SIMULATION AND COMPENSATION OF IONOSPHERIC PHASE
SCINTILLATION NOISE IN SPOTLIGHT SAR DATA
by
Brian Chang Chi Hsueh
A thesis submitted in conformity with the requirementsfor the degree of Master of Applied Science
Graduate Department of Electrical and Computer EngineeringUniversity of Toronto
Copyright c© 2009 by Brian Chang Chi Hsueh
Abstract
Simulation and Compensation of Ionospheric Phase Scintillation Noise in Spotlight SAR Data
Brian Chang Chi Hsueh
Master of Applied Science
Graduate Department of Electrical and Computer Engineering
University of Toronto
2009
This thesis addresses the problem of refocusing smeared SAR images caused by ionosphere
phase scintillation noise. A SAR data is smeared when the received signal experiences phase
irregularities due to platform orbit deviation, target movement, or, in this thesis, ionospheric
scintillation noise due to trans-ionosphere propagation is analyzed.
A SAR simulator is constructed to generate stripmap and spotlight data that satisfy the-
oretically predicted performances under ideal conditions. The simulator is incorporated with
ionospheric phase scintillation models to analyze the broadening effect on system’s PSF. De-
graded simulation spotlight data are used to test the proposed compensation algorithm.
This thesis proposes a two-dimensional polynomial phase fitting algorithm to compensate
scintillation noise. This work discusses some requirements of the scene in order to carry out
the compensation and what is gained and lost in the process. A successful application of the
proposed algorithm to TerraSAR-X data is also presented.
ii
Acknowledgements
I would like to first thank my supervisors, Dr. Raviraj Adve and Dr. Georgia Fotopoulos,
for their patience and continuous support during my graduate studies. I have learned more
than just engineering knowledge under their supervision. I would also like to thank my thesis
committee, Dr. Dimitrios Hatzinakos and Dr. Konstantinos Plataniotis, for their comments
toward the final version of this thesis.
Special thanks to my colleges in the Geomatics group for their time and effort that went
into the proofreading and the editing of this thesis.
Lastly, I am most grateful for my family for the endless support that made all this possible.
iii
Contents
1 Introduction 1
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Objectives and Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Synthetic Aperture Radar 6
2.1 Simulator Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Stripmap Range Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3 Stripmap Azimuth Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.4 Spotlight Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.5 Phase Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.5.1 Deterministic Phase Noise . . . . . . . . . . . . . . . . . . . . . . . . 20
2.5.2 Stochastic Phase Noise . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3 Target Models 25
3.1 Radar and Clutter Cross Section . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2 Hard Targets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.3 Distributed Targets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.4 Rayleigh Clutter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.5 Imaging Effects of SAR Systems . . . . . . . . . . . . . . . . . . . . . . . . . 31
iv
3.6 SAR Imaging Process as a LTI System . . . . . . . . . . . . . . . . . . . . . . 32
3.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4 Simulator 34
4.1 Simulator Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.2 Simulator Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.2.1 Input Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.2.2 Computational Modules . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.2.3 Post Processing Modules . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.3.1 Resolution Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.3.2 Phase Noise Simulation Results . . . . . . . . . . . . . . . . . . . . . 40
4.3.3 Rayleigh Clutter Simulation Results . . . . . . . . . . . . . . . . . . . 40
4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
5 Ionosphere Modeling 46
5.1 The Ionosphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
5.2 Ionospheric Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
5.3 Dispersive Ionosphere Effects on SAR . . . . . . . . . . . . . . . . . . . . . . 49
5.3.1 Range Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
5.3.2 Azimuth Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
5.4 Phase Screen Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
5.5 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
5.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
6 Phase Noise Compensation 59
6.1 Background on Phase Compensation Techniques . . . . . . . . . . . . . . . . 59
6.2 Phase Gradient Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
6.2.1 Brightest Point Detection and Center Shifting . . . . . . . . . . . . . . 61
v
6.2.2 Windowing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
6.2.3 Inverse Fourier Transform and Phase Estimation . . . . . . . . . . . . 63
6.2.4 Compensation and Iteration . . . . . . . . . . . . . . . . . . . . . . . 64
6.3 Application to Ionospheric Noise . . . . . . . . . . . . . . . . . . . . . . . . . 64
6.4 Proposed Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
6.4.1 Non-parametric Algorithm . . . . . . . . . . . . . . . . . . . . . . . . 72
6.4.2 Parametric Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
6.5 Remarks on the Proposed Algorithm . . . . . . . . . . . . . . . . . . . . . . . 82
6.6 Application to TerraSAR-X Data . . . . . . . . . . . . . . . . . . . . . . . . . 83
6.6.1 Selection of TerraSAR-X Data . . . . . . . . . . . . . . . . . . . . . . 83
6.6.2 Remarks on Estimation Results . . . . . . . . . . . . . . . . . . . . . 84
6.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
7 Conclusions and Future Work 87
7.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
7.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
A Simulator Design in Matlab 91
A.1 Simulator Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
A.2 Compensation Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
A.3 Other Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
Bibliography 98
vi
Chapter 1
Introduction
1.1 Background
SAR is an active remote sensing device that collects the Earth surface’s response or reflectivity
to electromagnetic illumination and maps the target to platform distance. The received signal
is sampled and stored digitally as complex numbers onboard the platform. After downloading
and post-processing, implemented spaceborne SAR technologies have been shown to provide
horizontal meter-level mapping and relative centimeter-level vertical mapping of the scene
[33].
SAR is able to achieve this resolution by periodically transmitting and receiving modulated
radio frequency (RF) signals and maintaining precise timing between transmissions and recep-
tions. This precise timing information is used to post-process these pulses in order to achieve
an enhanced resolution as if the ground was imaged by a very large conventional aperture radar.
Therefore, timing of the pulses is crucial to the generation of clear SAR images; in other words,
the phase of the received signals is crucial to the image quality and to the quality of subsequent
products derived from SAR data such as Interferometric SAR (InSAR). However, like all other
coherent systems, received signals are subject to timing errors which create phase aberrations
in the received signal. This thesis analyzes possible ionosphere phase abberations experienced
1
CHAPTER 1. INTRODUCTION 2
by spaceborne SAR systems only.
The ionosphere extends from approximately 50km to 500km into the atmosphere and it is
composed of ionized gases and charged particles as a result of complex reactions of solar UV
rays with the molecules of the atmosphere. These charged particles are trapped in the Earth’s
magnetic fields and they significantly affect the propagation of electromagnetic waves. Aaron
[1] gives a comprehensive overview of ionosphere activity on a global scale and Hall [15]
summarizes the effects of the ionosphere on communication systems to be the following :
1. Dispersion: frequency-dependent delay that distorts the signal’s amplitude and phase.
Broadband systems operating at lower frequencies are affected more than narrowband
systems at higher frequencies.
2. Faraday rotation: polarization-dependent delay, ordinary and extraordinary polarizations
experience different phase delays which cause an amplitude reduction in the received
signal.
3. Ray bending: the curving of the signal’s traveling path as a result of continuously chang-
ing refractive index.
4. Ionosphere scintillation: random amplitude and phase fluctuation of electromagnetic
waves in time and space. Phase scintillation is the main concern of this thesis.
Signal propagation in the ionosphere has been studied over decades, and the modeling of
such phenomena is a task that can range from simple empirical models to complex physical
models. The works in [35] and [34] offer comprehensive comparisons of commonly used
models. In this thesis, ionosphere-like phase noise is created using the following models:
• Dispersion in trans-ionosphere signals is modeled using the Hartree-Appleton equation
as commonly found in Global Position System (GPS) literature [22].
CHAPTER 1. INTRODUCTION 3
• Phase scintillation is described by a phase screen model with spatially varying electron
density in order to simulate ionosphere-like phase scintillation and scintillation-caused
imaging effects [2].
1.2 Thesis Outline
This thesis is outlined as follows:
Chapter 2 reviews the basic concepts of radar imaging using linearly frequency modulated
pulses and matched filtering and the effect of deterministic and stochastic noise in the system.
This chapter provides the theoretical performance limitations on both stripmap and spotlight
mode in modern SAR systems and describes how these limitations are affected by phase noise.
Chapter 3 reviews the target models used in the simulator including natural targets such as
woodland and grassland and artificial point targets such as corner reflectors.
Chapter 4 introduces the design of the simulator including input, computation, and process-
ing modules. The simulated products are tested against the theoretical predictions presented in
the previous chapters, including resolution, phase noise broadened PSF, and clutter statistics.
Chapter 5 describes the ionosphere models used in this thesis in two parts, deterministic
and stochastic. The deterministic ionosphere model assumes that the electron density is static
and ionosphere is only dispersive. The stochastic ionosphere model assumes the electron den-
sity follows a wide-sense stationary random process in space and its phase noise effects are
described by a phase screen model. Both parts are incorporated into the simulator and the
resulting degraded SAR data are demonstrated.
Chapter 6 reviews existing phase compensation algorithms, both parameterized and non-
parameterized, and presents an extension of these algorithms in order to compensate for the
ionosphere phase noise in spotlight data. First, the processing of an existing autofocusing
algorithm, the Phase Gradient Algorithm (PGA), is described. Second, an extension to the
PGA is described by using a two-dimensional polynomial phase fitting algorithm called the
CHAPTER 1. INTRODUCTION 4
Phase Differenceing Algorithm (PDA) to estimate scintillation noise. This chapter also shows
improved simulated and measured TerraSAR-X (TSX) data. The TSX system provides the
background and motivation of this thesis. As will be demonstrated, the testing of the simula-
tor uses TSX parameters and the proposed algorithms are developed in an attempt to refocus
measured data.
Finally, the conclusions highlight the key results and recommend further applications and
extensions to this work.
1.3 Objectives and Contributions
The main objectives of this thesis are:
• Construction of a SAR simulator capable of generating stripmap and spotlight data.
• Modeling of Ionospheric phase noise in the simulator.
• Derivation of the effect of phase noise on the SAR point spread function (PSF).
• Proposition of a phase compensation algorithm for ionosphere phase noise.
Overall, this thesis provides an open-source SAR simulation package that is used to gener-
ate spotlight and stripmap data with trans-ionosphere propagation effects in this thesis. How-
ever, under the framework of this simulator, more realistic imaging environments could be
incorporated such as ground target movement, realistic SAR orbit and orbital perturbations,
movements of the Earth, and troposphere weather effects. Under these conditions, various
kinds of compensation algorithms can be tested and evaluated by researchers. In this thesis,
the simulator mainly provides spotlight data for the verification of theoretical performances of
SAR and of effects of multiplicative phase noise. Furthermore, these simulation data are used
to evaluate the performance of the existing and the proposed ionospheric scintillation phase
noise compensation algorithms. As a result, this thesis could show that ionospheric scintilla-
tion noise can not be removed by existing estimation algorithms. However, with the proposed
CHAPTER 1. INTRODUCTION 5
algorithm, ionospheric phase noise could be better estimated and removed to achieve SAR
image refocusing in both simulated and measured data.
Chapter 2
Synthetic Aperture Radar
This chapter reviews stripmap and spotlight SAR imaging modes and describes these modes
as transfer functions of a linear time-invariant (LTI) system. These transfer functions provide
insights to the theoretical limitations of SAR and tools to analyze phase noise in the system.
Both deterministic and stochastic phase noise are analyzed in the second part of this chapter,
and the broadening effect of phase noise is shown to be related to phase noise correlation. The
discussion in this Chapter is taken largely from [4], [18], [27]; the reader is referred to those
sources for additional details.
2.1 Simulator Geometry
A X-band spaceborne radar platform flies over an area to reconstruct a ground scene’s re-
flectivity. To describe such a system mathematically, the following coordinate systems are
constructed.
1. Local coordinate system (xl, yl, zl)
A right-handed Cartesian coordinate system centered at the scene centre is used to de-
scribe the relationship between the target and the platform as shown in Figure 2.1.
It is also convenient to define spherical coordinates, θ, φ, and R based on the target-
6
CHAPTER 2. SYNTHETIC APERTURE RADAR 7
lx
lz
φθ
Platform Position
ly
Platform Velocity Vs
R
incφ
lookφ
Scene Centre
Figure 2.1: Local Coordinate System
platform geometry. φ is often referred to as grazing angle and R as slant range; moreover,
look angle, φlook, and incidence angle, φinc, are also defined to supplant the spherical
coordinate system. Note that −π < θ < π and 0 < φ < π/2.
2. Platform coordinate system (xp, yp, zp)
During one satellite transmission, the satellite is assumed to be fixed in space. This fixed
location is defined as the origin of the platform coordinate system as shown in Figure
2.2.
In this coordinate system, the satellite flies in the xp direction without any perturbations
such that its rotational angles (attitude), roll, pitch, and yaw, are set to zero throughout the
simulation in order to test the system’s best performance under unperturbed conditions.
However, slight variations of these angles are expected in real systems and they pose
difficulties in post-processing. The compensation techniques for these difficulties are
analyzed and discussed in [4]. For notational brevity, it is common to use range or cross-
track direction to describe the direction of the yp axis and azimuth or along-track for the
xp axis in the figure.
CHAPTER 2. SYNTHETIC APERTURE RADAR 8
Yaw
py
lx
lzPitch Roll
px
pz pypx
pz
lyScene Centre
Platform Transmit Position
Platform Transmit Position
Figure 2.2: Platform Coordinate System
3. Antenna coordinate system (xa, ya, za)
The antenna coordinate system is specifically designed to find the relative angles between
a target and the antenna in order to calculate the beam pattern. The simulator parameters
are designed to match a recently launched SAR platform, TerraSAR-X (TSX) [42], and
important platform parameters are summarized in Table 2.1. With the help of the simu-
lator, one can test how different parameters influence the system and the imaged scenes,
and, with appropriate parameters, this simulator can be modified to model any proposed
SAR mission. The aperture antenna onboard TSX is directed to the broadside of the
platform at a 45o angle as shown in Figure 2.3 and its array factor (AF) and directivity
(Do) are approximated as follows [36]:
AF (θa, φa) ≈ sinc
(θaLa
λ
)sinc
(φaLr
λ
)(2.1)
Do(θa = 0, φa = 0) ≈ 4π
(LaLr
λ2
)(2.2)
where θa and φa are the spherical coordinates with respect to the antenna, La and Lr
CHAPTER 2. SYNTHETIC APERTURE RADAR 9
Carrier Frequency, fo 9.6 GHz
Bandwidth 50MHz
Range Antenna Width, Lr 4.8m
Azimuth Antenna Length, La 0.784m
Pulse Repetition Rate (PRR) 2700 Pulses Per Second
Satellite Height 514 Km
Incidence Angle 150 to 600
Table 2.1: TerraSAR-X Parameters [36]
are the antenna length along the azimuth direction and width along the range direction.
