Along-Track Resolution Enhancement Forwide-Bandwidth, Low-Frequency SAR by Accounting for the Wavelength Change over the Bandwidth

Abstract

Common methods for processing SAR data, such as the Range-Doppler Algorithm (RDA) and the Chirp Scaling Algorithm (CSA), make certain approximations about the SAR signal. As the transmit frequency drops, the bandwidth grows, and the beamwidth increases, the approximations miss important factors required to precisely process the data. This paper shows that the approximations correspond to keeping lower order terms of the expansion of the SAR transfer function. We demonstrate the limits for focusing low frequency SAR data with these approximations. Previous methods for correcting the approximation errors are shown and a new method for including an arbitrary number of terms in processing the data is discussed. The concepts presented are verified using simulated SAR data.

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ALONG-TRACK RESOLUTION ENHANCEMENT FOR WIDE-BANDWIDTH,
LOW-FREQUENCY SAR BY ACCOUNTING FOR THE WAVELENGTH CHANGE OVER THE
BANDWIDTH
Evan C. Zaugg and David G. Long
Brigham Young University Microwave Earth Remote Sensing Laboratory
459 Clyde Building, Provo, UT 84602 801-422-4884 zaugg@mers.byu.edu
ABSTRACT
Common methods for processing SAR data, such as the Range-
Doppler Algorithm (RDA) and the Chirp Scaling Algorithm (CSA),
make certain approximations about the SAR signal. As the transmit
frequency drops, thebandwidth grows, and thebeamwidth increases,
the approximations miss important factors required to precisely pro-
cess the data. This paper shows that the approximations correspond
to keeping lower order terms of the expansion of the SAR transfer
function. Wedemonstrate thelimitsforfocusing low frequency SAR
data with these approximations. Previous methods for correcting the
approximation errors are shown and a new method for including an
arbitrary number of terms in processing the data is discussed. The
concepts presented are verified using simulated SAR data.
1. INTRODUCTION
New synthetic aperture radar systems operating with wide band-
widths at low frequencies attract attention due to the potential for
improving the data quality of existing applications and investigating
new uses. At low frequencies the approximations made in formu-
lating a number of SAR processing algorithms, such as the Range-
Doppler Algorithm (RDA) and the Chirp Scaling Algorithm (CSA),
are no longer good approximations. The errors caused by many
of these approximations have been addressed, together with sug-
gested remedies, in previous works. Also at low frequencies, a wider
beamwidth is required for high azimuth resolution, this causes prob-
lems with the center beam approximation used in motion compen-
sation [1] and in the chirp scaling process used in the CSA [2]. The
widebandwidth means thatthewavelengthchanges appreciably over
the radar chirp, but the approximations fail to fully account for the
widely varying wavelength, resulting in azimuth de-focusing [3].
Processing for low frequency SAR’s is typically done with the
wavenumber domain Omega-K [4] algorithm or time domain pro-
cessing methods [5]. These methods avoid the approximations that
make RDA and CSA problematic at low frequencies and with wide
beamwidths. Unfortunately these methods are much less computa-
tionally efficient. The ω-k processing requires a costly interpolation
to perform the Stolt mapping, and the algorithm makes it difficult to
implement a range-dependent motion compensation method. Time
domain methods can be very precise for all SAR configurations, but
are very inefficient. Efforts have been made to modify the CSA to
efficiently handle the effects of the wide aperture and varying wave-
lengths [1], to extend the ω-k algorithm to handle motion compen-
sation [4], and to streamline the time domain methods.
This paper explores the effects of the approximations made in
SAR processing. In Section 2, the general SAR signal is derived
and in Section 3, the approximations are analyzed. Simulated data is
presented to demonstrate the effects of the approximations. Section
4 explains the previous efforts to efficiently handle the approxima-
tion errors, and Section 5 presents the possibility of a generalized
algorithm that accounts for an arbitrary number of terms in the ap-
proximation is discussed.
2. THE GENERAL SAR SIGNAL
For our analysis, we consider only the phase functions of the SAR
signal, ignoring the initial phase. As in the development presented
in [6] we can describe the phase of the demodulated baseband SAR
signal as
Φ0 = −4πf0R(η)/c + πKr(τ − 2R(η)/c)2
(1)
where f0is the carrier frequency. R(η) is the range to a given target
at slow time η. Kris the chirp rate and τ is fast time.
