y)/r
2
m
with respect to raw data generated
by a time-domain simulator. This error manifests itself in the
azimuth direction and after processing will degrade the image
quality and may not allow higher aperture angle simulations. In
order to reduce this error, the concept of amplitude modulation
of chirp signals presented in [9] and [16] can be used. Accord-
ing to this concept, if the variation of motion errors is small, raw
data can be expressed in 2-D wavenumber domain as follows:
S(k, k
y
) = P(k)

m,n
(r
m
, y
n
)S
emn
(k, k
y
)
exp
_
jr
m
_
4k
2
k
2
y
jk
y
y
n
_
(10)
where S
emn
(k, k
y
) is related to S
emn
(k, y) via
y = y
n

k
y
_
4k
2
k
2
y
r
m
. (11)
Raw data can be processed by the range migration algorithm
[9], [17] or the Omega-k algorithm [15], [18], summarized by
I(k
r
, k
y
) = P

(k) [F(k, k
y
)S(k, k
y
)]

4k
2
k
2
y
k
r
(12)
where F(k, k
y
) is a matched lter for a reference range given by
F(k, k
y
) = P

(k) exp
_
jr
ref
_
4k
2
k
2
y
_
(13)
and
_
4k
2
k
2
y
k
r
represents interpolation. I(k
r
, k
y
) is the
resulting processed image in the interpolated 2-D domain, and
in case of no motion errors, i.e., S
emn
(k
r
, k
y
) = 0, it can be
written as
I(k
r
, k
y
)=|P(k)|
2

m,n
(r
m
, y
n
)exp(jk
r
r
m
jk
y
y
n
). (14)
This represents the bandlimited version of 2-D FT of the reec-
tivity map (r, y), where this bandlimiting (assuming rectan-
gular antenna patterns) is given in the transmitted wavenumber
direction by |P(k)|
2
and in the azimuth wavenumber direc-
tion by |k
y
| 2k sin(
y
/2) due to the limited aperture angle

y
. The chirp-scaling algorithm can also be used to process the
raw data by approximating the interpolation in (20).
Raw data can be generated efciently by manipulating (12)
to obtain the following equation:
S(k, k
y
) = [I(k
r
, k
y
)]
k
r

4k
2
k
2
y
P(k)F

(k, k
y
). (15)
Thus, an image can be used to generate raw data by means
of interpolation and 2-D ltering. This process is described in
[6] as equivalent to inverse image formation. Eldhuset [5] also
describes this simulation for satellite data using the inverse-
extended exact transfer function. However, to take into account
the motion errors, the input image must contain this extra
information. For this purpose, (10) can be used, and in order to
796 IEEE GEOSCIENCE AND REMOTE SENSING LETTERS, VOL. 8, NO. 4, JULY 2011
maintain the equivalence of the scaling approach, motion errors
can be divided in two steps [9]: narrow beamwidth motion
errors for a reference range r
ref
S
eref
(k, y) = exp {j2kr
ref
(y)} (16)
and differential motion errors
S
dmn
(k
r
, k
y
) = S
emn
(k
r
, k
y
) S
eref
(k
r
, k
y
). (17)
Considering narrow BW approximation, i.e., k k
c
, (12) can
be approximated as
I(k
r
, k
y
) |P(k)|
2

m,n
(r
m
, y
n
) exp(jk
r
r
m
jk
y
y
n
)
exp [j2k
c
{r
emn
(y) r
eref
(y)}] (18)
where
y y
n

k
y
2k
c
r
m
(19)
r
erefn
(y) = r
emn
(y)|
m=ref
. (20)
As a result of this approximation, an azimuth FT of the modied
reectivity map can be obtained as
(r, k
y
) =

n
(r, k
y
) (21)
where
n
(r, k
y
) is given by (22), shown at the bottom of
the page.
Equation (23)
r
ern
(.) =