Using these parameters, TSX has a 3dB beamwidth of 0.035o in range, 0.00586o in
azimuth, and a maximum directivity of 46dB.
px
py
pz
ax ay
az
Planar Radar
45o
aθ aφ
Figure 2.3: Antenna Coordinate System
CHAPTER 2. SYNTHETIC APERTURE RADAR 10
2.2 Stripmap Range Imaging
At the beginning of the observation, the antenna transmits a signal and illuminates ground
targets within its beamwidth. This transmitted signal is a linearly frequency modulated (LFM)
signal given by:
href (t) = cos(wot + πKrt
2)rect
(t
Tr
)(2.3)
θref (t) = wot + πKrt2
where wo is the carrier frequency in radians, Tr is the duration of the pulse in seconds, and Kr
is called the chirp rate in Hz/s2 that controls the rate at which frequency increases as a function
of time.
Using the principles of linearity, the ground can be modeled as the superposition of many
point targets. The received signal bounced back from a point target located at the origin with
unit amplitude reflectivity, is given by
hreceive (t) = cos(wo(t − to) + πKr(t − to)
2)rect
(t − to
Tr
)(2.4)
θreceive(t) = wo(t − to) + πKr(t − to)2 (2.5)
where to = 2R/c is the round trip time, R is the platform-target distance, and c is the speed of
light in a vacuum.
This received signal is coherently demodulated, low pass filtered, and digitized using the
appropriate A/D converter to a sequence of numbers as follows:
hout (t) =1
2exp
(jθreceive (t) − jwot (t)
)rect
(t − to
Tr
)
=1
2exp
(j(2π
toλ
+ πKr (t − to)2 ))rect(t − to
Tr
)(2.6)
These digitized samples represent the projected reflectivity of a scene within the antenna
beamwidth on the slant range. The projection is a consequence of the side-look imaging geom-
etry of SAR; since the antenna sends out spherical wavefronts, all equidistant targets will be
received at the same time as if they have been projected on the slant range as Figure 2.4 shows:
CHAPTER 2. SYNTHETIC APERTURE RADAR 11
Far Range Close Range
Spherical Wavefront
Antenna Pattern
Transmitted Pulse
Platform Position
Slant Range
3dB Beamwidth
lz
lx
Figure 2.4: Observed Target Response on Slant Range
To complete the processing, the ground station downloads and compresses this digital infor-
mation by taking its autocorrelation with the time-inversed reference signal (matched filtering)
as [4]:
hPSF (t) = hout (t) � h∗ref (−t) ≈ Tsinc
(KrT (t − to)
)(2.7)
Intuitively, in order to locate a signal that has bounced back after some unknown time, a moving
window (convolution) is applied to find the shifted maximum peak (sinc) at the round-trip time
to and this operation is equivalent to pulse compression.
The slant range resolution, ρr, in seconds is simply the inverse of the sinc bandwidth taken
from (2.7).
ρr =0.886
KrT(s) (2.8)
Resolution measured in meters is (2.8) multiplied by a factor of c/2 as:
ρ′r = ρr
c
2(m) (2.9)
Therefore, increasing the bandwidth of LFM signals, either by increasing the chirp rate or
extending the pulse duration, one can improve the resolution. A typical range resolution for
the TSX system with 50Mhz bandwidth is 3m.
CHAPTER 2. SYNTHETIC APERTURE RADAR 12
2.3 Stripmap Azimuth Imaging
So far SAR imaging in range has been described using LFM signal and matched filtering.
In this section, azimuth imaging is accomplished by a very similar modulation-demodulation
process, however, azimuthal modulation is generated by the motion of the platform (Doppler
shift) and not by the hardware itself.
After one transmission, the satellite moves to the next location and repeats the same oper-
ation until a target leaves the radar’s field of view which is the 3dB beamwidth of the antenna
as shown in Figure 2.5.
slantθ minR
Scene Centre
Transmit Antenna Beam Pattern
Beam Pattern Experienced by Target
Platform Positions
Slant Range
Figure 2.5: Azimuth Imaging and Target Area.
where θslant is the projected spherical coordinate θ in Figure 2.3 onto the slant range plane and
Rmin is the minimum of all the slant range distances R across the aperture.
To model the change in satellite position in time, variable τ is used. To distinguish between
t and τ , t is often referred to as range time or fast time because it is on the order of 10−6
seconds, and τ as azimuth time or slow time in the order of seconds. Furthermore, during
one transmission and reception the satellite is assumed to be stationary and that τ does not
change with respect to t, hence making t and τ independent. As the satellite moves along the
CHAPTER 2. SYNTHETIC APERTURE RADAR 13
yl direction, the satellite to target range R(τ) changes as a function of τ as:
R(τ) =
√R2
min + (Vsτ)2 (2.10)
≈ Rmin +V 2
s τ 2
2Rmin
Substituting (2.10) into (2.6) and modifying to = 2R(τ)/c as a function of τ , a two dimen-
sional transfer function is obtained that describes the phase change, in both azimuth and range,
of a single point target with unit amplitude reflectivity in all directions.
himp(t, τ) = exp
(−j
4πRmin
λ− jπKaτ
2 + jπKr(t − 2R(τ)/c)2
)(2.11)
Hence, Ka is the azimuth frequency modulation rate given as a function of satellite velocity
and target distance as:
Ka =2V 2
s
cRmin
(2.12)
Therefore, when a target is observed by the platform, due to the Doppler effect, this target
experiences frequency modulation as a function of azimuth time τ .
The observation time, Ta, is the amount of time a target stays within the target area as a
function of target range and velocity as follows:
Ta = 0.886λ
La
Rmin
Vs(2.13)
where the factor 0.886 λLa
is the 3dB width of the sinc function used in (2.1) defined in radians.
A factor of Rmin/Vs is the beamwidth projection on the ground expressed in seconds. The
bandwidth in azimuth is simply the product of Ka and Ta as:
Ba = Ta × Ka = 0.8862Vs
La(2.14)
From (2.14), this azimuth bandwidth is independent of target parameters; it is only a function
of platform parameters, velocity and antenna size. After similar compression as in the range
case, the achievable azimuth resolution is:
ρa =0.886
Ba=
La
2Vs(2.15)
CHAPTER 2. SYNTHETIC APERTURE RADAR 14
In terms of distance, (2.15) is multiplied by velocity Vs as:
ρ′a = ρavs =
La
2(2.16)
which is one half of the antenna width in the azimuth direction regardless of the target position
and satellite speed.
To explain the independent property of azimuth resolution on target position, one can notice
that the closer a target, the smaller the Doppler FM rate, but wider the beamwidth; the further
a target, the larger the Doppler bandwidth, but smaller beamwidth. Combining these facts,
every target under illumination is imaged by signals of the same bandwidth, hence making
SAR capable of reconstructing the entire scene with equal details.
After matched filtering in the azimuth and range domain, the overall 2-D SAR PSF is
hPSF (t, τ) ≈ const × sinc
(t
ρr
)sinc
(τ
ρa
)(2.17)
So far, a particular imaging mode called the stripmap mode has been explained. In this
operating mode the onboard antenna remains fixed and points to a direction orthogonal to the
flight path. As the system flies over the target scene, the antenna sweeps out a large portion of
the ground, and a point target located on the ground would have a phase signal in both range
and azimuth time that can be summarize as a transfer function as follows:
hout(t, τ) = f � himp(t, τ) (2.18)
hPSF (t, τ) = hout(t, τ) � h∗ref(−t,−τ) (2.19)
where f is the overall ground reflectivity that will be described in more details in Chapter 3,
and h∗ref(−t,−τ) is the complex conjugate of the time reversed LFM signal in (2.4) in both
range and azimuth time with Kr being the range frequency modulation rate in range time t and
Ka being the azimuth frequency modulation rate in τ .
CHAPTER 2. SYNTHETIC APERTURE RADAR 15
2.4 Spotlight Mode
This section introduces another SAR operation mode called the spotlight mode in which the
target exposure time is increased by rotating the antenna to keep the target in sight for an
extended period of time, and effectively extending the antenna beamwidth that, in turn, results
in improved azimuthal resolution. Consider the received signal from another target located at
(x1, y1, z1) from the origin as shown in Figure (2.6) as follows:
lz
0R1R
Point target 1 1 1( , , )X Y Z Scene Centre 0 0 0( , , )X Y Z ly
lx
Figure 2.6: Spotlight Imaging Mode
hreceive (t) = cos(wo(t − t1) + πKr(t − t1)
2)rect
(t − t1
Tr
)(2.20)
θreceive(t) = wo(t − t1) + πKr(t − t1)2 (2.21)
where t1 is the round trip time to point target 1.
Upon reception the platform in spotlight mode uses a different demodulation scheme in
which the reference signal used in demodulation becomes the received signal from the scene
centre as follows:
θref = wo(t − t0) + πKr(t − t0)2 (2.22)
where t0 is the round trip return time to the scene centre as defined previously. This demod-
ulation scheme reduces the A/D sampling rate [18] by using a prior digital elevation model
CHAPTER 2. SYNTHETIC APERTURE RADAR 16
(DEM) of the scene. Imprecise prior knowledge or orbital drift from the expected trajectory
causes demodulation error and leads to blurring. Chapter 6 discusses existing compensation
techniques for such kinds of blurring.
After demodulation, the signal phase is the difference between the received phase and the
reference phase as follows:
hout(t) = exp(j (θreceive − θref)
)= exp
(− j
2
c(wo + 2πKrto)(R1 − R0) + j
4πKr
c2(R1 − R0)
2)
(2.23)
After a change of notation, (2.23) becomes:
hout(K, θslant) = exp(− jK (R1 − R0) + j
4πKr
c2(R1 − R0)
2)
(2.24)
where K = −2c
(wo + 2Kr
(t − 2Ro
c
) )which denotes the scaled and offset time sample index
t; the offset term is the Ko = 2wo/c term and the scaling term is 4πKr/c.
Placing the two targets from (xo, yo, zo) and (x1, y1, z1) on the slant range plane as shown
in Figure 2.7, R1 can be rewritten as a function of R0 and the projected distance ρ and the
projected angle γ.
R21 = R2
o + ρ2 − 2ρRo sin(θslant + γ) (2.25)
Furthermore, with the following approximations
R1 − R0 ≈ −ρ sin(θslant + γ) +ρ2
2R0cos2(θslant + γ) (2.26)
(R1 − R0)2 ≈ ρ2 sin2(θslant + γ)
The demodulated phase is rewritten as:
θspotlight ≈ Kρ sin(θslant + γ) − Kρ2
2R0cos2(θslant + γ) +
4Krρ2
c2sin2(θslant + γ) (2.27)
The second and third term of (2.27) are undesired phase signals called the range curvature
effect and deramping phase residual. Detailed correction methods of these terms can be found
CHAPTER 2. SYNTHETIC APERTURE RADAR 17
Platform Positions 1
ρ γ
slantθΔ
Scene Centre
oR
1R
Platform Positions 2
Figure 2.7: Spotlight Slant Range Geometry
in [18]. The only ideal phase signal is the first term, it can be rewritten as follows:
Kρsin(θslant + γ) = (ρcosγ)(K sin θslant) + (ρ sin γ)(K cos θslant)
= Kxxo + Kyyo (2.28)
This equation states that, during one transmission, the received phase can be expressed as linear
frequencies mapped onto a fictitious two-dimensional plane. This plane is often referred to as
the phase plane with orthogonal axis Kx and Ky as shown Figure 2.8. The slopes of these
linear frequencies, xo and yo, are determined by the polar coordinates of the target projected
on the slant plane γ and ρ. These slopes do not change with respect to the positions of the
satellite and target. In other words, as the platform moves to another position, the system takes
more samples of the same two-dimensional phase function as shown in Figures 2.8 and 2.9. In
Figure 2.8 the dots denote the samples made by the satellite from different slant angles and at
different sampling times, and Figure 2.9 shows that the phase function sampled is a 2-D plane
with a slope of xo with the Kx axis and a slope of yo with the Ky axis, and this function is
sampled by the sampling points in the previous figure.
It is apparent from the form of (2.28) that to compress the signal, a two-dimensional Fourier
CHAPTER 2. SYNTHETIC APERTURE RADAR 18
yK
oK
xK
K slantθ
Figure 2.8: Sample Point in 2-D Signal Space
XK
yK
Slope oy
Slope ox
Spotlight 2-D Phase Function
Sampling Points
Figure 2.9: 2-D Phase Function
Transform is sufficient. If the scene centre is imaged, the linear frequency coefficients are zero,
which would yield a two-dimensional sinc at the centre of the image after compression. If
some other target is imaged, its projected location, γ and ρ, will determined xo and yo. After
a Fast Fourier Transform (FFT), these linear frequencies effectively shift the sinc function
proportional to the projected distance of the target with respect to the scene centre, hence,
making this target distinguishable from other targets. In addition to the shifting due to linear
phases, this target also retains a constant phase value that can be found by setting t = 0 and
CHAPTER 2. SYNTHETIC APERTURE RADAR 19
θslant = 0 in (2.27) which is the intercept of the 2-D phase function with the origin of the phase
history plane as follows:
θconst =2wo
c(R1min − R0min) +
4πKr
c2(R1min − R0min)2 (2.29)
This phase constant is the difference in minimum slant range between the target R1min and
the scene centre R0min modulus 2π. This phase value enables the spotlight data to be further
processed using InSAR methods to reconstruct a DEM of the scene [18] and should be pre-
served after any compensation. From the properties of the FFT, the more samples of the signal
in the time domain, the more precise the frequency domain representation will be. The same
principle applies to spotlight mode imaging, the range time samples are determined by the
hardware sampling rate and the pulse duration as it is in the stripmap case. However, azimuth
resolution is improved in spotlight by extending θslant and making more measurements using
a wider synthetic aperture. Therefore, the achievable azimuth resolution is then the inverse of
the azimuth FM rate in (2.12) multiplied by the extended exposure time ΔθslantRo/Vs:
ρa = 0.886λ
2VsΔθslant
(2.30)
ρ′a = 0.886
λ
2Δθslant
(2.31)
To complete this analysis, the spotlight imaging is summarized as follows:
hout(t, τ) = f � himp(t, τ) (2.32)
hPSF (t, τ) = FFT(hout(t, τ)) (2.33)
where the impulse response in spotlight is a two dimensional linear phase function with slopes
xo, yo, and a constant θconst in (2.28).