The first term describes the azimuth modulation, it consists of
the phase left over after demodulation. It is purely a function of the
carrier frequency and the changing range to a target. The second
term in Eq. (1) is the transmit chirp delayed by the two-way travel
time to the target. If we were to reduce our bandwidth to a single
frequency, the second term would go to zero, but we would still have
the same azimuth modulation.
The approximations made in SAR processing algorithms are
madetothesignal inthewavenumber, ortwo-dimensional frequency
domain. To obtain an expression for the signal in this domain, we
take the range and azimuth Fourier transforms of Eq. (1). We ap-
proximate the Fourier transforms using the principle of stationary
phase (POSP), which is valid expect in the extreme case of having
radar frequencies near zero.
As shown in [6], the wavenumber domain expression is
−4πR0(f0+ fτ)
c
Φ1RA
=
s
1 −
c2f2
η
4v2(f0+ fτ)2−πf2
τ
Kr
=
−4πR0f0
c
s
D2(fη) +2fτ
f0
+f2
τ
f2
0
−πf2
Kr
τ
(2)
where
D(fη) =
s
1 −
c2f2
4v2f2
η
0
,
(3)
R0 is the range of closest approach, v is velocity, fτ is range fre-
quency, and fηis azimuth frequency.
Eq. (2) is the phase of the SAR signal in the wavenumber do-
main. For a target at a given range Rref, the target can be ideally
focused with the reference function multiply
HRFM =4πRreff0
c
s
D2(fη) +2fτ
f0
+f2
τ
f2
0
+πf2
Kr
τ
(4)
This works regardless of squint, beamwidth, and chirp bandwidth.
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Fig. 1. 500 MHz bandwidth SAR data is simulated for a single point target at three frequencies with two processing algorithms. The columns
from left to right are the center frequencies, 9.75 GHz, 1.75 GHz, and 500 MHz. The top row shows images processed with the Omega-K
algorithm and at the bottom the same data is processed with the CSA. The simulation parameters are identical for each data set except for
the beamwidth which varies with the center frequency to maintain the same theoretical azimuth resolution for each example. The measured
azimuth resolutions obtained by these trials are compared in Table 1.
3. SAR APPROXIMATIONS
The Omega-K algorithm uses the exact representation of Eq. (2) and
applies Eq. (4) for a reference range. Stolt interpolation is done to
correct for the other ranges. This makes the ω-K algorithm a good
choice for systems with low-frequency, a large beamwidth, and a
largebandwidth. Thisprecision comes at thecost of high complexity
and high processing time compared to the CSA and RDA.
Other algorithms use a Taylor series approximation of Eq. (2).
The square root term can be expanded as
Υ(fτ)=
s
D2(fη) +2fτ
f0
+f2
τ
f2
0
≈
Υ(0) +Υ
?(0)
1!
fτ +Υ
??(0)
2!
f2
τ+Υ
???(0)
3!
f3
τ...(5)
RDA keeps only the 0th order term
ΦRDA ≈ −4πR0f0
c
· [D(fη)] −πf2
τ
Kr
(6)
which makes the algorithm relatively simple.
Eq. (6) is the azimuth modulation, corrected in the range-Doppler
domain during “azimuth compression.” The second term is the chirp
modulation corrected in the “range compression” step. The range-
cell migration (RCM) correction is an interpolation that makes up
for the neglected RCM term and the secondary range compression
compensates for neglected higher order terms.
The CSA keeps up to the second order terms
The first term of
Table 1. Simulated along-track resolution for a 500 MHz band-
width SAR with theoretic along-track resolution of 24 cm, as show
in Fig. 1. The beamwidth varies with the center frequency to main-
tain the same azimuth resolution for each test.
Center Frequency
1.75 GHz
0.289 m
0.290 m
20.56◦
Algorithm
Omega-K
CSA
500 MHz
0.243 m
0.437 m
77.30◦
9.75 GHz
0.294 m
0.294 m
3.67◦
Azimuth Beamwidth
ΦCSA
≈−πf2
»
τ/Kr− 4πR0f0/c ·
fτ
f0D(fη)+D2(fη) − 1
D(fη) +
2f2
0D3(fη)f2
τ
–
(7)
In the square brackets, the first term is the azimuth modulation, the
second term is the range-cell migration, and the third term is “cross-
coupling” between the range and azimuth frequencies.
