l-band insar estimates of greenland ice sheet …l-band insar estimates of greenland ice sheet...

L-BAND INSAR ESTIMATES OF GREENLAND ICE SHEET

ACCUMULATION RATES

A DISSERTATION

SUBMITTED TO THE DEPARTMENT OF ELECTRICAL

ENGINEERING

AND THE COMMITTEE ON GRADUATE STUDIES

OF STANFORD UNIVERSITY

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

Albert C. Chen

December 2013

http://creativecommons.org/licenses/by-nc/3.0/us/

This dissertation is online at: http://purl.stanford.edu/mh091sq5213

© 2013 by Albert Conan Chen. All Rights Reserved.

Re-distributed by Stanford University under license with the author.

This work is licensed under a Creative Commons Attribution-Noncommercial 3.0 United States License.

ii



http://purl.stanford.edu/mh091sq5213

I certify that I have read this dissertation and that, in my opinion, it is fully adequatein scope and quality as a dissertation for the degree of Doctor of Philosophy.

Howard Zebker, Primary Adviser


Sigrid Close


Ivan Linscott

Approved for the Stanford University Committee on Graduate Studies.

Patricia J. Gumport, Vice Provost for Graduate Education

This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file inUniversity Archives.

iii

Abstract

The Greenland Ice Sheet contains nearly 3 million cubic kilometers of glacial ice. Were

the ice to completely melt, that would cause the sea level to rise about 7 meters. There

is thus considerable interest in monitoring the mass of ice in the Greenland Ice Sheet.

Each year, the ice sheet gains ice from snowfall and loses ice through iceberg calving

and other ablation mechanisms. Thus assessing the ice sheet’s mass balance (annual

net gain/loss of ice) requires accurate spatial mapping of accumulation rates (mean

annual snowfall); however this is currently based on sparse in-situ observations. In

this thesis, we examine how recent satellite radar remote sensing data can be used to

supplement in-situ accumulation rate estimates in the inner regions of the Greenland

Ice Sheet.

We present a method using interferometric synthetic aperture radar (InSAR) data

to obtain estimates of snow accumulation in Greenland. InSAR is a technique that

provides images of the Earth from radar data collected by a spacecraft. The complex

phase of InSAR images is often used to measure Earth surface topography and de-

formation. We show that the second-order phase statistics (coherence) is related to

subsurface structure, which, in the inner dry-snow zone of the Greenland ice sheet, is

related to accumulation rate.

We have implemented software to form and geocode InSAR images of Greenland

and correct for ionospheric inhomogeneity, which has limited the accuracy of longer-

wavelength measurements of the Earth’s polar regions. We developed a geophysical

iv

model to relate accumulation rate to Greenland firn (glacial ice) structure, and an

electromagnetic scattering model to relate firn structure to InSAR measurements. By

inverting the model we obtain estimates of Greenland ice sheet accumulation rates.

We show a comparison of our results with in-situ measurements over a 1, 400 km

strip spanning the entire dry-snow zone, and demonstrate that they follow the in-

situ measurements more accurately than state-of-the-art results derived from radar

amplitude measurements alone.

v

Acknowledgments

I wish to thank my advisor, Prof. Howard Zebker, for all his guidance, advice, help,

and support as I worked on the research in this thesis. Howard taught and showed

me so much about how to develop engineering software, how to think scientifically,

how to do good research, how to publish results, and how to be a good collaborator.

I am very grateful to have had the opportunity to be in his group. I also thank the

other thesis readers and members of my oral defense committee: Sigrid Close, Ivan

Linscott, Joe Kahn, and Tony Fraser-Smith. Earlier mentoring by Prof. Fawwaz

Ulaby, who first got me interested in SAR as an undergraduate at the University of

Michigan, is also appreciated.

I would like to thank all the colleagues at Stanford who helped me, gave me advice,

and offered their encouragement as I pursued my graduate studies. Lin Liu, who was

a post-doctoral scholar in Prof. Zebker’s group during the latter years of my Ph.D.

offered particularly valuable and insightful advice on Greenland and the cryosphere.

Piyush Agram spent much time patiently teaching me about InSAR processing and

answering my questions during the earlier years. Jingyi Chen was a valuable collabo-

rator as I worked on the portion of this thesis related to the ionosphere. I would also

like to thank the various graduate students with whom I’ve shared an office during

my time at Stanford: Ana Bertran Ortiz, Shadi Oveisgharan, Lauren Wye, Jaime

Lien, Qiuhua Lin, Will Woods, Hrefna Marin Gunnarsdottir, and Fraser Thomson.

Other members of Prof. Zebker’s group whose help is deeply appreciated are Cody

vi

Wortham and Jessica Reeves.

This work was supported in part by the Reed-Hodgson Stanford Graduate Fel-

lowship.

Many amazing classmates, friends, and roommates have made my time at Stanford

deeply memorable and meaningful, including Michael Chen, Andrew Spann, Eleanor

Lin, Jason Bau, Tim Yoo, Gerwin Hassink, Miranda Shen, Nii Okai Addy, Tayo

Oguntebi, Neeraj Sonalkar, and many others.

Finally, I thank my wife Yi-Ching, as well as my parents, my sister, and all my

uncles, aunts, and cousins for their support and encouragement.