CHAPTER 2. SYNTHETIC APERTURE RADAR 20
2.5 Phase Noise
2.5.1 Deterministic Phase Noise
The proper pulse compression of SAR requires the received signal to have linearly modulated
phase history in stripmap data and 2-D linear phase history in spotlight data. This section ana-
lyzes the system’s response in the presence of deterministic phase noise. More specifically, the
effect of additive noise at the receiver and multiplicative noise during transmission is analyzed.
Consider a noisy LFM signal as follows:
hreceive(t) = exp(jθSAR(t)
)× exp(jφe(t)
)+ N (2.34)
θreceive(t) = θSAR(t) + φe(t) + θN (2.35)
where θSAR(t) is the desired system modulated phase, stripmap or spotlight, φe(t) is some
arbitrary multiplicative phase noise, and N is thermal noise, and θN is the angular component
of thermal noise. Under high SNR conditions, it has Gaussian distribution with variance equal
to half of the variance of the thermal noise [38].
Assume φe(t) is some small arbitrary function that can be approximated using Taylor series
expansion as follows:
φe(t) ≈ φe0 + φe1t + φe2t2 + φeit
i where i = 3, 4, ...,∞ (2.36)
In spotlight imaging, (2.35) is Fourier transformed during pulse compression. Using prop-
erties of the FFT, the imaging effect of noise is as follows:
• additive white noise, N , does not affect pulse compression or cause any blurring, but
rather adds a low intensity random signal to the image. i.e., the PSD of noise is a constant
with amplitude σ2n.
• multiplicative noise, exp(jφe), is multiplied with exp(jθSAR) in the time domain which
is equivalent to a convolution in the frequency domain, therefore, the PSF of SAR in the
CHAPTER 2. SYNTHETIC APERTURE RADAR 21
presence of multiplicative noise is the convolution product of noiseless PSF with the FFT
of exp(jφe(t)) as follows:
hout(t) = sinc
(t
ρr
)� FFT
[exp (jφe(t))
]. (2.37)
Furthermore, using Taylor Series expansion, the imaging effect of each term is as fol-
lows:
– φe0 does not affect the image quality.
– φe1 controls the amount of linear shift of the PSD location, but does not cause
spreading.
– φe2 controls the amount of spreading in the PSD.
– the higher order terms also control the spreading of the PSD.
Similar results can also be obtained for stripmap imaging; suppose the demodulated and
low-pass filtered stripmap signal h(t) = rect(
tT
)exp(jπKt2) experiences some phase noise
exp(jφe(t)
)and is matched filtered with the reference signal href (t) = rect
(tT
)exp(jπKt2)
as follows:
g(t) = hout � href
=
∫ ∞
−∞rect
( u
T
)rect
(t − u
T
)exp(jπKu2
)exp(jφe(u)
)exp( − jπK(t − u)2
)du
= exp(jπKt2
) ∫ ∞
−∞rect
( u
T
)rect
(t − u
T
)exp(jπKu2)exp
(jφe(u)
)du
The integral only has contribution where the two rect functions overlap, following the sim-
plification used in [4], the integral can be split into two parts where one is to the left and one to
right of the matched filter to be
g(t) = exp(−jπKt2)(rect
(t + T/2
T
)∫ t+T/2
T/2
exp(j2πKtu)exp(jφe(u)
)du
+rect
(t − T/2
T
)∫ T/2
t−T/2
exp(j2πKtu)exp(jφe(u)
)du)
(2.38)
CHAPTER 2. SYNTHETIC APERTURE RADAR 22
where the first term is the right shifted version and the second term is the left shifted version of
the response∫ T/2
t−T/2exp(j2πKtu)exp
(jφe(u)
)du. The response itself can be easily seen as the
Fourier Transform of the product of the received phase and the phase noise and this integral is
equal to the convolution of the FFT of individual terms.
2.5.2 Stochastic Phase Noise
This section will analyze the systems’ response to random phase noise. The works in [37]
and [3] have provided an analytical derivation for degraded antenna power patterns and the
amount of shifting and resolution degradation under the influence of stationary Gaussian dis-
tributed phase noise in conventional linear phased array antennas. This section will review
these derivations. Suppose φe(t) is some correlated phase noise with some ACF Rφe(Δt); in
spotlight processing the output g(t) is as follows:
g(u) =
∫ ∞
−∞hout(t)he(t) exp(−j2πtu)dt (2.39)
he(t) = exp(jφe(t))
The measured intensity is given by:
E[|gout(u)|2] = gout(u)g∗
out(u) (2.40)
=
∫∫hreceive(t1)h
∗receive(t2)he(t1)h
∗e(t2)
× exp(j2π(t1 − t2)u)dt1dt2 (2.41)
After substitution by parts, t = (t1 + t2)/2 and ζ = (t1 − t2),
E[|gout(u)|2] =
∫ ∞
−∞Rg(ζ)Rhe(ζ) exp(j2πζu)dζ (2.42)
Rg(ζ) =
∫ ∞
−∞hreceive(x +
1
2ζ)h∗
receive(x − 1
2ζ)dζ (2.43)
Rhe(ζ) = E[he(x +1
2ζ)h∗
e(x − 1
2ζ)] (2.44)
CHAPTER 2. SYNTHETIC APERTURE RADAR 23
where the ACF of h, Rh(ζ), can be related to the ACF of phase noise, Rφe(ζ), by
Rhe(ζ) = E[exp(jRφe(t)) exp(−jRφe(x + ζ))
]= E
[(cos φe(x) + j sin φe(x)
)(
cos φe(x + ζ) − j sin φe(x + ζ))]
= E[(
cos φe(x) cos φe(x + ζ))]
(2.45)
As seen in (2.44), the average PSF is the FFT of the product of Rg and Rh which is equal
to the convolution product of the FFT of each individual term.
E[|gout(u)|2] =
∫v
PG(u − v)PH(v)dv (2.46)
PG(v) =
∫ ∞
−∞Rg(ζ) exp(j2πζv)dζ (2.47)
PH(v) =
∫ ∞
−∞Rhe(ζ) exp(j2πζv)dζ (2.48)
Therefore, the average noisy PSF is the convolution product of the noiseless PSF with the
PSD of the phase noise. A perfectly correlated noise case, whose FFT is a delta function,
would not affect the PSF after convolution. A completely uncorrelated noise, whose transform
is a constant, results in a flattened PSF.
2.6 Summary
This chapter has expressed SAR system as a two-dimensional LTI system in which the input
has been described as a constant representing a target’s reflectivity. The output of the system
is sampled at the hardware sampling rate in range and at the pulse repetition rate in azimuth.
The achievable range and azimuth resolution is then dependent on the bandwidth in range
and beamwidth in azimuth, but not the location of the target. All targets under illumination
have identical resolutions. In this section, two operating modes were introduced, stripmap
and spotlight, to demonstrate the trade-off between resolution and coverage. This chapter has
CHAPTER 2. SYNTHETIC APERTURE RADAR 24
also established the broadening effects of both deterministic and random phase noise to be the
convolution product of the noiseless PSF with the PSD of the multiplicative phase noise.
Chapter 3
Target Models
Up to this point, SAR has been described as a LTI system without specifying the inputs. In
this chapter, these inputs will be described as different target responses. More specifically,
responses from artificial objects such as corner reflectors and natural scenes, such as grassland
and woodland, are described.
3.1 Radar and Clutter Cross Section
In Chapter 2, a unit amplitude was used to represent a target’s reflectivity. More practically, the
standard measurement of reflectivity is the radar cross section (RCS) of the target. It measures
the strength of the returned signal power after illumination, and it can be thought of as the
effective area seen by the receiver at a particular angle. RCS is defined as:
σ = limR→∞
4πR|εr|2|εt|2
= limR→∞
4πRPr
Pt
(3.1)
where εt and εr are the transmit and received electric field strength and R is distance from the
receiver to target.
25
CHAPTER 3. TARGET MODELS 26
Incorporating RCS with Frii’s formula, transmit power, receiver power, the overall received
power are related as follows:
Pr =Gt(θat , φat)Gr(θar , ϕar)
(4π)3R2t R
2r
Pt × σ + N (3.2)
where Pr is the received power, Gt and Gr are transmit and receive antenna factors multiplied
by the antenna directivity in (2.1) and (2.2) respectively evaluated at some transmit angles,
θat , φat , and receive angles, θar , ϕar , with respect to the antenna coordinate system, Rt and
Rr are distances from the transmitter and the receiver respectively to the target, and Pt is the
transmit power. N is thermal noise as discussed previously with variance σ2n = 4KbTδf , here
Kb is Boltzmann’s constant, T is temperature in Kelvins, and δf is the bandwidth in Hertz.
3.2 Hard Targets
Artificial targets or hard targets have a well-defined geometry and they give consistent returns
as function of transmitter and receiver angles. These returns form a consistent RCS profiles
and can be obtained through experiments. The precise modeling of geometric objects in SAR
is not the focus of this research work but can be found in [9] and [26]. In our simulator, point
targets with a constant RCS in all directions such as corner reflectors are simulated. Because
of this convenient property of corner reflectors that reflection angle is equal to incident angle
in every direction, corner reflectors are often the preferred ground instrument to calibrate radar
systems.
3.3 Distributed Targets
Distributed targets are an extension of point targets; due to terrain irregularities, most natural
targets do not have a well-defined reflection geometry, but rather they tend to have many scat-
tered reflection points. Distributed targets are comprised of many elementary point scatterers,
CHAPTER 3. TARGET MODELS 27
where each scatterer has a random reflection amplitude, but the superposition of these random
amplitudes will result in the total RCS for that distributed target. Since different satellite sys-
tems have beamwidth and processing biases, a standardized measure of clutter RCS is the radar
cross section coefficient, σo, which is the average RCS per unit area or the averaged calibrated
RCS of a terrain target:
σo =σ
Resolution Area=
∣∣∣∣E[ N∑n=1
an exp(jθn)]∣∣∣∣
2
Resolution Area(3.3)
where E is he expectation operator and the variables are defined as follows:
• N is the number of scatterers within a resolution cell.
• an is the random reflected signal amplitude of the nth point scatterer.
• θn is the phase component of the nth scatterer which is determined by 2R/k where 2R
is the platform-scatter round-trip distance and k is the wavenumber.
• σo is the average RCS coefficient of the entire resolution cell.
The four basic variables in the distributed target model have been defined and the summing
process is performed when these scatterers are simultaneously illuminated by the antenna (see
Figure 3.1). Upon reception, every returned signal within a resolution cell will contribute to
the total signal return for that resolution cell, ε, as follows:
ε =N∑
n=1
anexp(jθn) (3.4)
CHAPTER 3. TARGET MODELS 28
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X X
X X
X
X X
1
n
N
nn
ja e θε=
=∑
Azimuth Range
Figure 3.1: Summing Process of Point Scatterers in a Resolution Cell
The received intensity is then:
E[I] = E[|ε|2]
= E[ε × ε∗
]= E
[ N∑i=1
N∑j=1
aiejθia∗
je−jθj]
=N∑
n=1
E[|an|2
]= NE
[|an|2]
(3.5)
The derivation assumes that the received amplitude and phase is independent between scatterers
and phase is uniformly distributed between 0 and 2π due to the large target-platform distances.
The amplitude of an individual point scatterer, an, is not observable in any radar system. In
this simulator, the distribution of an is assumed to follow a Rayleigh distribution as follows:
f(an, α) =an
α2exp
(−a2n
2α2
)(3.6)
CHAPTER 3. TARGET MODELS 29
Since the expected value of received intensity, E[I], should be the RCS of the distributed target,
σ, therefore, the simulator uses a Rayleigh distribution as follows:
σ = NE[|an|2] = N(E[|an|]2 + var[an]
)= N2α
α =σ
2N(3.7)
where var[an] = α2(2 − π/2) and E[|an|] = α√
π/2.
Therefore, in the simulator, a random scatterers’ amplitude is a Rayleigh distribution and it
is only chacterized by the RCS of the area and the number of scatterers. More precise modeling
of the behavior of elementary scatterers as a function of incident angles and polarization states
can be found in [10]. In this thesis, only Rayleigh distribution is assumed and, as will be shown
later in this chapter, the development of the Rayleigh clutter is insensitive to the distribution of
an.
3.4 Rayleigh Clutter
Starting with Equation (3.1) , the summation can be broken down into the summation of the
real and imaginary parts as follows:
εRE =
N∑n=1
|an| cos(θn) (3.8)
εIM =N∑
n=1
|an| sin(θn) (3.9)
If N is large and the resolution cell contains no dominant scatterers, then by the Central Limit
Theorem, the real and imaginary part form Gaussian distributions with the following mean and
CHAPTER 3. TARGET MODELS 30
variance. The individual mean and covariance are calculated as follows:
E[εIM ] = E[εRE ] =N∑
n=1
E[an]E[cos(θn)] = 0
Σ2 = V AR[εIM ] = V AR[εRE ] =
N∑i=1
N∑j=1
E[|ai||aj|]E[cos(θi) cos(θj)]
=N∑
n=1
E[|an|2]2
=σ
2(3.10)
(3.11)
Furthermore, the joint Gaussian distribution, PεRE ,εIM, intensity distribution, PI , and amplitude
distribution, PA, are as follows:
COV [εRE , εIM ] = E[εREεIM ] − E[εRE ]E[εIM ] (3.12)
=N∑i
N∑i
E[aiaj ]E[cos(θi) sin(θj)] = 0
PεRE ,εIM=
1
2πΣ2exp
(−ε2
RE + ε2IM
2Σ2
)(3.13)
PA(A) =A
Σ2exp
(− A2
2Σ2
)A ≥ 0 (3.14)
PI(I) =1
2Σ2exp
(− I
2Σ2
)I ≥ 0 (3.15)
Therefore, σo completely characterizes the behavior of Rayleigh clutter regardless of the dis-
tribution of individual an. Furthermore, since the real and imaginary parts are uncorrelated,
the phase information of the Rayleigh clutter is uniformly distributed and is informationless.