m
r
emn
(.)(r r
m
) (23)
is a map of differential motion errors for all the range positions
and each azimuth position. The result is
I(k
r
, k
y
) = |P(k)|
2
F
r
{(r, k
y
)} . (24)
Therefore, a modied reectivity map with embedded motion
errors can be generated for every azimuth position and all the
range positions and added together using the expression given
in (21) that consists of summation for each azimuth position.
The result is then utilized for generating an image I(k
r
, k
y
)
containing the motion errors in 2-D wavenumber domain by
multiplying F
r
{(r, k
y
)} with a bandlimiting lter or 2-D
convolution with a reference image. Subsequently, raw data are
generated from (15) and multiplied with S
eref
(k, y). Apart
from generating raw data, as a rst step, an image containing
differential motion errors is also obtained by this procedure.
TABLE II
SIMULATION PARAMETERS (STRIPMAP SAR WITHOUT ANY SQUINT)
Fig. 1. Simulated motion errors.
This process results in computational savings compared to a
time-domain simulation, as the summation in range direction
is taken care of by FTs. The total computations may be ap-
proximated as n
y
N
r
N
y
that are less than those required by a
time-domain simulator but more than the narrow azimuth beam
and arbitrary deviation approach of [8]. Please note that, if we
use both narrow BW and narrow beamwidth approximations
for differential motion errors, (15) can be used to generate
raw data efciently as described in (25), shown at the bottom
of the page. In fact, this is equivalent to the narrow azimuth
beam and moderate deviation approach presented in [8] but is
computationally much more efcient, requiring approximately
5N
r
N
y
computations.
IV. ERROR SOURCES
The simulation method presented in Section III consists of
three approximations.
1) exp{jk(y
n
y)
2
r
em
(y)/r
2
m
} is ignored in the az-
imuth direction due to narrow BW approximation. The
resulting phase error is less compared to that in [8]
and allows for higher aperture angle simulations. This
is so because in [8], the ignored term is exp{jk(y
n

y)
2
r
em
(y)/r
2
m
}, whereas we include the dependence on

n
(r, k
y
) = exp(j2k
c
r jk
y
y
n
)(r, y
n
) exp
_
j2k
c
_
r
ern
_
y
n

k
y
2k
c
r
_
r
eref
_
y
n

k
y
2k
c
r
ref
___
(22)
S(k, k
y
) =F
r
_
F
y
_
F
1
y
_
I(r, k
y
) exp
_
j
_
4k
2
c
k
2
y
r
__
S
em
(k
c
, y)
_
exp
_
j
_
4k
2
c
k
2
y
r
_
_
k
r

4k
2
k
2
y
P(k)F

(k, k
y
)
(25)
KHWAJA et al.: EFFICIENT STRIPMAP SAR RAW DATA GENERATION 797
Fig. 2. (a), (b) Contour plots for NR and FR using raw data simulated by the proposed approach with azimuth aperture = 10

. (c), (d) Contour plots for NR and

FR using raw data simulated by (8) with azimuth aperture = 10

.
Fig. 3. (a), (b) Contour plots for NR with original and reduced motion errors using raw data simulated by the proposed approach with chirp bandwidth =
200 MHz. (c), (d) Contour plots for NR with original and reduced motion errors using raw data simulated by the proposed approach with carrier frequency =
9.6 GHz.
k
c
. Thus, the phase error arising in the azimuth direction
is reduced by a factor of k/(k
c
+ k).
2) An error of exp{j2k(r
em
(y) r
eref
(y))} arises in
the time direction that causes a mispositioning of the
compressed image in the range direction leading to a
resolution loss in azimuth direction. For this error to
remain small, the following constraint should be satised:
max [2k {r
emin
(y) r
eref
(y)}] 1. (26)
This error is the same as for the narrow azimuth beam
and moderate deviation approach in [8]. The misposi-
tioning varies with the distance between range position
and reference range and is more severe for near range
(NR) than for far range (FR). The reason is that r
em
=
d
e
(y) sin(
m
+
xz
(y)), which is greater in amplitude to-
ward the NR compared to the FR. The resulting phase er-
ror can be ignored as long as max{r
emin
(y) r
eref
(y)}
is small and a reasonable limit is less than 1/8th of
pixel size in range direction. This has been veried by
simulations and can be inferred from the discussion on
interferometry in [10].
3) The scaling in (19) is based on the fact that, for a
chirp function, multiplication in time or distance domain
by a function is equivalent to multiplication by another
function in wavenumber domain. The relation between
the two functions is given by the scaling that is valid if
the BW of the simulated error is much smaller than that
of the azimuth chirp, i.e.,
BW [exp {j2k
c
(r
emn
(y) r
eref
(y))}]
BW [exp {j2k
c
y
2
/(2r
m
)}]
1. (27)
A practical limit for the aforementioned ratio veried by
simulation results is less than 2%. This error is more
severe for NR than for FR.
V. EXAMPLES
In order to demonstrate the effectiveness of the proposed
approach, simulation was carried out using the parameters
given in Table II and the platform motion errors having a
maximum amplitude of 6.5 m shown in Fig. 1. These errors
are severe and will cause signicant image quality degradation
if not compensated. The azimuth aperture angle was chosen
as 10

