S C I PHASE SCINTILLATION N SPOTLIGHT SAR DATA...Chapter 2 reviews the basic concepts of radar imaging using linearly frequency modulated pulses and matched ﬁltering and the effect

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].


• 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


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


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.


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


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


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:


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.


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


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)


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 .


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


(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


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


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


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).


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


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)


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)


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


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,


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)


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)


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


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.


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


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


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


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.


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.


Range Index

Azi

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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


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

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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


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


−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


(b) Averaged Mainlobe Width

Figure 4.6: Average SAR PSF under Random Phase Noise


−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


(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





−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


(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





Range Index

Azi

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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.


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

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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


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.


( , )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


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)


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-


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-


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


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


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.


Azi

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h In

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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


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


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


Azi

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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


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.


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


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.


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)


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,


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.


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Figure 6.3: PGA Estimation Result of a Highly Correlated Phase Noise


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Figure 6.4: Residual Imaging Effect after PGA Compensation


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Figure 6.5: PGA Estimation Result of a Correlated Phase Noise


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Figure 6.7: PGA Estimation Result of a Highly Decorrelated Phase Noise


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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


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


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


(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


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.


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Figure 6.12: PDA Estimation Results


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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.


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(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


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)

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Figure 6.16: Phase Noise in Two part, Blurry-Indecent and Blurry-Dependent



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Figure 6.17: Imaging Effect of Blurry-Indecent and Blurry-Dependent Part



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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


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.


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.


(a) Full TerraSAR-X Scene over Antarctica

Range Index

Azi

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50 100 150

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(b) A Section of the Previous Figure

Figure 6.20: Original TerraSAR-X Data over Antarctica


Range Index

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(a) Original TSX Selection

Range Index

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(b) PGA Result

Range Index

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(c) PDA Result

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


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


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


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-


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.


– 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


– 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.


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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Documents

S C I PHASE SCINTILLATION N SPOTLIGHT SAR DATA...Chapter 2 reviews the basic concepts of radar imaging using linearly frequency modulated pulses and matched ﬁltering and the effect