Therefore, the Rayleigh clutter model is valid if elementary point scatterers are numerous,
have no single dominant scatterer, independent, randomly scattered, and immobile from one
scan to the other; when these conditions are violated, the distribution tends to deviate away
from Rayleigh. The works in [27] have summarized empirical distributions, such as Weibull,
log-normal, and K distribution, that are able to fit a wider range of data sets. However, the
general Rayleigh distribution will be the only distributed surface this thesis shall concern itself
with.
CHAPTER 3. TARGET MODELS 31
If a new variable, s, is defined to be intensity over RCS, s = I/σo, then (3.15) becomes:
Ps(s) = e−s s ≥ 0 (3.16)
Hence, the observed pixel intensity at each point can be regarded as a unit deterministic RCS
value multiplied by unit exponentially distributed speckle noise. This also implies that the
multiplicative noise distribution can be made identical to any terrain clutter regardless of the
transmit power.
3.5 Imaging Effects of SAR Systems
So far the statistical behavior of a pixel intensity has been derived from elementary point scat-
terers without characterizing the radar system itself. In this section, the full impulse response
and imaging effects of the SAR system will be included in the derivation. From (3.3), the
strength of a pixel is the superposition of the scatterers on the ground. Suppose every point
scatterer is now located at some coordinate (xo, yo) such that the summation of the electric
field is modified as a two-dimensional integral as follows:
ε(xo, yo) =
∫xl
∫yl
f(xl, yl)hPSF (xo − xl, yo − yl)dxldyl (3.17)
where f(xl, yl) is a complex 2-D function with amplitude equal to the scene’s reflectivity and
phase proportional to the target-satellite distance, and hPSF (xl, yl) = sinc(
xl
ρ′r
)sinc
(yl
ρ′a
)is
the processed SAR PSF without the constant in (2.17) expressed in range and azimuth coor-
dinates. This formula is equivalent to a two-dimensional convolution of the SAR PSF with a
complex reflectivity which is identical to the concept of a SAR transfer function in the previ-
ous chapter. The input to the system is a random uncorrelated Gaussian random process for
Rayleigh clutter and a delta function for point targets. After convolution, the output is sam-
pled, and the sampling indices, xo and yo, are determined by the hardware in range and pulse
repetition rate in azimuth as discussed in Chapter 2. When one sample is measured at xo = 0
and yo = 0, it is equivalent to taking a discrete sample at the origin of the scene. By (3.17), this
CHAPTER 3. TARGET MODELS 32
return is the weighted sum of many point scatterers around the scene centre and the weighting
is determined by the SAR impulse response as described before. In terms of imaging effect,
convolving two-dimensional sinc with an uncorrelated random field will incur some correlation
and smooth out the underlying random field. The amount of smoothing is determined by the
width of the SAR PSF. If the system had infinite bandwidth, the imaged random field would be
unchanged.
3.6 SAR Imaging Process as a LTI System
The following processing chain summarizes, the Rayleigh clutter model, SAR transfer func-
tion, and post-processing into a complete context,
[ . ]l mg AzimuthSampRange
ling
N
* ( , )refh t τ− − ( , )r impP h t τ )( ,l lf x y
Figure 3.2: Complete SAR LTI system
where f(xl, yl) is a two-dimensional uncorrelated random field representing Rayleigh clutter,
Rf(xl,yl)(Δxl, Δyl) = Σ2δ(Δxl)δ(Δyl) where Σ2 = σ/2. xl and yl are spatial coordinates of
the clutter which can be easily transformed into time indices t and τ by calculating the round
trip time and satellite pass time.
3.7 Summary
In this chapter, a SAR image data has been explained in two parts, first is the randomness of
reflectivity of Rayleigh clutter and second is the smoothing effect of convolving SAR impulse
responses of this reflectivity. The randomness of reflectivity can only be quantified using prob-
abilistic terms; over multiple scans of the same terrain clutter, the amplitude returns form a
CHAPTER 3. TARGET MODELS 33
Rayleigh distribution and intensities form an exponential distribution, and both distributions
are characterized by the average RCS. During one scan, one realization of the random reflec-
tivity is convolved with the SAR transfer function, stripmap or spotlight, and this convolution
product is sampled at different positions and intervals determined by the system’s physical
parameters. Moreover, to explain a SAR image in a communication context, an uncorrelated
Gaussian random process with variance equal to the RCS of the scene is the input to the SAR
LTI system, and with Frii’s radar formula and the Nyquist noise equation, a complete the char-
acterization of SAR systems, both signal and noise is presented.
Chapter 4
Simulator
This section introduces the design of the simulator and shows verified simulation results includ-
ing SAR stripmap and spotlight data, broadening effects of phase noise in PSF, and Rayleigh
clutter statistics.
4.1 Simulator Review
Many SAR simulators have been developed in the past with different purposes including effi-
cient frequency domain simulators in [18], complex SAR and InSAR simulators with surface
backscattering clutter model with shadowing and polarization effects in [10] and [8] in the fre-
quency domain and [25] in the time domain, and volume backscattering simulators such as the
signals from forests in [23], and a hard target simulators in [9] that performs precise ray-tracing
signals from 3-D man-made structures. Furthermore, some research have focused on orbital
simulations such as the works in [20] for bi-static (two concurrent platforms) and multi-static
satellite formations and in [17] developed by the Canadian Space Agency to test the platform’s
performance at different frequencies and polarizations. This thesis presents a time domain raw
data generator and processor capable of simulating simple point targets and Rayleigh surface
clutter in both stripmap and spotlight mode. This open source simulator provides a framework
for developers to simulate perturbations in the system such as platform orbital drifts, satellite
34
CHAPTER 4. SIMULATOR 35
attitude deviations, and target movements. In this thesis, this simulator is used to generated
realistic stripmap and spotlight data in the presence of ionospheric dispersion and phase scin-
tillation as will be described in detail in Chapter 5.The main purpose of these data is to evaluate
performances of the algorithms in Chapter 6.
4.2 Simulator Modules
This section gives an overview of the simulator modules as follows:
Target Parameters
Stripmap / Spotlight Signal Generation
Platform Parameters
Antenna Pattern
Azimuth and Range Compression
Ionosphere Model
Raw Data
Phase Noise Compensation
Computation Modules
Input Modules
Post Processing Modules
Focused Data
Figure 4.1: Simulator Modules
4.2.1 Input Modules
The inputs to the simulator are target parameters and platform parameters. Target parameters
are the positions and the backscattering strength of the corner reflectors and the elementary
scatterers. Corner reflectors have identical returns in all direction and the returns’ RCS is sig-
nificantly higher than the background clutter RCS. Clutter RCS is the summation of random
CHAPTER 4. SIMULATOR 36
small backscatter strengths of the elementary scatterers, which is Rayleigh distributed as de-
scribed in (3.7).
The other inputs are the platform parameters such as orbital height, velocity, antenna size,
and incident angle. As discussed in Chapter 2, these parameters are taken from the TerraSAR-
X mission [36] and they will determine the quality of the imaged scene. In the computations,
the platform is assumed to move in a direction perpendicular to the imaged scene and not in
a elliptical orbit. This is because in this rectangular coordinate system, one can better design
a target grid using the scene centre as the origin instead of using the Earth’s centre for the
purpose of verifying our simulation results.
4.2.2 Computational Modules
After defining the target position and backscatter strength maps, the simulator generates re-
turned signals on a target-by-target and pulse-by-pulse basis in the time domain based on the
two input maps. In each iteration, one target is imaged at a time and superimposed to the
returns from previous targets. Also, at each transmitter position, the platform-target geome-
try is calculated in the three coordinate systems described, these coordinate information will
be used to calculate the antenna pattern, ionospheric scintillation noise, path attenuation, and
generate raw stripmap or spotlight signals using (2.6) and (2.23) respectively. Note that the
generated signal is the demodulated stripmap and spotlight raw signal, and not the full impulse
response. This is because, in the simulator, range time t and azimuth time τ are not indepen-
dent, in other words, the satellite is not fixed in space during one transmission. By calculating
the precise transmit and receive positions (timing), different platform parameters can be tested
to see the effects on the position of the imaged scene. The computation of these modules can
be accomplished sequentially on one station or simultaneously using a cluster of processors.
CHAPTER 4. SIMULATOR 37
4.2.3 Post Processing Modules
SAR post processing and error compensation encompasses a vast list of topics and active re-
search. This thesis will focus on the Range-Doppler Algorithm (RDA) for stripmap processing
and FFT for spotlight data and ionospheric phase noise compensation in Chapter 6. A detailed
implementation of RDA can be found in [4]. At the output of the simulator is the focused SAR
image or the observed reflectivity map by the platform.
4.3 Simulation Results
In this section, theoretical limitations of SAR are tested using the simulator.
4.3.1 Resolution Limits
Stripmap Data Simulation Results
Two stripmap simulation results are shown below. Figure 4.2 shows the processed 2-D sinc
PSF of one single point target located in the scene centre.
Figure 4.3 shows the resolving power of SAR in both the range and the azimuth direction.
Four targets are placed on a grid separated by a distance five times the resolution and, in the
simulation result, they are separated by five pixels, hence achieving the desired resolution.
Spotlight Data Simulation Results
In spotlight data simulation, two simulation results are also shown below. Figure 4.4 shows the
2-D sinc PSF of spotlight mode.
Figure 4.5 shows four identical targets in stripmap simulation with a doubled exposure
time of 1.2s. The extended exposure time effectively decreases the azimuthal resolution by a
half while keeping the range resolution unchanged. The results show that the point targets are
separated by ten pixels in the azimuth direction but unchanged in range.
CHAPTER 4. SIMULATOR 38
Range Index
Azi
mut
h In
dex
10 20 30 40 50 60 70 80 90 100
10
20
30
40
50
60
70
80
90
100
Figure 4.2: Stripmap Point PSF
Range Index
Azi
mut
h In
dex
10 20 30 40 50 60 70 80 90 100
10
20
30
40
50
60
70
80
90
100
Figure 4.3: Simulated Stripmap Data with 4 Targets
CHAPTER 4. SIMULATOR 39
Range Index
Azi
mut
h In
dex
10 20 30 40 50 60 70 80 90 100
10
20
30
40
50
60
70
80
90
100
Figure 4.4: Spotlight PSF
Range Index
Azi
mut
h In
dex
10 20 30 40 50 60 70 80 90 100
10
20
30
40
50
60
70
80
90
100
Figure 4.5: Spotlight Data with 4 Targets
CHAPTER 4. SIMULATOR 40
4.3.2 Phase Noise Simulation Results
In this section, the simulator is used to test the effect of stochastic phase noise in stripmap
data. More specifically, Monte Carlo simulation results will be compared to analytical results
in (2.48). Using the TSX parameters, a LFM signal with a bandwidth of 50Mhz is multiplied
by the phase noise of various correlation lengths. A signal of length 1000 samples is multiplied
by phase noise with 1000, 500, and 100 samples correlation length and the results are shown
in Figures 4.6, 4.7, and 4.8. In these figures, the averaged normalized PSFs are plotted for the
case of errorless PSF, simulated noisy PSF, and numerical integrated PSF.
4.3.3 Rayleigh Clutter Simulation Results
In this section, the simulator is used to simulate Rayleigh clutter as background noise. The
terrain RCS coefficient, -5.3dB, is taken from [39] for a vegetated terrain at a receive angle of
45o. Moreover, as stated in the same work, Ulaby claims that clutter statistics can be correctly
simulated with as few as 10 elementary scatterers in a resolution cell, therefore, in our simula-
tion, resolution cells have 2 elementary scatterers per meter, given the resolution of TSX, there
are more than 20 scatterers per cell. A sample clutter image is shown below in Figure 4.9.
As mentioned in Chapter 3, terrain clutter behaves as Gaussian noise with a small amount
of correlation between pixels induced by the SAR PSF, hence, producing ”grainy” noise, com-
monly called speckled noise. To test the terrain statistics of speckle noise, the terrain signal
distribution from multiple SAR images of the same scene should be plotted. However, taking
multiple scans of the same scene is impractical. Therefore, it is common to assume that the
system is ergodic in the literature [13] and that spatially averaging across many pixels is equiv-
alent to temporal averaging across many scans. Therefore, amplitude and intensity distribution
of the imaged pixels are shown in Figure 4.10. The mean intensity distribution is around 5.5
which is equal to the RCS coefficient multiplied by the expected system’s resolution.
To simulate realistic SAR images, point targets with stronger RCS are placed on top of a
CHAPTER 4. SIMULATOR 41
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
x 10−5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Time (us)
Nor
mal
ized
Ant
enna
Pat
tern
Errorless ResponseSimulation ResponseNumerical Integrated Response
(a) Averaged PSF under Highly Decorrelated Phase Noise
−6 −4 −2 0 2 4 6
x 10−7
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Time (us)
Nor
mal
ized
Ant
enna
Pat
tern
Errorless ResponseSimulation ResponseNumerical Integrated Response
(b) Averaged Mainlobe Width
Figure 4.6: Average SAR PSF under Random Phase Noise
CHAPTER 4. SIMULATOR 42
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
x 10−5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Time (us)
Nor
mal
ized
Ant
enna
Pat
tern
Errorless ResponseSimulation ResponseNumerical Integrated Response
(a) Averaged PSF under Correlated Phase Noise
−6 −4 −2 0 2 4 6
x 10−7
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Time (us)
Nor
mal
ized
Ant
enna
Pat
tern
Errorless ResponseSimulation ResponseNumerical Integrated Response
(b) Averaged Mainlobe Width
Figure 4.7: Average SAR PSF under Random Phase Noise
CHAPTER 4. SIMULATOR 43
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
x 10−5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Time (us)
Nor
mal
ized
Ant
enna
Pat
tern
Errorless ResponseSimulation ResponseNumerical Integrated Response
(a) Averaged PSF under Highly Correlated Phase Noise
−6 −4 −2 0 2 4 6
x 10−7
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Time (us)
Nor
mal
ized
Ant
enna
Pat
tern
Errorless ResponseSimulation ResponseNumerical Integrated Response
(b) Averaged Mainlobe Width
Figure 4.8: Average SAR PSF under Random Phase Noise
CHAPTER 4. SIMULATOR 44
Range Index
Azi
mut
h In
dex
550 560 570 580 590 600 610
145
150
155
160
165
170
175
180
185
190
195
Figure 4.9: Rayleigh Clutter Background Image
clutter background as Figure 4.11 shows.