, and two points were simulatedone toward the FR

at a distance of 500 m in range from the scene center and
the other is 500 m in range from the scene center toward the
NR. The reference point was chosen to be at a distance of
150 m from the center range toward NR. The reason is that
the errors arising due to approximations vary strongly toward
the NR, and choosing the reference point in the middle of the
scene can cause signicant error in the NR simulated data.
Raw data were simulated using the proposed approach. Mo-
tion errors were compensated completely followed by standard
image formation. The resulting contour plots for both NR and
FR are shown in Fig. 2(a) and (b). Difference in integrated
sidelobe ratio (DISLR) and impulse response widening (IRW)
are also calculated with respect to reference images generated
by time domain simulated and processed data, so that they can
be used as criteria for judging the effectiveness of the proposed
method. The satisfactory limits for DISLR and IRW are chosen
as between 10% and 12% and 5% and 7%, respectively. This
is based on the discussion in [15] concerning quadratic phase
error and its effect on a point response. The gures show that
these differences remain within satisfactory limits; therefore,
798 IEEE GEOSCIENCE AND REMOTE SENSING LETTERS, VOL. 8, NO. 4, JULY 2011
the proposed method is well suited for high aperture angle
simulations. To demonstrate improvement in the performance
offered by the proposed method, contour plots and correspond-
ing values for DISLR and IRW using the narrow azimuth beam
and arbitrary deviation approach presented in [8] are shown in
Fig. 2(c) and (d). It can be seen that the DISLR and IRW values
are much higher, and thus, the simulation approach presented
in this letter outperforms the existing approach.
For further conrmation, range BW was doubled. The cor-
responding contour plot is shown in Fig. 3(a). The results ob-
tained using the simulated error give a DISLR and IRW of 6%
and 11% toward the NR, respectively. Subsequently, the maxi-
mum amplitude of the simulated motion errors was reduced by
25%. The contour plot toward the NR given in Fig. 3(b) shows
that the difference decreases. Thus, the errors in the simulation
increase with the increase of BW due to the second approxima-
tion. Similarly, the carrier frequency was increased to 9.6 GHz,
and the azimuth aperture angle was decreased to 4

. In this
case, BW of the motion errors increases due to higher carrier
frequency. When motion errors given in Fig. 1 were simulated,
the DISLR and IRW values of 17% and 2% were obtained to-
ward the NR, as shown in Fig. 3(c). The corresponding contour
plot obtained by simulating reduced motion errors is shown
in Fig. 3(d). It can be seen that the DISLR and IRW values
decrease.
Please note that the errors in the simulation vary according to
the distance from the reference range as discussed in Section IV
and are not dependent on azimuth positions. However, in partic-
ular, for squint-mode geometry, errors in azimuth direction for
points at the edges of a simulated scene may occur due to the
interpolation-based raw data simulation scheme. These errors
can be compensated by increasing the quality of the interpolator
or using modied interpolation as proposed in [19].
VI. CONCLUSION
In this letter, a new method for SAR raw data simulation
in the presence of sensor motion errors has been described.
This approach is based on narrow BW approximation and has
two main advantages: First, it is more efcient than a time-
domain simulator, as it replaces 2-D summation with a 1-D
summation, owing to the use of Fourier transforms. Second,
it allows simulation of higher aperture angles compared to the
existing simulation approach. This has been demonstrated by
examples where simulated raw data were motion compensated
and processed, and the subsequent contour plots were shown.
Hence, it can be used to simulate motion errors for an extended
scene such as an urban scene and check the effectiveness of
different motion compensation techniques. It is also shown that
the proposed approach is sensitive to chirp BW and carrier
frequency.
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