“The same thrill, the same awe and mystery, come again and again

when we look at any problem deeply enough. . . . we turn over each new

stone to find unimagined strangeness leading on to more wonderful

questions . . . ”

Richard P. Feynman, The Pleasure of Finding Things Out, Chap. 6.

vii

Contents

Abstract iv

Acknowledgments vi

1 Introduction 1

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.2 Objectives of this Study . . . . . . . . . . . . . . . . . . . . . 4

1.2 Previous Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.4 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 The Greenland Ice Sheet 9

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.1.1 Coordinate System . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2 Structure of the Greenland Ice Sheet . . . . . . . . . . . . . . . . . . 12

2.2.1 Surface Elevation and Ice Thickness . . . . . . . . . . . . . . . 12

2.2.2 Ice Facies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2.3 Dry-snow zone . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3 Mass Balance of the Greenland Ice Sheet . . . . . . . . . . . . . . . . 15

viii

2.3.1 Accumulation Rate . . . . . . . . . . . . . . . . . . . . . . . . 19

3 InSAR Remote Sensing 22

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.2 Signal Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.2.1 SAR Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.2.2 Interferometric SAR (InSAR) . . . . . . . . . . . . . . . . . . 28

3.2.3 InSAR Coherence . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.2.4 Geocoding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.3 Electromagnetic Scattering Models . . . . . . . . . . . . . . . . . . . 38

3.3.1 Backscatter Power . . . . . . . . . . . . . . . . . . . . . . . . 39

3.3.2 InSAR Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.3.3 InSAR Coherence . . . . . . . . . . . . . . . . . . . . . . . . . 42

4 Spatially Varying Ionosphere 46

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.2 Theory: Ionospheric Effects in InSAR . . . . . . . . . . . . . . . . . . 49

4.2.1 Microwave Propagation in the Ionosphere . . . . . . . . . . . . 49

4.2.2 Ionospheric phase advance . . . . . . . . . . . . . . . . . . . . 53

4.2.3 Azimuth Offsets due to Ionospheric Inhomogeneities . . . . . . 54

4.3 Accurate Coregistration Method . . . . . . . . . . . . . . . . . . . . . 56

4.3.1 Accurate Coregistration Algorithm . . . . . . . . . . . . . . . 58

4.3.2 Ionospheric Phase Estimate . . . . . . . . . . . . . . . . . . . 61

4.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.4.1 Greenland test scene results . . . . . . . . . . . . . . . . . . . 62

4.4.2 Iceland processing results . . . . . . . . . . . . . . . . . . . . . 68

4.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.5.1 Quantifying the Improvement in Coherence . . . . . . . . . . . 74

ix

4.5.2 Limitations and Conclusions . . . . . . . . . . . . . . . . . . . 77

5 Dry-Snow Zone Firn Modeling 80

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5.2 Geophysical firn model . . . . . . . . . . . . . . . . . . . . . . . . . . 84

5.2.1 Firn Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

5.2.2 Firn grain radius . . . . . . . . . . . . . . . . . . . . . . . . . 86

5.3 Electromagnetic scattering model . . . . . . . . . . . . . . . . . . . . 88

5.3.1 Scattering model . . . . . . . . . . . . . . . . . . . . . . . . . 90

5.3.2 Numerical results and discussion . . . . . . . . . . . . . . . . . 92

6 Accumulation Rate Case Study Results 98

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

6.2 Case Study: Radar Mosaic . . . . . . . . . . . . . . . . . . . . . . . . 99

6.3 Regularized Inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

6.3.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . 102

6.3.2 Uncertainty Analysis . . . . . . . . . . . . . . . . . . . . . . . 102

6.4 Case Study: Dry-Snow Zone Transect . . . . . . . . . . . . . . . . . . 104

6.4.1 Data and InSAR processing . . . . . . . . . . . . . . . . . . . 104

6.4.2 Forward Model Results . . . . . . . . . . . . . . . . . . . . . . 107

6.4.3 Inversion Results . . . . . . . . . . . . . . . . . . . . . . . . . 110

7 Conclusions 113

7.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

7.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

Bibliography 117

x

List of Tables

1.1 Contributions to current global sea-level rise. . . . . . . . . . . . . . . 3

1.2 Letter designations for common microwave bands. . . . . . . . . . . . 4

2.1 Greenland mass budget . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.1 ALOS PALSAR parameters . . . . . . . . . . . . . . . . . . . . . . . 25

6.1 Transect PALSAR data . . . . . . . . . . . . . . . . . . . . . . . . . . 104

xi

List of Figures

1.1 Greenland and the Greenland Ice Sheet . . . . . . . . . . . . . . . . . 2

1.2 Global sea level time series . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1 Ice sheet dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2 Elevation, bedrock elevation, and ice thickness . . . . . . . . . . . . . 13

2.3 Ice sheet morphology . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.4 Dry-snow zone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.5 Photos of Summit Camp, Greenland . . . . . . . . . . . . . . . . . . 17

2.6 Mass balance of the Greenland Ice Sheet . . . . . . . . . . . . . . . . 18

2.7 Greenland accumulation rate map . . . . . . . . . . . . . . . . . . . . 21

3.1 SAR geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.2 SAR data and focused SAR image . . . . . . . . . . . . . . . . . . . . 29

3.3 Removing topographic phase . . . . . . . . . . . . . . . . . . . . . . . 31

3.4 InSAR imaging geometry . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.5 Normalized coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.6 Coherence estimator . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.7 Geocoded interferogram. . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.8 Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.1 ALOS orbit and Earth’s ionosphere . . . . . . . . . . . . . . . . . . . 48

xii

4.2 InSAR coregistration coordinate systems and geometry . . . . . . . . 59

4.3 Test scenes for our method. . . . . . . . . . . . . . . . . . . . . . . . 63

4.4 Greenland interferograms . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.5 Greenland coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.6 Azimuth and range offsets for Greenland scene . . . . . . . . . . . . . 67

4.7 Greenland coherence histogram . . . . . . . . . . . . . . . . . . . . . 68

4.8 Estimated ionospheric phase . . . . . . . . . . . . . . . . . . . . . . . 69

4.9 Estimated ionospheric phase in Greenland scene . . . . . . . . . . . . 70

4.10 Iceland interferograms . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.11 Iceland coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.12 Azimuth and range offsets for Iceland scene. . . . . . . . . . . . . . . 73

4.13 Iceland spline results . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.14 Coherence vs. misregistration . . . . . . . . . . . . . . . . . . . . . . 76

4.15 Greenland coherence improvement . . . . . . . . . . . . . . . . . . . . 77

5.1 Inverse problem formulation . . . . . . . . . . . . . . . . . . . . . . . 82

5.2 B-26 site map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5.3 Firn density at B-26 . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