4.4 Summary
This chapter has explained the modules in the developed simulator and compared it with other
available simulators. The simulator in time domain is developed to test the performance of the
system in relation with different design parameters and to simulate realistic Rayleigh clutter
background given a particular RCS coefficient. The purpose of this simulator is to produce
realistic SAR images for evaluating compensation techniques later in this thesis.
CHAPTER 4. SIMULATOR 45
0 10 20 300
0.05
0.1
0.15
0.2
Rayleigh Clutter Intensity Values
Freq
uenc
y
Figure 4.10: Statistics of the simulated Rayleigh Clutter
Range Index
Azi
mut
h In
dex
550 560 570 580 590 600 610
145
150
155
160
165
170
175
180
185
190
195
Figure 4.11: Simulation SAR Image with Corner Reflectors and Clutter Background
Chapter 5
Ionosphere Modeling
In this chapter, two models are used to describe the ionosphere as a random dispersive medium.
First, electron density is assumed to be same in space and only dispersive effect is analyzed
using the Hartree-Appleton approximation. Second, electron density is modeled as a correlated
two-dimensional random field and the effect of ionospheric phase scintillation is analyzed. By
using these models, we one can study the phase irregularity during trans-ionosphere propa-
gation and its smearing effect on SAR data as discussed in Chapter 2. The objective of this
chapter is to model generic ionosphere-induced phase variations.
5.1 The Ionosphere
The ionosphere is a region in the upper atmosphere that contains charged particles due to the
ionization of molecules. The amount of ionization varies with respect to geographical area,
time, height, and solar activity, generally active packs of turbulence appears during evening
hours around the magnetic equator and high-latitude regions and have destructive effects on
the communication system [1]. The modeling of such phenomena is still an active research
area, over the years, many prediction models have been established. The most complex phys-
ical models rely on many global ionospheric weather models to calculate complex chemical
reactions to produce a global electron density model. Recent research has assimilated various
46
CHAPTER 5. IONOSPHERE MODELING 47
data sources into these models as inputs to predict future ionosphere weather phenomena in
three dimensions [35]. Statistical ionospheric models have also been developed to study the
ionosphere’s response to a particular observed chemical process. These models permit the pre-
diction, as a function of time, on a regional scale where measurements are available. Rino
[31] has proposed a family of analytical models where turbulent ionosphere is assumed to be
contained in one or multiple thin layers. This phase screen approach provides the means to
analytically calculate the TEC variations, phase changes within the phase screen as a function
of input statistical electron density.
In order to determine the performance of a trans-ionosphere system, ionosphere weather
models must be combined with a set of propagation equations to form a complete ionosphere
propagation model. Generally from 3-D electron content models, ray bending effect using ray
tracing methods, attenuation models, dispersion model, and Faraday rotation model can be de-
rived [15] . Since a SAR system is subject to small phase abberations, this thesis only considers
the change in signal traveling paths across the aperture due to changes in electron density using
the phase screen model and due to dispersion using the Hartree-Appleton approximation.
5.2 Ionospheric Dispersion
The Hartree-Appleton approximation is a commonly used equation in trans-ionosphere com-
munication literature, such as GPS [22], that relates the refractive index, n, to electron density,
Ne, in an ionized medium as follows:
n =
√1 − f 2
N
f 2(5.1)
≈ 1 − 1
2
f 2N
f 2− 1
8
(f 2N
f 2
)2+ ..
≈ 1 − NI
CHAPTER 5. IONOSPHERE MODELING 48
where f is the instantaneous operating frequency in Hertz, fN is the plasma frequency defined
as:
f 2N =
e2Ne
4πεome≈ 80.6Ne (5.2)
where e = 1.6 × 10−19c is the elementary electron charge , εo = 8.854 × 10−12F/m is the
permittivity of free space, me = 9.1 × 10−31kg is the electron mass, Ne is the electron density
in electrons/m3, and NI is the equivalent index of refraction of ionosphere only.
Equation (5.2) shows that the refractive index decreases and the phase of a signal advances
in the ionosphere; the amount of advancement is inversely proportional to the operating fre-
quency, but directly proportional to the electron density. A higher frequency in a less dense
ionosphere travels closer to the speed of light in a vacuum. Therefore, the difference in travel-
ing velocity in a vacuum and in the ionosphere at any point in space would be
δVp = c/NI − c = cNI (5.3)
To know exactly the total change in velocity through the ionosphere, this formula must be
integrated along the entire path. 3-D mapping of the ionosphere is still an active research area
and it is almost impossible to reconstruct a tomographic map with a resolution comparable to
SAR [43]. Therefore, it is common in literature to assume a slab of constant vertical electron
distribution or to apply Chapman’s formula [2]. In this thesis, constant electron density is
assumed.
Using this assumption, the difference in travel time in the entire ionosphere can be calcu-
lated by integrating (5.3) over time to get the path difference in meters, then dividing by c to
get the time difference in seconds as follows:
Δt(f) =1
c
∫s
NI(s)ds ≈ 1
c
(40.3TEC
f 2
)(5.4)
Here s represents the traveling path in the ionosphere. TEC is the total electron content in a
column of unit area in electrons/m2.
CHAPTER 5. IONOSPHERE MODELING 49
( , )r impP h t τ
N
* ( , )refh t τ− − Azimuth
SampRange
ling
)( ,l lf x y
( , )ionG t τ
[ . ]l mg
Figure 5.1: SAR Transfer Function with Ionospheric Delay Model
Typical TEC values can range from 1016 to 1018 electrons/m2 depending on the time of
the day and geographical location. However, under our static assumption, pulses across the
aperture experience the same amount of TEC and do not affect the resolution. If the electron
density varies in space, it is possible to create broadening in SAR [7] as it is discussed later in
this chapter.
5.3 Dispersive Ionosphere Effects on SAR
5.3.1 Range Effects
During one pulse transmission, the modulated frequency increases linearly as a function of time
causing the signal to experience different amounts of phase changes despite the short duration
of the pulses. To analyze this effect in range, an additional 2-D time delay filter is constructed
in the SAR system chain as follows:
Gion(t) = exp(−j4πftΔt(ft)) ≈ exp(j(co + c1t + c2t
2))
(5.5)
where ft is the instantaneous frequency of the transmitted signal in Hertz, which can be derived
by taking the time derivative of (2.4) as ft = fo + Krt, and with a factor of −4π to convert Hz
CHAPTER 5. IONOSPHERE MODELING 50
to radians with an additional factor of two for round-trip phase difference, and a sign change
to indicate phase advancement.
Approximating (5.4) using Taylor Series about the operating frequency, fo, the coefficients
can be found as follows:
c0 = −4π40.3TEC
c
1
fo
c1 = −4πKr40.3TEC
c
3
f 2o
c2 = −4πK2r
40.3TEC
c
8
f 3o
This formula suggests that the dispersion induced phase advancement can be analyzed us-
ing polynomials in which a lower coefficient is 1/fo times greater than the higher one, i.e.,
the linear phase term is 1/fo greater than the quadratic term and the quadratic term is only
noticeable at lower frequencies such as VHF and HF, typical numbers of TSX are c0 = 53,
c1 ≈ 10−1, and c2 ≈ 10−3Kr which is small compared to the chirp rate.
5.3.2 Azimuth Effects
Dispersion is a function of both frequency and TEC. During one transmission in the range
direction, the signal’s frequency increases but TEC stays constant. However, when the platform
moves in the azimuth direction, the amount of TEC changes as a function of satellite to target
geometry as shown in Figure 5.2 where TEC(θslant) is the slant angle TEC as a function of
slant angle, θslant and TECmin which is the TEC experienced at the point of closest approach
as follow:
TEC sec(θslant) ≈ TECmin(1 +y2
l
2R2min
) (5.6)
ΔTEC = TECmin
(1 +
y2l
2R2min
)− TECmin
=
(v2
sτ2
2R2min
)TECmin (5.7)
CHAPTER 5. IONOSPHERE MODELING 51
lylx
lz
minTEC
s ( )( ) ecslant min slantTECTEC θ θ=
Satellite Position 2
Satellite Position 1
Figure 5.2: Geometry-related TEC variations
Combining this analysis with the range dispersion equations, one can construct a full iono-
sphere time delay filter as follows:
Δt(t, τ) =1
c
(40.3TECmin
f 2t
v2τ 2
2R2o
)(5.8)
Gion(t, τ) = exp(−j4πftΔt(ft, τ)) (5.9)
The above results suggest that in range the change in traveling path is caused by dispersion;
in azimuth it is caused by the change in geometry. In range, the dispersive effects can be
parameterized into polynomials, and each term has its unique imaging effect as discussed in
Section 2.5. At the X-band, the blurry effect of the dispersive medium is trivial. In azimuth, the
change in geometry can be approximated as a quadratic phase term that could potentially affect
the azimuth resolution. To combat ionosphere dispersive phase abberation, most spaceborne
platforms operate at the C-band such as the Canadian Radarsat mission or the X-band such
as the German TSX mission with carrier frequency above 9 GHz and a very narrow antenna
pattern that are able to collect data in a very short span. For TerraSAR-X, the change in TEC
is less than 0.05% for a τ ≤ 0.5 seconds.
In this section, the dispersive effect of a static ionosphere has been established to be min-
CHAPTER 5. IONOSPHERE MODELING 52
imal in modern SAR systems, however, in some cases, platform imaging through a highly
irregular ionosphere region have been observed to experience loss of resolution [7]. Therefore,
it is necessary to model the electron density as a random medium in order to fully analyze
effects of realistic trans-ionospheric propagation.
5.4 Phase Screen Model
In the previous sections, electron density was assumed to be spatially and temporally static, and
the phase noise created by dispersion is identical among all targets. In this section, electron
density will be modeled as a random process in space. More specifically, electron density is
modeled as a wide-sense stationary random field that distorts the spherical wavefront causing
each target on the ground to experience slightly different phase noise. To describe the variations
of the observed electron density in space, it is common to use a power-law or power-spectrum
model [16] that describes the PSD of Ne in 3-D space as follows:
RNe(k) = Cs|k|−β (5.10)
where Cs is called the strength of turbulence that controls the amount of the variations and K is
the vector wavenumber in 1/m, and β is the spectral component that controls the smoothness of
the variations. Such a distribution is always isotropic and wide-sense stationary (homogenous).
In many studies, the ionosphere is further assumed to be a thin two-dimensional screen
or multiple screens between the transmitter and the receiver and only within this screen the
electron density varies. By this assumption, researchers have been able to analytically provide
predictions of phase changes from ground measurements [31], [32], and [2]. The simulator
designs this 2-D distribution by using the following ACF:
RNe(Δxl, Δyl) = exp
(−√
Δx2l + Δy2
l
ρl
)(5.11)
where ρl is the correlation length that controls the smoothness of the 2-D random field. This
particular form of ACF is chosen because it can easily be simulated using Turning Band Meth-
CHAPTER 5. IONOSPHERE MODELING 53
ods [24] and it has an analytical Fourier transform solution that can be used to approximate
(5.10). Figure 5.3 incorporates this phase screen model with the original SAR geometry as:
lx
lz
2h
ly
Point target 1 1 1( , , )X Y Z hΔ
1h
R
Figure 5.3: SAR Geometry with Phase Screen Model
where h1 is the distance from the ground to the screen, h2 from the screen to the platform, and
Δh is the thickness of the screen.
Combing this electron density random field with the Hartree-Appleton approximation, one
can relate the ionosphere phase change to NI and the platform and target position (x1, y1, z1)
using the following approximation [7].
θε(x1, y1, z1, τ) =RΔh2π
λ
hNI
(h1vsτ + h2yl
h,h2xl
h
)(5.12)
where Δh is the thickness of the turbulent phase screen, and h = h1 + h2 is the height of the
satellite, and (vsτ ) is the satellite position at different azimuthal times.
This formula suggests that when the phase screen decreases in height, h1 → 0, h2 → h, the
parameters do not change with respect to satellite positions, hence there is no phase change.
However, when the phase screen is at the same height as the satellite h1 = h2, the phase
variations are maximized. Also, from the satellite-target geometry, targets located on the same
CHAPTER 5. IONOSPHERE MODELING 54
range line will experience very similar phase noise but targets on different azimuth lines will
not. In other words, two targets parallel to the flight path will experience almost identical
phase noise; whereas two target perpendicular will not. In the simulation, the thickness of
turbulent layer is 10km at height of 350km, which is the height of the most volatile layer in the
ionosphere [1].
5.5 Simulation Results
This section will illustrate the imaging effect of random phase noise using the developed sim-
ulator. Under spotlight imaging mode of aperture length 1000m, multiple targets are imaged
on different range lines. For each target, all interception points of the signal with the phase
screen are calculated and the relative distance between these points are used to generate a 2-D
wide-sense stationary random field representing the change in electron density. After which,
the varying electron density are converted to phase changed as described in the chapter using
Hartree-Appleton approximation and (5.12). Three random fields are generated with 100m,
500m, and 1000m correlation lengths and plotted below in Figures 5.4, 5.5, and 5.6. These
figures show the phase change become more variant and the imaging effect of it become more
broadened as the correlation length decreases, and that phase changes more in the azimuth
direction than in the range direction simply due to the geometry of the system.