More generally, we expand Eq. (5) to an arbitrary number of
terms:
fτ
f0D(fη)+D2(fη) − 1
−D2(fη) − 1
2f3
From this equation we can explore how the approximations ef-
fect the signal at different frequencies and bandwidths. Each term
Υ(fτ)
≈
D(fη) +
2f2
0D3(fη)f2
τ
(8)
0D5(fη)f3
τ−5 − 6D2(fη) + D4(fη)
8f4
0D7(fη)
f4
τ...
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−600−400−2000 200400 600
0
0.5
1
Center Frequency: 1.75 GHz − Bandwdith: 500 MHz
Doppler Frequency
−600−400−2000 200400600
0
5
10x 10
−3
Center Frequency: 1.75 GHz − Bandwdith: 500 MHz
Doppler Frequency
0th order
1st order
2nd order
3rd order
4th order
5th order
6th order
−600−400−2000 200 400600
−100
0
100
Doppler Frequency
Squint Angle
Angle vs Doppler Frequency
60 degree beamwidth
30 degree beamwidth
−150 −100−50 050100 150
0
0.5
1
Center Frequency: 0.5 GHz − Bandwdith: 500 MHz
Doppler Frequency
−150−100−500 50100150
0
5
10x 10
−3
Center Frequency: 0.5 GHz − Bandwdith: 500 MHz
Doppler Frequency
−150−100−50050100150
−100
0
100
Doppler Frequency
Squint Angle
Angle vs Doppler Frequency
60 degree beamwidth
30 degree beamwidth
Fig. 2. The maximum magnitudes of the different order terms of Eq. (9) for a 500 MHz bandwidth at L-band (left) and UHF (right). The
top row shows the orders 0-6 while the middle row focuses on just the terms of order 2-6. The bottom row shows the relationship between
beamwidth and Doppler bandwidth for the velocity of 50 m/s used in the calculations. As the center frequency decreases, the higher order
terms become more important at smaller beamwidths. In addition, at lower frequencies a much larger beamwidth is required to maintain the
same azimuth resolution, making it doubly important to account for the higher order terms.
of order n in this expansion has a fn
more important asthe change in wavelength over thechirp increases.
When large enough, we must account for these higher order terms to
properly focus the image.
Simulated data is used to evaluate the azimuth focusing with the
different approximations for three different center frequencies (see
Fig. 1). For purposes of comparison between different center fre-
quencies, the simulation parameters are identical except for the an-
tenna beamwidth which changes with the center frequency to main-
tain the same azimuth resolution. In the simulation, there is a single
target at a known range. With the Omega-k algorithm, Eq. (4) is
used to “perfectly” focus the target. This is compared to process-
ing the same data with CSA approximations of Eq. (7). The mea-
sured azimuth resolutions are compared in Table 1. It is clear that
for lower frequencies, higher bandwidths, and larger beamwidths,
the neglected higher order terms of Eq. (9) become more important
(see Fig. 2). In fact, Eq. (9) shows that the challenges of processing
high-squint, large beamwidth, high bandwidth, and low frequency
SAR all are manifestations of the same root cause: approximations
that neglect the higher order terms.
τ/fn
0 term, which becomes
4. PREVIOUS COMPENSATION METHODS
There have been a couple of attempts to efficiently address the is-
sue of large changes in wavelength. In [1] a correction is suggested
for a large change in wavelength using the CSA. After the azimuth
matched filter is applied in the range-Doppler domain the data is
transformed back into the wavenumber domain by separating the
data into small blocks in range and performing a range FFT on each
block. A corrective phase function is applied that corrects for the
change in wavelength
ΔHAZ(fη,fτ) = e
„
j4πRb
„
Dλ(fη,fτ)
λτ
−D(fη)
λ0
««
(9)
where λτ is the wavelength corresponding to frequency (f0+ fτ),
Rbis range of the middle of the block being processed, and the λ-
dependent range migration function is
Dλ(fη,fτ) =
s
1 −
„λτfη
2Vr
«2
.
(10)
While easy to implement, this method fails to address the true
cause of the errors, and is thus unreliable. Simulations show that in
certain situations it improves the focusing, while in other situations
it makes things worse.
The second option is the Non-linear CSA algorithm proposed in
[7] which keeps up to the 3rd order term of Eq. (9). Compared to the
CSA, this method requires two additional range FFT’s and a phase
multiply. If terms higher than the 3rd order are needed for proper
focusing, we would need something more than what the non-linear
CSA can provide.