5.4 Grain radius at B-26 . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5.5 Volumetric radar cross-section . . . . . . . . . . . . . . . . . . . . . . 93

5.6 Modeled volume correlation . . . . . . . . . . . . . . . . . . . . . . . 94

5.7 InSAR data vs. accumulation rate . . . . . . . . . . . . . . . . . . . . 95

5.8 Objective function from simulated data . . . . . . . . . . . . . . . . . 96

6.1 Greenland SAR mosaic images . . . . . . . . . . . . . . . . . . . . . . 100

6.2 Scatter plots of predicted versus measured data . . . . . . . . . . . . 101

6.3 Transect case study location . . . . . . . . . . . . . . . . . . . . . . . 105

6.4 InSAR perpendicular baseline . . . . . . . . . . . . . . . . . . . . . . 106

xiii

6.5 Transect interferogram . . . . . . . . . . . . . . . . . . . . . . . . . . 106

6.6 Summit, Greenland . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

6.7 Predicted and measured σ0 for transect. . . . . . . . . . . . . . . . . 108

6.8 Predicted and measured coherence for transect. . . . . . . . . . . . . 108

6.9 Penetration depths for transect . . . . . . . . . . . . . . . . . . . . . 109

6.10 Accumulation rate estimated from σ0 data . . . . . . . . . . . . . . . 110

6.11 Accumulation rate estimated from ρvol . . . . . . . . . . . . . . . . . 111

xiv

List of Important Symbols

Atotal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Total ice sheet accumulation from snowfall

A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Specific accumulation rate

Bcrit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Critical baseline

B⊥ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Perpendicular baseline

~B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Earth’s magnetic field

c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Speed of light in free space

~d. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Data vector for inverse problem

e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Charge of an electron

E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Activation energy of grain-growth process

Ex, Ey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Electric field phasors

fB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cyclotron frequency (gyrofrequency)

fc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Transmitted pulse carrier frequency

fp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Plasma frequency

g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ground reflectivity

h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Elevation relative to WGS-84 ellipsoid

hraw . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Raw SAR data due to a point on the ground

i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Imaginary unit,√−1

Kr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . FM rate of transmitted pulse

xv

L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Effective number of looks in coherence estimation

Ls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Synthetic aperture length

~m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Model parameter vector for inverse problem

me . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mass of an electron

N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Number of pixels in coherence estimation window

n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Thermal noise; also index of refraction

ne . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Electron concentration in ionosphere

~p . . . . . . . . . . . . . . . . . . Geophysical and imaging parameters vector for inverse problem

rg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Average grain radius

R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Range (distance from radar to target)

R0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Range at closest approach

si . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Complex SLC pixel value for ith SAR image

sij . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Interferogram formed from SLC’s si and sj

sraw . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Raw SAR data

t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Time

T . . . . . . . . . . . . . . . . . . . . . . Transmitted pulse duration; also mean annual temperature

TEC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Total electron content

x. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Azimuth distance

xp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Position coordinate, polar-stereographic projection

yp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Position coordinate, polar-stereographic projection

γ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mass density of firn

δR . . . . . . . . . . . . . . . . . . . . . . . Line-of-sight deformation between two SAR acquisitions

δR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Range resolution

δx . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Azimuth resolution

δθr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Incidence angle difference for refracted wave

xvi

∆R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Range offset between two SAR acquisitions

∆x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Azimuth offset between two SAR acquisitions

∆R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Range pixel spacing

∆x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Azimuth pixel spacing

ε . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Relative dielectric constant

ε0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Permittivity of free space

κa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Absorption coefficient

κe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Extinction coefficient

κs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Scattering coefficient

σ0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Normalized radar cross-section

σv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Volumetric radar cross-section

θ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Incidence angle

θr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Incidence angle for refracted wave

λ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Wavelength

ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Plasma particle collision rate

ρ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . InSAR coherence

σ0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Radar backscatter brightness

τ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Propagation delay

φ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . InSAR phase

Φ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Latitude

Ψ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Wave polarization

xvii

Chapter 1

Introduction

1.1 Background

1.1.1 Motivation

The cryosphere is the part of Earth’s surface containing frozen water. Some com-

ponents of the cryosphere vary seasonally, such as snow-covered land and sea ice.

Other components are more permanent, such as permafrost, glaciers, and ice sheets

[Barry and Gan, 2011]. As part of the global climate system, the cryosphere plays

numerous roles. For example, the relatively high albedo of ice means that it reflects

solar radiation, thus reducing the greenhouse effect, and lowering temperature. Of

consequence to coastal regions, any melting of glaciers and ice sheets directly causes

a rise in global sea level.

This dissertation focuses on an important component of the cryosphere called the

Greenland Ice Sheet. This ice sheet covers about 85% of Greenland, a large island in

the arctic (see Figure 1.1). The Greenland Ice Sheet consists of about 2.9× 106 km3

of ice and would cause about 7.1 m of sea level rise if it completely melted [Marshall,

2011].

1

CHAPTER 1. INTRODUCTION 2

(a) (b)

Figure 1.1: The map in (a) shows Greenland (indicated by the orange arrow) andits location in the northern hemisphere, based on [Dahl-Jensen et al., 2009]. Theapproximate extent of the Greenland Ice Sheet is shown as the yellow region in (b)[Bales et al., 2009].


Figure 1.2: Relative global mean sea level. Uncertainty is indicated by the light-blueinterval. Data is from [Jevrejeva et al., 2008].

Source Sea level rise (mm/yr) Percentage of totalMelting of Greenland Ice Sheet 0.2 6Melting of Antarctic Ice Sheets 0.2 6Melting of other Glaciers 0.8 26Ocean thermal expansion 1.6 52Other 0.3 10Total 3.1 100

Table 1.1: Contributions to current global sea-level rise. From [United Nations En-vironment Programme, 2007].