5.6 Summary
This chapter describes two modeling methods of the ionosphere, the Hartree-Appleton equa-
tion and the phase screen model; the former describes the dispersive property of the ionosphere
while the latter describes the random phase error. The dispersive effect is related to the car-
rier frequency and TEC, and it affects systems operating at lower bands more severely but it
has minimum effects on modern SAR systems operating above 9GHz. The effect of random
CHAPTER 5. IONOSPHERE MODELING 55
medium is highly related to the correlation length and thickness of the screen. The imaging
effects can be severe when the ionosphere is highly variant as demonstrated in the simulation.
CHAPTER 5. IONOSPHERE MODELING 56
Azi
mut
h In
dex
Range Index
20 40 60 80 100 120 140 160 180 200
20
40
60
80
100
120
140
160
180
200 −20
−18
−16
−14
−12
−10
−8
−6
−4
−2
(a) Scintillation Phase Noise with 1000m Correlation Length
Azi
mut
h In
dex
Range Index20 40 60 80 100 120 140 160 180 200
20
40
60
80
100
120
140
160
180
200
(b) Imaging Effects of Scintillation Noise
Figure 5.4: Highly Correlated Ionosphere Scintillation Noise and Its Imaging Effects
CHAPTER 5. IONOSPHERE MODELING 57
Azi
mut
h In
dex
Range Index
20 40 60 80 100 120 140 160 180 200
20
40
60
80
100
120
140
160
180
200
−20
−18
−16
−14
−12
−10
(a) Scintillation Phase Noise with 500m Correlation Length
Azi
mut
h In
dex
Range Index20 40 60 80 100 120 140 160 180 200
20
40
60
80
100
120
140
160
180
200
(b) Imaging Effects of Scintillation Phase Noise
Figure 5.5: Correlated Ionosphere Scintillation Noise and Its Imaging Effects
CHAPTER 5. IONOSPHERE MODELING 58
Azi
mut
h In
dex
Range Index
20 40 60 80 100 120 140 160 180 200
20
40
60
80
100
120
140
160
180
200−15
−10
−5
0
5
10
15
20
25
(a) Scintillation Phase Noise with 100m Correlation Length
Azi
mut
h In
dex
Range Index20 40 60 80 100 120 140 160 180 200
20
40
60
80
100
120
140
160
180
200
(b) Imaging Effects of Scintillation Phase Noise
Figure 5.6: Highly Decorrelated Ionosphere Scintillation Noise and Its Imaging Effects
Chapter 6
Phase Noise Compensation
This chapter reviews existing phase noise compensation algorithms for one-dimensional phase
noise and attempts to correct for ionospheric noise using these methods. This chapter also
proposes a new refined approach to combat ionospheric noise when there is no redundancy of
phase noise among targets. Finally, compensated simulation and measured data are presented.
6.1 Background on Phase Compensation Techniques
Chapter 2 showed that post-processing of SAR depends on integrity of the received phase, both
spotlight and stripamp. Chapter 3 showed that Rayleigh clutter behaves as Gaussian noise and
the imaging of it is not affected by phase noise, but a point target will experience a broadened
PSF as the convolution product of errorless PSF with the Fourier Transform of the phase noise.
Chapter 5 has established that the signal propagating through the ionosphere is prone to phase
noise of various sorts; dispersion in range, geometry related TEC variations in azimuth, and
ionospheric scintillation phase noise for all targets.
Therefore, to compensate for phase noise, SAR data must have the following:
• There must be a sufficient number of point targets with broadened PSF from which the
FFT of the phase noise is extracted, then the phase noise itself.
59
CHAPTER 6. PHASE NOISE COMPENSATION 60
• Point targets must be far enough away from each other such that their PSFs do not inter-
fere with each other.
• Rayleigh clutter response must be low compared to the RCS of point scatter, since clutter
behaves as noise that overshadows the point targets and impedes any estimation.
With these requirements in mind, one can now starts to understand the steps of the existing
autofocusing algorithms. Autofocus of SAR signals is a long-researched topic that attempts
to maximize the mainlobe to sidelobe ratio. Generally, these methods can be categorized into
non-coherent and coherent methods [27]. Non-coherent methods operate solely on the intensity
values and do not preserve any phase information while coherent methods try to correct for the
phase noise before compression. In order to use the data for further processing such as Inter-
ferometric SAR, only coherent methods are investigated. Coherent methods have been applied
in various applications such as removing moving target smearing by Prominent Point Process-
ing (PPP) [41], spotlight polar-to-rectangular resampling residuals by Two-Dimensional Phase
Gradient Algorithms (2DPGA) [40]1, and demodulation errors and orbital deviations by Phase
Gradient Algorithms (PGA) [6], [19]. Among these coherent methods, there are generally
two categories of estimation techniques; parameterized and non-parameterized. Parameterized
methods model the phase noise as a polynomial as in (2.36). By estimating the coefficients,
these algorithms find the best polynomial fit for the phase noise, exp(jθe(t)). However, since
the exponential operator on the phase noise is not linear, i.e., phase noise is wrapped between
−π and π, these algorithms often require phase unwrapping before they can be carried out [28],
[5], [38] or some other fourier domain techniques [30], [29]. Because of these extra processing
steps, these algorithms are rarely applied at low SNR conditions [38]. Non-parametric algo-
rithms include a number of algorithms belonging to the family of Phase Gradient Algorithms
[6], [12], [19]. These algorithms are iterative wherein every iteration, the algorithm finds de-
1Warner calls this method 2-D PGA, however, the phase noise are actually in two directions, azimuth andrange, in each direction, phase noise is identical among all targets, hence this estimation process is different fromthe 2-D phase noise of this work.
CHAPTER 6. PHASE NOISE COMPENSATION 61
tectable changes in phase noise; over many iterations, these small phase changes are integrated
to give the full phase noise. Works in [14] provide a theoretical comparison of these methods
and found that non-parameterized models generally yield lower estimation variance than pa-
rameterized methods on the quadratic term at high SNR, but perform worse when estimating
higher order terms. Often the estimation of small higher order terms, parameterized algorithms
often result in over-estimation, and the variations due to thermal noise are modeled instead of
the true phase noise itself. To model higher order terms, PGA is often the better choice.
These autofocusing methods provide the basic framework for our proposed algorithm to
combat ionospheric phase scintillation. Unlike orbital deviation and demodulation error, phase
scintillation noise is not common to all targets. So far, we have not found any phase noise
compensation methods that combat arbitrary two dimensional phase noise. In the following
sections, we propose to use both non-parameterized and parameterized methods, and show that
a non-parameterized model is not feasible. Hence, this thesis adopts a parameterized compen-
sating algorithm that approximates scintillation phase noise as two-dimensional polynomials
and show what is gained and lost from using this algorithm.
6.2 Phase Gradient Algorithm
In order to understand the general steps of phase estimation, the processing steps of PGA are
described below and in Figure 6.1.
6.2.1 Brightest Point Detection and Center Shifting
The input to PGA is range compressed spotlight data; since there is minimum phase noise in
the range, the image will be focused in range but blurred in the azimuth . Since all targets on the
same range line have the same phase noise, finding the brightest point target would maximize
the phase noise signal. After detecting the brightest point, it is shifted to the centre of each
range line to accomplish two purposes, first, to line up the point returns for further processing
CHAPTER 6. PHASE NOISE COMPENSATION 62
Windowing
Detection & Shifting
Estimation
Compensation
IFFT
Threshold υ
Range-Compressed Signal
υ> υ<
Exit
Figure 6.1: Phase Gradient Algorithm Flowchart.
and, second, to set the linear coefficient of phase noise to zero. As described in Section 2.5.1,
linear coefficients control the amount of shifting in the image; by centering the brightest point,
the linear coefficients is set to zero, hence, removing the need to estimate this term.
6.2.2 Windowing
After centering the strongest peak, a damping window is enforced around this centre peak.
This step isolates the strongest copy of the degraded PSF in every range line and suppresses
undesired signal from other bright targets or terrain clutter. To accomplish this effectively,
the window width must encompass the noise PSF but filter out as much noise as possible.
Currently, the optimal window size is still an active research topic and, in most algorithms,
it is still not an automatic procedure. However, as stated in [12], if one degraded PSD is
encompassed in the first iteration, the algorithm will eventually converge to the correct solution.
CHAPTER 6. PHASE NOISE COMPENSATION 63
6.2.3 Inverse Fourier Transform and Phase Estimation
After the degraded PSF has been isolated to the centre of each range line, an inverse Fourier
Transform is carried out. This is to transform the degraded PSF from the image domain into the
phase noise signal in the phase domain. Suppose that in a data set there are L azimuth samples
and M range samples as shown in Figure 6.2.3. where θ[l,m] is the angular component of the
[1,1] [1,1] [1] [1,1]conste Nθ φ θ θ= + + … [1, ] [1, ] [ ] [1, ]M M const M N Meθ φ θ θ= + +
… [ , ] [ , ] [ ] [ , ]l m l m const me N l mθ φ θ θ= + +
…
[ ,1] [ ,1] [1] [ ,1]L L conste N Lθ φ θ θ= + + … [ , ] [ , ] [ ] [ , ]L M L M const Me N L Mθ φ θ θ= + +
Figure 6.2: Phase Signal Diagram with One-Dimensional Phase Noise
centre shifted signal phase, g[l,m], and it consists of three components; φe[l] is the noise signal in
azimuth direction only and it is common among all range lines, θN is the angular component of
thermal noise that assumes Gaussian distribution, and θconst[m] is the range line phase constant
and it is a geometry-related phase value as described in (2.28). Due to large distances to
the different point targets used in estimation, θconst can be regarded as uniformly distributed
across many range lines. More importantly, these phase constants should be preserved after
compensation in order to perform InSAR, in which a target’s elevation is derived from these
values [18].
To preserve these phase constants and to find the phase gradient of the phase noise θe[l], the
following identity is used:
φle[l,m] = φe[l,m] − φe[l+1,m] =
Im(ˆgl
[l,m] g∗[l,m]
)|g[l,m]|2 (6.1)
CHAPTER 6. PHASE NOISE COMPENSATION 64
where ˆgl[l,m] is the derivative of g[l,m] and φl
e[l,m] is the change in phase noise in the azimuth
direction (L direction). Furthermore, when this identity is applied to every column, range
line phase constants are removed by differentiation and only the changes in phase noise are
detected. After the phase gradient operator, there are M copies of phase noise derivatives from
each column. Intuitively, under Gaussian noise, the optimal solution is to average out these
noise. In [6], it is shown that averaging is the optimal linear minimum variance combination
of phase noise derivatives. More specifically, the estimator is defined as follows:
ˆφle[l] =
∑Mm=1 Im
(gl[l, m]g∗[l, m]
)∑M
m=1 |g[l, m]|2 (6.2)
6.2.4 Compensation and Iteration
After one iteration, the estimated phase noise derivative is integrated and removed from the
original data as follows:
gz+1[l] = gz[l] exp(−jφe[l]) (6.3)
where z is used here as an iteration index. Therefore, estimated phase noise is removed from
the input signal of the current iteration by taking the complex conjugate. The product of this
operation is fed into the algorithm in the next iteration until convergence. Convergence in
PGA is achieved by setting a lower bound on ˆφe[l].
ˆφe[l] is reduced when more and more
changes in phase noise are removed and its derivative approaches zero; at this point, PGA has
accomplished phase noise compensation and preserved the phase constants.
6.3 Application to Ionospheric Noise
In Figure 6.4, PGA was used to compensate for the phase noise in Figure 5.4. It is apparent
from the derivations above that PGA will only pick out the common phase noise patterns among
M range lines. In the case of a highly correlated phase noise, PGA can actually estimate most
of the phase noise and improve the image resolution. However, in the case of active ionosphere,
CHAPTER 6. PHASE NOISE COMPENSATION 65
attempts to use PGA to estimate a highly variable 2-D noise will fail simply because there is no
redundancy in the data. As shown in Figures 6.6 and 6.7 are the compensation results applied
to phase noise in Figures 5.5 and 5.6. In these two cases, since phase noise among targets are
different, it is insufficient to estimate all the phase noise using a 1-D signal and leaving much
residuals uncompensated for using PGA.
CHAPTER 6. PHASE NOISE COMPENSATION 66
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40
60
80
100
120
140
160
180
200 −20
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−16
−14
−12
−10
−8
−6
−4
−2
(a) Original Phase Noise
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20
40
60
80
100
120
140
160
180
200
2
4
6
8
10
12
14
16
18
(b) PGA Estimation Result
Figure 6.3: PGA Estimation Result of a Highly Correlated Phase Noise
CHAPTER 6. PHASE NOISE COMPENSATION 67
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20
40
60
80
100
120
140
160
180
200
(a) Original Imaging Effect
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20
40
60
80
100
120
140
160
180
200
(b) Imaging Effect After PGA Compensation
Figure 6.4: Residual Imaging Effect after PGA Compensation
CHAPTER 6. PHASE NOISE COMPENSATION 68
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20
40
60
80
100
120
140
160
180
200
−20
−18
−16
−14
−12
−10
(a) Original Phase Noise
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20
40
60
80
100
120
140
160
180
200
1
2
3
4
5
6
7
8
9
10
(b) PGA Estimation Result
Figure 6.5: PGA Estimation Result of a Correlated Phase Noise
CHAPTER 6. PHASE NOISE COMPENSATION 69
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20
40
60
80
100
120
140
160
180
200
(a) Original Imaging Effect
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20
40
60
80
100
120
140
160
180
200
(b) Imaging Effect After PGA Compensation
Figure 6.6: Residual Imaging Effect after PGA Compensation
CHAPTER 6. PHASE NOISE COMPENSATION 70
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20
40
60
80
100
120
140
160
180
200−15
−10
−5
0
5
10
15
20
25
(a) Original Phase Noise
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20
40
60
80
100
120
140
160
180
200−35
−30
−25
−20
−15
−10
−5
0
(b) PGA Estimation Result
Figure 6.7: PGA Estimation Result of a Highly Decorrelated Phase Noise
CHAPTER 6. PHASE NOISE COMPENSATION 71
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20
40
60
80
100
120
140
160
180
200
(a) Original Imaging Effect
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20
40
60
80
100
120
140
160
180
200
(b) Imaging Effect After PGA Compensation
Figure 6.8: Residual Imaging Effect after PGA Compensation
CHAPTER 6. PHASE NOISE COMPENSATION 72
6.4 Proposed Algorithms
To compensate for a two-dimensional phase noise, a two-dimensional estimation kernel must
be used. In the following sections, both non-parametric and parameterized solution will be
proposed. However, it will be shown that only parameterized solution is feasible. Consider the
modified signal model as follows:
[1,1] [1,1] [1] [1,1]conste Nθ φ θ θ= + + … [1, ] [1, ] [ ] [1, ]M M const M N Meθ φ θ θ= + +
… [ , ] [ , ] [ ] [ , ]l m l m const me N l mθ φ θ θ= + +
…
[ ,1] [ ,1] [1] [ ,1]L L conste N Lθ φ θ θ= + + … [ , ] [ , ] [ ] [ , ]L M L M const Me N L Mθ φ θ θ= + +
Figure 6.9: Phase Signal Diagram with 2-D Phase Noise
where φe[l,m] is now a two dimensional phase function to be estimated.