5. DEVELOPMENT OF NEW POSSIBILITIES
The goal then is to develop a new general processing scheme which
efficiently accounts for as many higher order terms as dictated by the
SAR parameters and the desired precision. Ideally, each additional
termfrom Eq. (9) would add minimally to the computational burden.
A simple start is to apply a correction for higher order terms
in the wavenumber domain for a given reference range. This can
be done after an additional FFT at the beginning of processing, as
in the non-linear CSA [7], or together with the bulk range cell mi-
gration correction in the traditional CSA. For example, a correction
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including up to the fourth order term
H4th
=
−(4πRreff0/c) ·
„D2(fη) − 1
(11)
«
2f3
0D5(fη)f3
τ+5 − 6D2(fη) + D4(fη)
8f4
0D7(fη)
f4
τ
is applied to the data in the wavenumber domain. This only properly
compensates for the higher order terms at the reference range, Rref,
but the difference with other ranges is small and may be neglected.
Applying this concept to the UHF data in Fig. 1, third-order
and fifth-order compensations are used to generate the images in
Fig. 3. The third-order image has an azimuth resolution improved
by 16.2% over the CSA, while the fifth-order shows an improve-
ment of 33.7%. These improvements are accomplished without any
noticable increase in processing time.
Each term in the expansion of Eq. (9) is predictably computed
from Eq. (5). Thus, a recursive step that applies corrections for an
arbitrary number of terms can be implemented
Hnth = (4πRreff0/c) ·Υ(n)(0)
n!
fn
τ.
(12)
This is a key step in developing a generalized processing algorithm.
Future development of this algorithm includes concrete determina-
tion of how many terms are required for proper focusing for specific
radar parameters, corrections for the range dependence of the higher
order terms (when they are not neglibable), and inclusion of a range-
dependent velocity.
6. CONCLUSION
With new SAR systems pushing the frequencies lower, the band-
widths larger, and the beamwidths wider, the old approximations
used in SAR algorithms miss important factors necessary to pre-
cisely process the data. This paper has shown how the higher order
terms in the expansion of the SAR transfer function become more
important for lower frequencies. Previous efforts to address this is-
sue have been explained and the possibilities of a new generalized
method have been developed. An efficient algorithm accounting for
higher order terms can be used to increase the precision over exist-
ing algorithms or even be implemented in real time, thus extending
to utility of wide-bandwidth, low-frequency SAR.
7. REFERENCES
[1] A. Potsis,
N. Uzunoglou, “Improving the focusing properties of SAR pro-
cessors for wide-band and wide-beam low frequency imaging,”
in Proc. Int. Geosci. Rem. Sen. Symp., pp.3047-3049, vol.7,
2001.
[2] A. Potsis, A. Reigber, E. Alivizatos, A. Moreira, and
N.K Uzunoglou. “Comparison of Chirp Scaling and Wavenum-
ber Domain Algorithms for Airborne Low-Frequency SAR.” In
Francesco Posa, editor, SAR Image Analysis, Modeling, and
Techniques V, vol 4883, pp. 25-36, March 2002.
[3] J.M. Swiger, “Resolution limits of ultra wideband synthetic
aperture radar using a rectangular aperture for FFT processing,”
IEEE Trans. Aerospace and Electronic Systems, vol.30, no.3,
pp.935-938, Jul 1994.
[4] A. Reigber, E. Alivizatos, A. Potsis, and A. Moreira. “Extended
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tegrated motion compensation.” in Proc. IEEE Radar, Sonar nd
Navigation, Vol. 153, Iss. 3, pp.301-310, June 2006.
A. Reigber,J. Mittermayer,A. Moreira,and
Fig. 3. The UHF data from Fig. 1 is reprocessed with the proposed
method. The top image includes up to the third-order term and has
a measured azimuth resolution of 0.3658 m. The bottom image was
processed including up to the fifth-order term and has an azimuth
resolution of 0.2896, which is comparable to the results from the
Omega-K.
[5] A.F. Yegulalp,“Fast backprojection algorithm forsynthetic aper-
ture radar,” in The Record of the 1999 IEEE Radar Conference,
pp.60-65, 1999.
[6] I.G. Cumming and F.H. Wong, Digital Processing of Synthetic
Aperture Radar Data, Artech House, 2005.
[7] G.W. Davidson, I.G. Cumming, and M.R. Ito. “A Chirp Scal-
ing Aproach for Processing Squint Mode SAR Data,” in IEEE
Trans. on Aerospace and Electronic Systems, 32 (1), pp. 121-
133, Jan. 1996.
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