A time series of relative global mean sea level for the past 300 years shows a clear

upward trend (Figure 1.2). At present, the global sea-level is rising at 3.1 mm/yr

[Jevrejeva et al., 2008]. Several contributions to present sea level rise are listed in

Table 1.1. About 6% of current sea level rise is caused by melting of the Greenland

Ice Sheet [Lemke et al., 2007; United Nations Environment Programme, 2007].

Field campaigns to obtain in-situ data on the Greenland Ice Sheet are difficult and

expensive because of the low temperatures (averaging about −20o C), the darkness


Designation Wavelength rangeL-band 15 to 30 cmC-band 3.75 to 7.5 cmX-band 2.5 to 3.75 cm

Table 1.2: Letter designations for common microwave bands.

(especially in winter), and the ice sheet’s size and remote location. As a result, in-situ

data on the Greenland Ice Sheet are currently sparse, particularly in the interior of the

ice sheet. Remote sensing, which involves measuring the ice sheet using spaceborne

and airborne sensors, therefore plays an important role in filling in our knowledge of

the spatial and temporal variation of many key ice sheet parameters.

1.1.2 Objectives of this Study

In this study we aim to analyze properties of the Greenland Ice Sheet using interfer-

ometric synthetic aperture radar (InSAR). Spaceborne InSAR systems, such as the

one used here, carry a satellite radar instrument, which transmits pulses of electro-

magnetic radiation as the spacecraft orbits the Earth. These pulses hit the surface

of the Earth, and are reflected back to the sensor where their echoes are recorded

by the radar instrument. These recorded echoes form the raw data for this study.

In this study we use an InSAR instrument that transmits pulses with a wavelength

λ = 23.61 cm. In microwave engineering, this spectral location is commonly referred

to as “L-band.” Some common designations for various microwave wavelength ranges

referred to in this thesis are shown in Table 1.2.

Here we focus on the dry-snow zone of the Greenland Ice Sheet. In this region,

the ambient temperature stays below 0o C all year long and little to no melting oc-

curs. Therefore, yearly snowfall in this region forms successive layers of glacial ice.

Examining these layers tells us about the average annual snowfall; this is analogous to


examining tree rings to find the tree’s annual growth rate.1 Annual snowfall roughly

represents the amount of ice added to the ice sheet each year through precipitation

(although some of the snow that falls is lost by sublimation). This is called the accu-

mulation rate. The difference between total accumulation and total ice lost through

various loss processes (such as iceberg calving, melt from the periphery of the ice

sheet, etc.) represents the net amount of ice lost or gained from the ice sheet.

In summary, the objectives of this study are to develop signal processing methods

to form L-band InSAR images of the Greenland Ice Sheet, to formulate the esti-

mation of accumulation rate from InSAR data as a mathematical inverse problem,

and to compare the resulting remote sensing accumulation rate estimates with in-situ

estimates in a suitable study area.

1.2 Previous Work

One of the earliest studies of accumulation on the Greenland Ice Sheet was done

by Carl Benson, who conducted extensive field work from 1952 to 1955 [Benson,

1960]. Benson also approximately delineated the extent of the dry-snow zone. Earlier

radar remote sensing studies of Greenland Ice Sheet accumulation used the European

Research Satellites (ERS), a set of two satellites equipped with C-band SAR instru-

ments. Using SAR radar backscatter data from ERS, Forster estimated accumulation

rates along a short transect near Summit Station [Forster et al., 1999]. Later, Munk

used C-band SAR radar backscatter data from a large mosaic of C-band SAR images

acquired by ERS to study accumulation rates in the dry-snow zone [Munk et al.,

2003]. In 2001, Hoen studied accumulation rate using C-band InSAR data from ERS

1A rare melt event occurred in 2012 in which the majority of the ice sheet experienced melt fora couple days in the summer. This occurred after our data was acquired, so it does not affect thework in this thesis. We have included some relevant information and references in Chapter 7 forreaders who are interested.


[Hoen, 2001]. Later Oveisgharan did a more detailed C-band InSAR study [Oveis-

gharan, 2007], and more recently Linow studied accumulation rates in Greenland and

Antarctica using both C-band and X-band SAR radar backscatter and scatterometry

data [Linow, 2011].

In this study, we use data from the Advanced Land Observing Satellite (ALOS),

launched by the Japan Aerospace Exploration Agency (JAXA) in 2006. It carried an

instrument called the “phased array type L-band synthetic aperture radar” (PAL-

SAR) which acquired the data used here. The carrier wavelength for PALSAR,

λ = 23.61 cm, is about five times larger than the wavelengths used in previous C-band

studies. Because of the longer wavelength, radar pulses transmitted by PALSAR pen-

etrate farther beneath the surface of the Greenland Ice Sheet than pulses transmitted

by C-band instruments. Since most temporal changes in the ice occur at the surface

driven by wind, L-band InSAR studies using longer temporal spacings between pairs

of SAR images are feasible, which is favorable from a satellite design perspective.

A potential disadvantage for L-band SAR instruments is that the ionosphere affects

electromagnetic wave propagation of L-band signals much more than do C-band sig-

nals.2 We will show, however, that in this case the effects of the ionosphere need not

prevent us from using ALOS data to study accumulation rates of the Greenland Ice

Sheet. Compensating for ionospheric artifacts in the InSAR analysis is one major

contribution from this work.

1.3 Contributions

The main contributions of this thesis are as follows:

1. We implemented software to form geocoded InSAR images of long transects of

2On the other hand, attenuation due to clouds and precipitation is more of a problem at higherfrequencies. This can sometimes be observed at C-band and is usually more noticeable at X-band[Marzano et al., 2010; Moore et al., 1997].


the Greenland Ice Sheet by stitching together frames processed using a motion-

compensation InSAR processor. We geolocate pixels in a polar stereogrpahic

projection to avoid geometric distortion at high latitudes.

2. We show that ionospheric propagation effects can interfere with accurate mea-

surements of InSAR phase and coherence in L-band studies of the Greenland

Ice Sheet. We develop a signal processing technique for compensating for these

effects.