6.4.1 Non-parametric Algorithm
A non-parametric estimation kernel can be adopted from phase unwrapping algorithms [12]
where a wrapped 2-D signal is unwrapped with respect to all adjoint phase values. Considered
a simplified case, where M = 2, L = 2, and the system is noiseless as follows:
[1,1] [1,1] [1]conse tθ φ θ= + [1,2 [1,2] [2]conse tθ φ θ= +
[2,1] [2,1] [1]conse tθ φ θ= + [2,2] [2,2] [2]conse tθ φ θ= + [1,1]
lθΔ
[2,1]mθΔ
[1,1]mθΔ
[1,2]lθΔ
Figure 6.10: Phase Differences Observations
CHAPTER 6. PHASE NOISE COMPENSATION 73
where Δθl is the phase differences in L direction and Δθm in M direction defined as follows:
Δθl[l,m] = Δθ[l+1,m] − Δθ[l,m] 1 ≤ l ≥ L − 1, 1 ≤ m ≥ M (6.4)
Δθm[l,m] = Δθ[l,m+1] − Δθ[l,m] 1 ≤ l ≥ L, 1 ≤ m ≥ M − 1 (6.5)
From these computed phase differences, one can formulate this problem as a least square fitting
problem, Δθ = P S, where Δθ is arranged as follows:
Δθ =
⎡⎢⎢⎢⎢⎢⎢⎢⎣
Δθl[1,1]
Δθl[1,2]
Δθm[1,1]
Δθm[2,1]
⎤⎥⎥⎥⎥⎥⎥⎥⎦
(6.6)
and S is a matrix of parameters to be estimated as follows:
S =
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
φe[1,1]
φe[1,2]
φe[2,1]
φe[2,2]
θconst[1]
θconst[2]
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
(6.7)
and P is the design matrix that relates the observations (Δθ) to the estimation parameters (S)
as follows:
P =
⎡⎢⎢⎢⎢⎢⎢⎢⎣
1 0 −1 0 0 0
0 1 0 −1 0 0
1 −1 0 0 1 −1
0 0 1 −1 1 −1
⎤⎥⎥⎥⎥⎥⎥⎥⎦(
(L−1)(M)+(L)(M−1))×(LM+M)
(6.8)
The dimensions of P are ((L− 1)(M) + (L)(M − 1))× (LM + M) where (L− 1)(M) is
the number of phase differences in azimuth, (M−1)(L) in range and (LM +M) is the number
of parameters which suggests that this system is over-determined when L > 3. However, the
CHAPTER 6. PHASE NOISE COMPENSATION 74
rank of P is always LM − 1, one less than the observations, regardless of the phase values.
This is because the path difference between two phase values is not unique, but it can also be
other linear combinations of any traversed path. i.e., Δθl[1,1] = Δθl
[1,2] + Δθm[1,1] + Δθm
[2,1]. As a
consequence, the system is always rank deficient by 1, and only relative phase difference can be
retrieved, i.e., the phase estimation results will be biased with a constant phase shift. Further-
more, with M additional column phase constants to estimate, the system is under-determined.
Therefore, parameterized algorithms are not feasible because of rank deficiency of M − 1 or-
der. Therefore, the only solution is to parameterize the model and to fit a two-dimensional
polynomial function on the phase noise function and reduce the estimation parameters.
6.4.2 Parametric Algorithm
Since non-parameterized methods can not be used due to rank deficiency, this section explores
the use of a parametric estimation method and proposes to use a 2-D polynomial phase es-
timation algorithm called the Phase Differencing Algorithm (PDA) [11] where a 2-D phase
polynomial of the following form is estimated:
vP,Q[l,m] = exp
(jΦ[l,m]
)(6.9)
Φ[l,m] =P∑
p=0
Q∑q=0
c(p, q)lpmq (6.10)
where vP,Q[l,m] is a unit amplitude with 2-D polynomial phase signal.
This particular form of phase polynomial is called the rectangular support polynomial func-
tion with degrees Q and P in [11] as shown in Figure 6.11.
PDA is also a step-wise algorithm where every coefficient is estimated one after another. In
the first iteration, PDA differentiates phase function P times in the L direction to remove the
dependence of the polynomial on L, leaving a one dimensional polynomial in M . Moreover, if
this 1-D polynomial is further differentiated Q− 1 times in the M direction, one can eliminate
the dependence of this polynomial on the first Q − 1 order terms, leaving us with a phase
signal that is dependent only on the highest order coefficient, c(Q, P ). By simple sinusoidal
CHAPTER 6. PHASE NOISE COMPENSATION 75
(0,0)C
0p =
1p =
2p =
0q = 1q = 2q =
Figure 6.11: Rectangular Support Polynomial with P=Q=2
frequency estimation, c(Q, P ) from the frequency of this complex sinusoid can be estimated.
After removing the phase contribution related to this coefficient and repeating this procedure
until all coefficients have been estimated. A summary of key mathematical development is as
follows:
Consider Phase Difference (PD) operators defined recursively as follows:
In the L direction:
PD0l [v[l,m]] = v[l,m] (6.11)
PD1l [v[l,m]] = v[l,m]v
∗[l,m+1] (6.12)
PDql [v[l,m]] = vPDq−1m[v[l,m]]PDq−1
m [v∗[l,m+1]] (6.13)
In the M direction:
PD0m[v[l,m]] = v[l,m] (6.14)
PD1m[v[l,m]] = v[l,m]v
∗[l,m+1] (6.15)
PDpm[v[l,m]] = PDp−1
m [v[l,m]]PDp−1m [v∗
[l,m+1]] (6.16)
The operators perform differentiation on the phase in either the L or M direction and if the
CHAPTER 6. PHASE NOISE COMPENSATION 76
PD operator is performed P and Q − 1 times on v[l,m].
PD(Q−1)m [PD
(P )l [v[l,m]]] = exp(jWpm + const) (6.17)
Wp =(−1)Q+P−1P !
Q!C(P, Q) (6.18)
After phase differencing operations, the 2-D phase signal is transformed into a 1-D complex
sinusoid whose frequency is a function of the highest coefficient. Using single-tone sinusoidal
estimation techniques [38] [30], the highest order coefficient is obtained. After this coefficient
is obtained, the phase variations due to this term is removed as shown here:
vP,Q−1[l,m] = vP,Q
[l,m] exp(−jC(P, Q)lP mQ) (6.19)
The algorithm then moves to estimate the next highest order polynomial until all coefficients
have been found.
Application to Ionosphere Phase Noise
Since ionospheric phase noise is correlated in space, it is reasonable to consider an approxi-
mation by a 2-D polynomial function. Estimation results of arbitrary correlated phase noise
are shown in Figures 6.12, 6.13, and 6.14. From Figures 6.12 and 6.13, the estimation results
are quite accurate even with only 9 coefficients, i.e., P = Q = 2, for the case of 500m and
1000m correlation length. However, when correlation length decreases to 100m in Figure 6.14,
the estimation accuracy is severely undermined; this is simply due to the fast variations in the
phase function that it cannot be approximated using a polynomial.
CHAPTER 6. PHASE NOISE COMPENSATION 77
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100
120
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200 −20
−18
−16
−14
−12
−10
−8
−6
−4
−2
(a) Phase Noise with 1000m Correlation Length
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20
40
60
80
100
120
140
160
180
200 −3
−2
−1
0
1
2
3
(b) Estimated Phase Noise using PDA
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20
40
60
80
100
120
140
160
180
200
(c) Estimated Phase Noise using PDA
Figure 6.12: PDA Estimation Results
CHAPTER 6. PHASE NOISE COMPENSATION 78
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40
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200
−20
−18
−16
−14
−12
−10
(a) Phase Noise with 500m Correlation Length
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40
60
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100
120
140
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180
200 −3
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−1
0
1
2
3
(b) Estimated Phase Noise using PDA
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20
40
60
80
100
120
140
160
180
200
(c) Estimated Phase Noise using PDA
Figure 6.13: PDA Estimation Results
CHAPTER 6. PHASE NOISE COMPENSATION 79
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40
60
80
100
120
140
160
180
200−15
−10
−5
0
5
10
15
20
25
(a) Phase Noise with 100m Correlation Length
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20
40
60
80
100
120
140
160
180
200 −3
−2
−1
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1
2
3
(b) Estimated Phase Noise using PDA
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20
40
60
80
100
120
140
160
180
200
(c) Estimated Phase Noise using PDA
Figure 6.14: PDA Estimation Results
Range line phase constants are also conveniently removed by the differentiation property
of PDA. The algorithm can correctly estimate higher order coefficients that best describe the
ionospheric phase noise and also try to find coefficients that best fit these range line phase
constants. Because of the random fluctuations of these phase constants, the estimation results in
the first row of the coefficient diagram (see Figure 6.11) are over-estimated. These coefficients
have increased to best approximate range line constants instead of true ionospheric phase noise.
The results in shown as in Figure 6.15.
CHAPTER 6. PHASE NOISE COMPENSATION 80
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20
40
60
80
100
120
140
160
180
200 −3
−2
−1
0
1
2
3
(a) Highly Correlated Phase Noise with Random Phase
Constants
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20
40
60
80
100
120
140
160
180
200 −3
−2
−1
0
1
2
3
(b) Over-Estimated PDA Results with Random Phase
Constants
Figure 6.15: PDA Estimation Results with Random Phase Constants
The proposed solution is simply to set these coefficients to zero to avoid this problem. By
leaving these coefficients uncompensated, the algorithm effectively separates a 2-D phase noise
into two parts where one only has constant phase values in each range line and one contains
all the other variations in the range and azimuth. Moreover, from Section 2.5.1, constant phase
values do not cause blurring, therefore, leaving these values uncompensated does not change
CHAPTER 6. PHASE NOISE COMPENSATION 81
the resolution. The variations that cause blurring will be estimated and corrected. The estima-
tion and compensation results and the imaging effect of the proposed solution are in Figures
6.16 through 6.18.
Phase Noise = Estimation (Blurry Dependent)
+ Estimation Residual (Blurry Independent)
Range Index
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40
60
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120
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160
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200-3
-2
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2
3
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40
60
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200-3
-2
-1
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3
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20
40
60
80
100
120
140
160
180
200
-2
-1.5
-1
-0.5
0
0.5
1
Figure 6.16: Phase Noise in Two part, Blurry-Indecent and Blurry-Dependent
Phase Noise = Estimation (Blurry Dependent)
+ Estimation Residual (Blurry Independent)
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40
60
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100
120
140
160
180
200
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20
40
60
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100
120
140
160
180
200
A
zim
uth
Inde
x
Range Index20 40 60 80 100 120 140 160 180 200
20
40
60
80
100
120
140
160
180
200
Figure 6.17: Imaging Effect of Blurry-Indecent and Blurry-Dependent Part
Phase Noise = Estimation (Blurry Dependent)
+ Estimation Residual (Blurry Independent)
Range Index
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40
60
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200-3
-2
-1
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3
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20
40
60
80
100
120
140
160
180
200 0.5
1
1.5
2
Figure 6.18: Phase Noise in Two part, Blurry-Indecent and Blurry-Dependent
CHAPTER 6. PHASE NOISE COMPENSATION 82
Phase Noise = Estimation (Blurry Dependent)
+ Estimation Residual (Blurry Independent)
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Figure 6.19: Imaging Effect of Blurry-Indecent and Blurry-Dependent Part
Phase constants are further used in InSAR processing; these uncompensated ionospheric
phase constants will becomes a dominant factor in InSAR accuracy in DEM reconstruction
[16]. However, in this thesis, leaving the phase constants uncompensated, the refocusing of
SAR PSF due to ionospheric noise is still accomplished.
6.5 Remarks on the Proposed Algorithm
While trying different simulation data sets with different geometries and correlation lengths,
some properties of the solution were noticed:
• Lack of redundancy:
One dimensional phase noise is redundant in all range lines, however ionosphere phase
noise does not have this property. Every strong scatter is crucial in the fitting process and
the lack of such strong point targets means the estimated coefficients will be erroneous.
Therefore, it is more important to have fewer but strong targets rather than use more
weak targets corrupted by noise. It is difficult to find such a scene that satisfied all these
conditions, hence limits the applications of such algorithm.
• Higher order term estimation in noise:
Approximating smaller variations by estimating higher order coefficients in noise is very
difficult. Noise often overshadows the true signal variation and causes an estimation
CHAPTER 6. PHASE NOISE COMPENSATION 83
error. Estimation errors of higher order terms are propagated in the algorithm and create
erroneous results. The algorithm has been found to be accurate only using up to third
order terms. i.e., P = Q = 2.
6.6 Application to TerraSAR-X Data
The proposed 2-D estimation algorithm will now be applied to measured observations collected
by the TSX platform in this section.
6.6.1 Selection of TerraSAR-X Data
A TSX spotlight data with strong and isolated point scatters are the basic requirements for
compensation. Finding such data with active ionospheric conditions is not an easy task. Since
ionosphere is most active around the magnetic poles and the magnetic equators around lo-
cal midnight, ionosphere turbulence also happens in small patches that extends from several
kilometers after sunset to few hundreds of kilometers at midnight [1]. In this thesis, a TSX
spotlight taken over Antarctica (67035′S, 69015′W) on November 24th, 2007 has been chosen.