3. We demonstrate that a volume-scatter model can be used to explain ALOS L-

band InSAR measurements of Greenland. We derive the first spaceborne InSAR

coherence measurements and estimates of accumulation rate from L-band SAR

data.

4. We present an error analysis for the measurement of accumulation rate from

InSAR coherence data. We also show that estimates of accumulation rate based

on L-band InSAR coherence measurements perform better than estimates based

solely on radar backscatter brightness.

5. We implemented a large-scale comparison between in-situ accumulation rate

measurements and L-band InSAR measurements. The comparison is done along

a transect that traverses the entire dry-snow zone of the Greenland Ice Sheet.

1.4 Overview

The remainder of this thesis is organized as follows:

We begin in Chapter 2 by presenting some background on the geophysics of the

Greenland Ice Sheet and explaining our problem formulation. We define accumula-

tion rate, which we aim to measure in this work, and explain how it contributes to


the overall Greenland Ice Sheet mass balance. We also show in-situ accumulation

rate data, which we use as “ground truth” for comparison. We explain how the mor-

phological structure of the Greenland Ice Sheet means that accumulation rate in a

region called the dry-snow zone can be studied using radar remote sensing data.

In Chapter 3 we explain how our radar instrument works. We overview the princi-

ples of synthetic aperture radar (SAR) imaging and explain how the phase difference

between pairs of SAR images can be calculated to form interferometric SAR (InSAR)

images. Next we define InSAR coherence as a normalized second-order statistic of In-

SAR pixel values, and show how this can be used for measurements of the subsurface

structure of the Greenland Ice Sheet.

In Chapter 4 we address the problem of imaging the Greenland Ice Sheet using

an L-band microwave radar instrument through a spatially-varying ionosphere. We

explain the physics of electromagnetic wave propagation through the ionosphere and

show how this affects our InSAR images. We show evidence of ionospheric artifacts

in our data and show how these can be alleviated using signal processing.

In Chapter 5 we describe the relationship between accumulation rate in the dry-

snow zone of the Greenland Ice Sheet and InSAR observations. We use a geophysical

model to relate accumulation rate to firn structure, and an electromagnetic scattering

model to relate firn structure to InSAR data. This forms the forward model relating

the geophysical parameter of interest (accumulation rate) and the observed data.

Inversion of the forward model presented in Chapter 5 is the topic of Chapter 6.

We explain how we formulate the inversion as a mathematical inverse problem and

how we introduce regularization to account for measurement uncertainty. We then

present our inversion results, uncertainty analysis, and comparison with in-situ data.

Finally, in Chapter 7 we summarize our conclusions and contributions and give

suggestions for future work on this topic.

Chapter 2

The Greenland Ice Sheet

2.1 Introduction

A glacier is a body of ice formed by accumulation of snowfall over many years.

Small mountain glaciers can be found at all latitudes, while continental-scale glaciers,

called ice sheets, are currently located only in Greenland and Antarctica. Thorough

introductions to the physical modeling of glaciers are given in [Cuffey and Paterson,

2010; Hooke, 2005]. This chapter presents geophysical concepts and datasets that we

have used in our study of accumulation rates on the Greenland Ice Sheet.

The Greenland Ice Sheet is a very dynamic system. Figure 2.1 shows a concep-

tual cross-section of the Greenland Ice Sheet that illustrates these ice sheet dynamics.

Each year, snow falls as precipitation and accumulates on the upper surface of the

Greenland Ice Sheet. Freshly fallen snow consists of snow grains separated by inter-

connected air pockets, with a density usually between 300 kg/m3 and 400 kg/m3. As

it is buried underneath successive years’ snowfall, the snow gradually transforms into

glacier ice, which consists of ice grains and isolated air bubbles, with a density of

around 850 kg/m3. Snow in the intermediate stages of this transformation is known

as firn.

9

CHAPTER 2. THE GREENLAND ICE SHEET 10

Figure 2.1: Conceptual cross-section of the Greenland Ice Sheet illustrating ice sheetdynamics.


Over thousands of years, individual grains of ice travel downwards and then hori-

zontally outwards toward the periphery of the ice sheet. Ice may eventually leave the

ice sheet by melting and flowing into the ocean or by becoming part of an iceberg

that breaks off the ice sheet and falls into the ocean. Note that in this study, we ne-

glect the outward component of the movement, which is relatively small in the inner

dry-snow zone.

2.1.1 Coordinate System

Because lines of longitude converge at the north pole, maps of Greenland drawn in

the latitude-longitude coordinate system have significant geometric distortion. For

example, one second of latitude corresponds to a north-south distance of about 31

meters throughout Greenland, whereas a second of longitude corresponds to an east-

west distance of about 15.4 meters at the southern tip of Greenland and about 3.76

meters at the northern coast. Thus, in the latitude-longitude coordinate system,

the Cartesian distance formula should not be used for computing distances on the

Greenland Ice Sheet.

We instead project all points into a north polar stereographic projection with

center longitude −45 E and true scale at 70 N [Snyder, 1987]. We denote projected

position coordinates by xp and yp. In the xp, yp-plane defined by this projection, the

north pole is at the origin, and the −45 E meridian lies on the y-axis. We calculate

distances in this coordinate system using the Cartesian distance formula. Within the

Greenland Ice Sheet, the error introduced by this method ranges from zero (for points

at 70 N) to 4%. Henceforth we assume that this distortion is negligible.


2.2 Structure of the Greenland Ice Sheet

2.2.1 Surface Elevation and Ice Thickness

The maps in Figure 2.2 show the geometry of the Greenland Ice Sheet. Figure 2.2a is

a digital elevation model (DEM) showing the elevation of the upper ice sheet surface

[Howat et al., 2012].1 The map was made using a combination of laser altimetry and

stereoscopic optical imaging data.

Figure 2.2b shows the elevation of the bedrock underneath the ice sheet [Bamber

et al., 2013]. This map was made by combining data from several ice-penetrating

radar instruments using carrier frequencies ranging from 2.5 MHz to 200 MHz. The

difference between the surface and bedrock elevations is the thickness of the ice sheet

(Figure 2.2c). These plots show that the most of the Greenland Ice Sheet is over

1, 000 m thick, and the maximum thickness is over 3, 200 m. The L-band microwave

signals used in this study penetrate at most hundreds of meters into the ice; therefore

we will treat the ice sheet as an infinite half-space.

2.2.2 Ice Facies

The Greenland Ice Sheet can be divided into several zones called facies based on

variations in the structure of the ice, as depicted in Figure 2.3. The coldest, inner

region is called the dry-snow zone. In this region, the temperature normally stays

below freezing all year round, the surface of the ice is relatively flat, and the subsurface

structure of the ice is determined predominantly by layers formed from successive

years of snowfall. We quantify the boundary of this region below, since it is the

study area for this thesis. Closer to the periphery of the ice sheet is the percolation

zone, in which melting on the surface occurs during the warmer summer months.

1All elevations in this thesis are referenced to the WGS-84 ellipsoid [National Imagery and Map-ping Agency, 2000].


(a) Digital elevation model

(b) Bedrock elevation (c) Ice thickness

Figure 2.2: Digital elevation model (DEM) (a), bedrock elevation (b), and ice thick-ness (c) of the Greenland Ice Sheet.


Figure 2.3: Conceptual illustration of the Greenland Ice Sheet illustrating the relativelocation and structure of the dry-snow and percolation zones.

Here, meltwater can percolate down into the ice sheet and then refreeze, forming ice

pipes and ice lenses of various orientations which result in increased radar backscatter

relative to the dry-snow zone. Finally, near the coast we find the wet-snow zone and

ice zone, in which extensive melting and refreezing during the summer months results

in large continuous masses of ice.

2.2.3 Dry-snow zone

As part of the NASA Program for Arctic Regional Climate Assessment (PARCA), Ab-

dalati et.al. collected passive microwave brightness temperature data on the Green-

land Ice Sheet. They used these data to estimate the extent of snow melt on the

Greenland Ice Sheet [Abdalati and Steffen, 1997]. These estimates were made every

other day from 1979 to 1987 and every day starting from August, 1987. Using their

estimates, we define the dry-snow zone as the set of all points in the Greenland Ice

Sheet for which the total number of estimated melt days from 1979 to 2007 was zero.

The outline of the dry-snow zone resulting from this definition is shown in Figure 2.4.

From comparison, we also show Benson’s estimate of the extent of the dry-snow zone


based on his earlier in-situ observations [Benson, 1960]. The relatively good agree-

ment between the results shows that although there may be some uncertainty in the

exact location of the boundary of the dry-snow zone, we can be fairly confident of its

general shape and extent.

To assist in visualizing the dry-snow zone, we have included two optical pho-

tographs in Figure 2.5. The photos were taken at Summit Station, located at the

highest point of the Greenland Ice Sheet at 72.6 N and −38.4 E. Figure 2.5a shows

a panorama illustrating the relatively flat and featureless surface of the ice sheet

(by David Noone, http://climate.colorado.edu). This explains the lack of easily-

distinguishable features in radar images of the ice sheet. In Figure 2.5b, the investi-

gator has dug a pit about two meters into the surface of the ice and is standing at the

bottom of the pit. This is called a snow pit. The photo shows one of the side walls

of the pit, illustrating the layers of firn resulting from past year’s snowfall (from Po-

larTREC, http://www.polartrec.com). We will discuss modeling of electromagnetic

waves from the firn in Chapter 5.

2.3 Mass Balance of the Greenland Ice Sheet

We have seen that the Greenland Ice Sheet gains mass each year as snow falls on

it as precipitation. It also loses mass each year as some of the ice melts and some

is discharged into the ocean in the form of icebergs. The net change in mass of the

ice sheet is called the mass balance. A negative mass balance means that the ice

sheet is experiencing a net loss of ice each year and therefore contributing to sea-level

rise. Mass balance for the Greenland Ice Sheet is usually given in gigatons per year

(Gt/yr), where one gigaton is 1012 kilograms.

There are two main approaches to measuring the mass balance of the Greenland

Ice Sheet. The first approach is to measure changes in the Earth’s gravitational field


Figure 2.4: Extent of the dry-snow zone. The coloring inside the Greenland Ice Sheetindicates the estimated number of days of melt from 1979 to 2007 based on melt datafrom [Abdalati and Steffen, 1997]. The red outline is Benson’s estimate of the extentof the dry-snow zone [Benson, 1960], while the green outline is our estimate based onthe PARCA melt data.


(a) Summit Camp, Greenland

(b) Snow pit at Summit Camp

Figure 2.5: Photos showing the surface of the Greenland Ice Sheet as seen fromSummit Station (a), and the inside of a snow pit near Summit Station (b).


Figure 2.6: Mass balance of the Greenland Ice Sheet, based on data from [Rignotet al., 2008].

resulting from changes in the total mass of the ice sheet. This has been accomplished,

for example, in the Gravity Recovery and Climate Experiment (GRACE) [Velicogna,

2009]. The second approach is to measure the various mass gains and losses and sum

them to determine the mass balance. In addition to providing independent validation

of results from the gravity method, this mass balance method allows us to understand

the processes contributing to glacier mass balance in greater detail. In this method,

mass balance is the sum of accumulation and ablation. Accumulation is the total mass

gained through all processes that add mass to the ice sheet, whereas ablation is the

total mass lost due to all processes that remove mass from the ice sheet. An example

of results from this method is shown in Figure 2.6 [Rignot et al., 2008]. This thesis

aims to contribute to our understanding of the accumulation term in this method.