In this spotlight data, numerous isolated icebergs were imaged. The entire scene and the se-
lected portion from the bottom right corner used in estimation are shown in Figure 6.20. The
scene exhibits blurred icebergs in the azimuth (column) direction and focused icebergs in the
range (row) direction indicating existence of azimuthal phase noise.
These isolated icebergs have strong returns and behave more as hard targets than radar
clutter. This phenomena might be explained because icebergs’ backscatter is a combination
of surface and volume scattering; it exhibits a radar profile that varies with incident angle, ice
type, age and volume, and SAR images are often used for iceberg detection and classification
[21]. Therefore, the refocusing of blurred icebergs using the proposed compensation algorithm
could assist in iceberg identification and detection and enhance ship navigation safety.
CHAPTER 6. PHASE NOISE COMPENSATION 84
6.6.2 Remarks on Estimation Results
The PGA compensation result in Figure 6.21(b) shows minimal change to the image. As
discussed in Chapter 6, this is due to targets experiencing different phase noise such that the
phase noise averages out to zero during estimation and does not affect the result at all. However,
after PDA compensation, the results in Figure 6.21(c) show more intensified returns in the
centre of each of the three icebergs. However, this result was only obtainable in this specific
section of the original data and after many trials with other sections of the same original data,
there were not any successful estimation results. This might be due to the lack of strong
scatters, since coefficient estimations are carried out across many range lines, contributions
from each range line is crucial. To eliminate low intensity returns, a lower threshold is enforced
on the scatter to be used in estimation. Therefore, to have a successful estimation, there should
be a sufficient number of strong scatters in each range line and they should also be sufficient
spaced apart such that their returns do not interfere one another. Such conditions are rarely
satisfied in real life observations.
6.7 Summary
This chapter has summarized the basic requirements for the coherent autofocus algorithm:
strong point scatters, high SNR, and isolated point scatterers. Existing 1-D phase compen-
sation methods and non-parameterized phase estimation kernels are not feasible to estimate a
highly variant ionospheric phase scintillation noise, and only a non-parameterized polynomial
function can be applied. By approximating a smooth phase noise of order three, some iono-
spheric phase noise can be removed and the compensated results show considerable resolution
improvements for all point targets, both simulation and measured data.
CHAPTER 6. PHASE NOISE COMPENSATION 85
(a) Full TerraSAR-X Scene over Antarctica
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CHAPTER 6. PHASE NOISE COMPENSATION 86
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Figure 6.21: Original TSX Data and Compensation Results
Chapter 7
Conclusions and Future Work
This final chapter summarizes the contributions made in this thesis and proposes a number of
ways this thesis could be further applied and extended.
7.1 Conclusions
Among the many design challenges and sources of error in SAR, this thesis chose to address
the problem of SAR smearing due to ionospheric phase scintillation noise. More specifically,
this thesis has demonstrated that a SAR image is smeared under ionospheric scintillation noise,
attempted to resolve ionospheric phase scintillating noise in the following steps:
• SAR simulator :
A software package was developed in Matlab that includes a raw data generator in the
time domain, ionosphere propagation models, and a SAR processor. A complete list of
files and functions are included in Appendix A. The simulator is designed in the time do-
main to allow the user to simulate different realistic perturbations such as moving target
simulation, platforms deviations, and Earth’s movements. In this thesis, the simulator is
used primarily to demonstrate and simulate random ionosphere scintillation phase noise.
– The effect of random phase noise:
87
CHAPTER 7. CONCLUSIONS AND FUTURE WORK 88
The effect of the random phase noise has been derived as the convolution product
of noiseless PSF and the PSD of the cosine of random phase for both spotlight and
stripmap data. The result has been demonstrated using the simulator and shows
good correspondence with past research on the effect of random phase error on
conventional linear phase arrays.
– Ionosphere scintillation and trans-ionosphere propagation simulation:
Trans-ionospheric SAR signals are simulated using well-established propagation
principles, including the Hartree-Appleton equation, a 2-D varying electron density
random field, and the phase screen model, to relate observed ionosphere phase noise
to platform-geometry. Depending on the degrees of electron density fluctuations
in the ionosphere, targets can experience highly variant phase noise, as shown in
Chapter 5.
• Phase noise compensation:
The second part of this thesis involves the compensation for ionospheric phase noise.
Compensation using the PGA algorithm has been shown to be successful given a slow-
varying phase noise is present i.e., estimation by finding common phase noise error
among targets. However, in order to deal with more variant phase noise, a parame-
terized 2-D polynomial phase function, the PDA, is used to approximate the correlated
phase noise, through a series of phase differentiation and estimation steps in the PDA al-
gorithm, an arbitrary function can be well-approximated using an order three rectangular
polynomial. By removing this estimated phase, the image can be refocused.
However, through our simulations, three insufficient aspects of the PDA algorithm were
noted:
– Blurry independent noise can not be estimated:
The algorithms can only estimate and remove phase noise that cause blurred PSF.
Although, this uncompensated phase noise does not cause any visible spreading on
CHAPTER 7. CONCLUSIONS AND FUTURE WORK 89
PSF, but it will affect on the accuracy of InSAR DEM as discussed.
– Insufficient number of strong targets:
PDA polynomial fitting algorithms uses all hard targets located on all range lines for
estimation, i.e., a fitting processing is carried out across many range lines. Hence,
having sufficient number of isolated scattered is crucial to accurate estimation. In
remote regions where the ionosphere is more active, this condition is rarely satis-
fied, hence limiting the application of the proposed algorithm.
– Higher order coefficients overshadowed by noise:
In PDA, a higher order coefficient can only be estimated after a series of differen-
tiations of the phase, however, the variance of white noise doubles after each dif-
ferentiation, and this poses difficulties in estimating small higher order coefficients
in low SNR conditions. So far, the program can only safely estimate a rectangular
polynomial of order three on both simulated data and measured TSX data.
Application of the proposed algorithm to measured TSX data has resulted in a more
focused image of the icebergs. A spotlight data over Antarctica contains both dynamic
ionospheric activities and strong returns from the icebergs. After applying the PDA
algorithm to a selected scene, some smeared images of the iceberg have shown a greater
contrast and reduced smearing.
7.2 Future Work
The bottleneck in estimating ionospheric noise is primarily the lack of strong scatterers and
the unknown conditions of the ionosphere in remote regions. Given the high resolution of
SAR data and the sensitivity of the phase changes, external sources of data such as ionospheric
tomography can not provide additional measurements of comparable resolution, hence external
sources do not assist the estimation process. However, in the future, this algorithm will be
applied to additional measured InSAR data sets to see the effects of this algorithm on InSAR
CHAPTER 7. CONCLUSIONS AND FUTURE WORK 90
correlation and other products derived from SAR data.
Appendix A
Simulator Design in Matlab
The detailed implementations of the developed simulator and the proposed compensation al-
gorithms are included in this Appendix.
A.1 Simulator Functions
A full list of raw data generation and processing functions is presented below:
• SVConstants.m
– Description: This file is the simulator parameter initialization file that contains
satellite antenna specifications, trajectory path, thermal noise parameters, and other
constants such as speed of light, radian to degree conversion..etc.
– Input: none
– Output: none
• SimConstants.m
– Description: This file defines target parameters or calculate parameters associated
with SVConstants.m such as baseline and geometry offsets. This file also loads tar-
gets’ location and backscatter strength files defined as DEM.mat and sigmaNot.mat
91
APPENDIX A. SIMULATOR DESIGN IN MATLAB 92
– Input: none
– Output: none
• AntennaGainPattern.m
– Description: This file calculates the antenna beam pattern based on the defined
antenna beam pattern profiles. If no profile is specified, a generic sinc function is
used. Different tilting angles are also used in order to simulated the side-looking
properties of TSX platform.
– Input: Antenna spherical system coordinates, theta, phi, and tilting angles about the
antenna coordinate system, xtilt, ytilt, ztilt
– Output: Antenna gain value
• PointScatterModelRawRanAz.m
– Description: This file generates raw signal in either stripmap or spotlight mode.
The signals generated are the demodulated signals in baseband and not the original
modulated signals. This simplification is used in order to speed up the computa-
tions. The main purpose of this function is to time delayed raw data by taking in the
satellite transmitting and receiving positions and the coordinates and the strength
of the point scatter on the ground.
– Input: Transmitting satellite position, SVTX, receiving satellite position, SVRX,
coordinates of the point scatterer, deltaX, deltaY, deltaZ, backscatter strength, Ak,
and operating mode, operatingMode.
– Output: time delayed raw data
• TBM.m
– Description: This file produces isotropic homogenous random fields by using the
Turning Band Method. This file is separated into two parts. A one dimensional
APPENDIX A. SIMULATOR DESIGN IN MATLAB 93
random process is first simulated in a particular direction, then this one dimensional
random process is repeated in different random direction. The total random fields is
the summation of these 1-D random processes. A 1-D random process is generated
by the FFT method and 2D random field is generated by TBM method as described
in [24]. This file takes in random coordinates upon which the random fields is
generated according to two input parameters. One parameter controls the variance
and another parameter controls the smoothness of the random field
– Input: 2-D Coordinates of a random field, xCoord, yCoord. Variance parameter, a,
and smoothness parameters, b.
– Output: one random field is produced based on the coordinates specified in the
input, x.
• ionoSim.m
– Description: This file calculates the geometry of the satellite with respect to the
targets and the phase screen. As the satellite is moving in the azimuth direction, the
transmitted signals penetrate the ionosphere at different points depending on the
phase screen height and the target location. This file takes platform orbital infor-
mation and the target locations and calculates these coordinates. These coordinates
are passed into TBM.m.
– Input: Satellite height and velocity, h and vs. Target spatial coordinates, deltaX,
deltaY, deltaZ. Phase screen height.
– Output: A random field produced by TBM is passed back to the main program.
• AzFilter.m
– Description:
This file performs matched filtering in the frequency domain in the azimuth direc-
tion as described in [4] Chapter 2. The implementation is chosen because of its ef-
APPENDIX A. SIMULATOR DESIGN IN MATLAB 94
ficiency and it can be carried simultaneously across many range lines. The matched
filtering in azimuth and in range can be carried out interchangeably because of the
independency of these time indices as discussed before.
– Input: Azimuth Raw Data, FullData,
– Output: Azimuth focused Data, focusedData
• RangeFilter.m
– Description:
This file performs matched filtering in the frequency domain in the range direction.
– Input: Range raw data, FullData,
– Output: Range focused data, focusedData
• FFT2.m
– Description:
Fast Fourier Transform in two dimension is taken directly from the built-in function
in Matlab without any alterations.
• multipleFacetSummationServer.m
– Description:
This is the main execution file in the simulator that combines the above functions
to generate raw data. This file also acts as a cluster server in which multiple targets
are split up among different cluster nodes. The function in server mode will create
subdirectories, and these subdirectories will be used individually by each cluster
node. The difference between operating in server mode or cluster node mode is the
availability of arguments passed in. If no arguments are given, then this function
will act as a server and will simply split the targets among the available number of
servers specified in SimConstants.m. However, if arguments are given, the program
will act as a cluster node, and go into different directories to produce raw data.
APPENDIX A. SIMULATOR DESIGN IN MATLAB 95
– Input: two inputs are used to specify the process ID, processID, and cluster Index,
clutterIndex, that this machine should run on.
– Output: raw data
A flow chart of the above described functions is presented below:
SimConstants.m
multipleFacetSummationServer.m
SVConstants.m
AntennaGainPattern.m
ionoSim.m
PointScatterModelRawRanAz.m
Focused Data
TBM.m
AzFilter.m FFT2.m
RangeFilter.m Compensation Module
Stripmap Data Spotlight Data
Compressed Data
Figure A.1: Dependency Chart of Simulator Functions and Files
A.2 Compensation Functions
Given a compressed data, either simulated or measured, the compensation algorithms will
reprocess them to generated more focused data. There are two main algorithms used, namely
the Phase Gradient Algorithm (PGA.m) and the Phase Differencing Algorithm (PDA).
• PGA.m
APPENDIX A. SIMULATOR DESIGN IN MATLAB 96
– Description: This file performances the iterative Phase Gradient Algorithm and
plots the compensation results at each iteration. The general processing steps are
introduced in [6]. The program takes a compressed image and produces a compen-
sated image.
– Input: Compressed Spotlight Data, f
– Output: Compensated Data, F
• ionoComp.m
– Description: This file performances the general steps of compensation algorithms
as described in Chapter 6, including detection and shifting, windowing, and com-
pensation. This file is to supplement the PDA algorithm by pre-procssing a com-
pressed image. This pre-processed result is then passed in to PDA algorithm for
estimation. The estimation results will be gathered to perform compensation.
– Input: Compressed Spotlight Data, f
– Output: Compensated Data, F
• multipleFacetSummationServer.m
– Description: This file implements the Phase Differencing Algorithm. A two dimen-
sional phase image is passed in and an array of coefficients that best represent the
received phase noise is returned. If no data is passed in, a fictitious random field
generated from TBM.m is used for estimation.
– Input: Pre-processed phase data, thetamn.
– Output: Estimated coefficient array, Cij.
A flow chart of the above described functions is presented in Figure A.2.
APPENDIX A. SIMULATOR DESIGN IN MATLAB 97
Compressed Data
Focused Data
ionoComp.m PGA.m
PDA.m
PDA Estimation PGA Estimation
Focused Data
Figure A.2: Compensation Functions
A.3 Other Functions
• phaseScintllationPlot.m
– Description: This function resamples the phase scintillation noise on a evenly
spaced grid in order to plot them. To plot phase noise experienced by different
targets at different locations, the phase noise is resampled on a rectangular grid.
– Input: Phase noise data from ionoSim, randomNoise
– Output: Resampled phse noise on a evenly spaced grid, randomNoiseResampled
• estimate3dBWidth.m
– Description: This file is to estimate the width of a sinc function given the desired
dB value. A typical 3dB width of a unit sinc function is approximated 0.886. This
functions will take one single argument, the dB value, and find the width of the
sinc function corresponding to the given dB value.
– Input: dB value in log, dB
– Output: sinc width, x
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