The largest contributor to accumulation is snowfall. Other processes that con-

tribute to accumulation are deposition (ice formed from water vapor at or near the

ice surface), basal freeze-on (liquid water freezing onto the bottom of the ice sheet),

and wind blowing snow from adjacent land onto the ice sheet. In this study, we are


Process Gt/yrNet surface accumulation +228± 34Ablation from ice discharge −495± 29Total −267± 38

Table 2.1: Approximate 2007 mass budget, based on data from [Rignot et al., 2008].

primarily concerned with snowfall. Ablation processes include melting, iceberg calv-

ing, and sublimation. Current estimates of accumulation, ablation, and mass balance,

as well as the relevant uncertainties, are shown in Table 2.1.

2.3.1 Accumulation Rate

The total accumulation per year from snowfall Atotal can be expressed as an integral

over the surface of the ice sheet:

Atotal =

∫∫ice sheet

A(xp, yp)dxpdyp (2.1)

The integrand is called specific accumulation rate, denoted A. It is the mean annual

snowfall per unit area and varies as a function of location on the ice sheet. Following

the convention in the literature, we will drop the word “specific” and simply refer to

this as “accumulation rate” except when required for clarity.

Accumulation rate is measured in grams of ice per year per square centimeter

(g/yr/cm2). Because one gram of ice is equivalent to one gram of liquid water, we

can write

1g ice

yr · cm2= 1

g ice

yr · cm2· 1 cm3water

1 g ice= 1 cm/yr w.e (2.2)

where “w.e.” stands for “water-equivalent.” Accumulation rates within the dry-snow

zone can be up to about 50 cm/yr w.e. The goal of this study is to obtain estimates


of A from InSAR data.

To compare our results with results derived from in-situ measurements, we will

use the results reported in [Bales et al., 2001]. In this study, Bales used data from

313 ice cores, including 168 within the dry-snow zone. They then used a geostastical

interpolation method called kriging to obtain an interpolated map of accumulation

rate, shown in Figure 2.7. The map shows a general gradient in the north-east direc-

tion and higher accumulation rates in the south than in the north. Geophysically, this

is due to the general atmospheric circulation pattern in the area, which carries water

vapor from the North Atlantic and Baffin Bay over Greenland, with precipitation

decreasing as air masses descend towards the north [Ohmura and Reeh, 1991].


Figure 2.7: In-situ measurements of accumulation rate [Bales et al., 2001].

Chapter 3

InSAR Remote Sensing

3.1 Introduction

In the previous chapter, we delineated the dry-snow zone and defined accumulation

rate. We now turn to a discussion of the InSAR instrument that we use to estimate

accumulation rate in the dry-snow zone. As we mentioned in Section 1.2, the data

for this study were acquired by the ALOS PALSAR instrument. Detailed technical

information on the instrument and its calibration are given in [Rosenqvist et al., 2007;

Shimada et al., 2009].

PALSAR is a SAR instrument, which means that the raw data that it acquires

can be processed using a SAR signal processing algorithm to form high-resolution

images of the Earth. In InSAR, we also use the phase difference between pairs of

SAR images, as well as the second-order statistics of this phase.

In Section 3.2 we explain the signal processing steps involved in generating the

various SAR image products from raw SAR data, and in Section 3.3 we explain the

physical interpretation of image pixel values in terms of electromagnetic scattering

phenomena. We focus on details specific to processing our PALSAR data and physical

22

CHAPTER 3. INSAR REMOTE SENSING 23

models that are needed for later chapters. Additional background and detailed dis-

cussions of SAR signal processing are given in [Cumming and Wong, 2005; Curlander

and McDonough, 1991]. Also, thorough reviews of InSAR are given in [Burgmann

et al., 2000; Hanssen, 2001; Rosen et al., 2000].

3.2 Signal Processing

3.2.1 SAR Imaging

Geometry

The geometry of a spaceborne SAR imaging system is shown in Figure 3.1. The space-

craft (the ALOS satellite in our case) travels along an orbit path about 700 km above

the Earth’s surface. The direction of the spacecraft’s motion is called the azimuth

direction, denoted x. The SAR instrument transmits pulses of electromagnetic radia-

tion which hit a portion of the Earth’s surface called the antenna beam footprint. As

the spacecraft orbits, it records the echoes (backscattered energy) from these pulses

as raw data. A SAR processor is usually a piece of digital signal processing software

that combines these raw data to form a high-resolution image of the swath of the

Earth’s surface traversed by the antenna beam footprint.

The distance from the spacecraft to a point on the ground is called the slant

range, denoted R. For simplicity, we will also refer to slant range as simply range. The

direction on the ground orthogonal to the azimuth direction is called the ground range

direction; distance along this direction is denoted Rg. Slant range can be mapped

to ground range using the relationship between slant range and ground range pixel

spacing

∆Rg =∆R

sin θ(3.1)

Here θ is called the incidence angle, and is defined as the angle between the normal


Figure 3.1: SAR geometry.

to the ground surface and the pulse propagation direction. SAR images presented as

functions of range and azimuth are said to be in radar coordinates. When each pixel

in a SAR image is put into a latitude-longitude or polar stereographic coordinate

system, the image is said to be geocoded.

Range and Azimuth Compression

As it moves along its orbit path, the SAR instrument transmits electromagnetic pulses

of the form

p(t) = rect(t/T ) cos(2πfct+ πKrt

2)

(3.2)

This is called a linear FM (or chirp) pulse with carrier frequency fc, FM rate Kr,

and duration T . The carrier frequency is related to the wavelength by fc = c/λ (c

denotes the speed of light in free space). For the ALOS data used in this study, the

values of fc, Kr, and T are given in Table 3.1. The approximate bandwidth of this


Parameter ValueWavelength 23.61 cmCarrier frequency 1.27 GHzLinear FM rate −8.7158× 1011Hz/secPulse duration 16.0 µsRange resolution 10.7 mAzimuth resolution 4.5 mRange to first pixel 758 kmIncidence angle 21.5

Orbit repeat time 46 days

Table 3.1: ALOS PALSAR system parameters. Values are given for the polarimetrymode in which much Greenland data was acquired during International Polar Year2007-2008.

type of linear FM pulse is given by BW = |KrT | (see, for example, [Cumming and

Wong, 2005], Sec. 3.2.2). The bandwidth for our data is about 14 MHz, resulting

in a fractional bandwith of about 1.1%. Since this can be considered a narrow-band

signal, we will consider the transmitted pulse to be a plane wave of frequency fc for

the purposes of electromagnetic scattering modeling.

The SAR instrument receives the echoes of the transmitted pulses and mixes them

to baseband using quadrature demodulation, resulting in a complex-valued signal,

which is sampled and quantized using 5-bit quantization. The digitized echo of each

pulse then forms a single line of the raw SAR data. Successive lines correspond to

successive spacecraft transmit locations in the azimuth direction. Since propagation

delay τ and range R are related by R = cτ/2, the data samples in each line correspond

to increasing range. We can thus think of raw SAR data as a 2-D function of azimuth

and range, which we denote sraw(x,R). An example of raw SAR data is shown in

Figure 3.2a. Although the data are complex-valued, the figure shows the magnitudes

of the samples as a grayscale image.

Consider a point P on the ground within the imaged swath at location (x0, R0). As

the radar passes by that point, it falls within the beam footprint for many successive


pulses, because the radar travels only a small fraction of the width of the beam

footprint between pulses. The distance traveled by the radar during which the point

is in the beam footprint is called the synthetic aperture length, Ls. The range to

the point, RP (x;x0, R0), is a function of both the radar’s azimuth position x and the

point’s position (x0, R0). The raw SAR data due to a point of unit amplitude located

at (x0, R0) is then a function of range and azimuth given by

hraw(x,R;x0, R0) = rect

(R−RP (x;x0, R0)

cT/2

)rect

(x− x0

Ls

)× exp

(−i4π

λRP (x;x0, R0)

)× exp

(iπKr

(2

c(R−RP (x;x0, R0))

)2)

(3.3)

If no further signal processing were applied, this would be the impulse response of the

SAR imaging system. The raw SAR data due to a general terrain with reflectivity

g(x,R) is then given by the 2-D superposition integral

sraw(x,R) =

∫ ∫hraw(x,R;x′, R′) · g(x′, R′)dx′dR′ + n(x,R) (3.4)

where n(x,R) represents thermal noise. In many cases,

hraw(x,R;x0, R0) ≈ hraw(x− x0, R−R0; 0, Rc) (3.5)

where Rc is the distance from the orbit path to the center of the imaged swath. Then

we can re-write Equation 3.4 as the convolution

sraw(x,R) = hraw(x,R; 0, Rc) ∗ g(x,R) + n(x,R) (3.6)

Thus SAR imaging is approximately, though not exactly, shift-invariant.


SAR processors use some form of matched filtering to sort the energy in raw SAR

data into range and azimuth bins and approximately recover g(x,R) from sraw(x,R).

Energy can be sorted into range bins by the two-way propagation time delay. In the

azimuth direction, energy in the echoes can be sorted into azimuth bins by Doppler

frequency, fdop, which is related to azimuth position by

fdop = − 2vx

λR0

(3.7)

Locations in front of the radar have positive fdop, while those behind the radar have

negative fdop. We will examine how this relationship is altered by a spatially varying

ionosphere in Chapter 4.

In this study, we use the SAR processor described in [Zebker et al., 2010]. The

resulting focused image is called a single-look complex (SLC) image, and denoted

s(x,R). Well-designed SAR processors restore (or almost exactly restore) shift-

invariance, so that the resulting image of g(x,R) can be written as the convolution

s(x,R) = W (x,R) ∗ g(x,R) (3.8)

where W (x,R) is the overall SAR impulse response, which is well modeled by

W (x,R) = sinc

(x

δx

)sinc

(R

δR

)(3.9)

The range resolution, δR, is given by

δR =c

2BW(3.10)


and the azimuth resolution, δx, is given by

δx =La2

(3.11)

where La is the physical length of the antenna. An example of the result of applying

the SAR signal processing algorithm to raw SAR data is shown in Figure 3.2b.

3.2.2 Interferometric SAR (InSAR)

Signal Processing

InSAR is a technique in which we measure the phase difference between two SAR

images of the same area, acquired at slightly different viewing angles and/or at dif-

ferent times. Given two SAR SLC’s, s1(x,R) and s2(x,R), we can examine the phase

difference between them by forming the cross-product

s12(x,R) = s1(x,R) · s2(x,R)∗ (3.12)

The ∗ symbol represents complex conjugation. This cross-product is called an InSAR

image, or an interferogram. Typically, s2 will have a different x,R coordinate system

from s1 because of the different orbit geometries. To form the interferogram, we

therefore must first resample s2(x,R) into the coordinate system of s1(x,R). Because

s12(x,R) = |s1(x,R)| |s2(x,R)| ei( 6 s1−6 s2) (3.13)

the phase of the interferogram is the phase difference between s1 and s2. We will

denote this phase by φ.

The InSAR phase is a sum of several contributions (see Section 3.3.2), one of

which is phase due to topographic height. In this study, we use the DEM (explained


(a)

(b)

Figure 3.2: Raw SAR data (a) and focused SAR image (b).


in Section 2.2.1) to calculate the topographic contribution, φtopo, using Equation 3.26.

We then form the flattened interferogram

s12,f lat = s1 · s∗2 · e−iφtopo (3.14)

This is useful because topographic phase is deterministic and can be calculated from

the relatively well-known DEM elevations.

Figure 3.3a shows an interferogram formed from two SLCs of the same area shown

in Figure 3.2. Figure 3.3b shows the corresponding flattened interferogram. (The

images shows a region on the western coast of Greenland near its capital city of

Nuuk.)

Perpendicular Baseline

To better understand the physical basis for InSAR, we now examine the geometry,

shown in Figure 3.4. As shown in Figure 3.4a, the two SLC’s are acquired from two

orbit paths that are nearly parallel. The distance between the two orbit paths is called

the spatial baseline, or simply baseline. Figure 3.4b shows the InSAR geometry in

the plane perpendicular to the orbit paths. Points A1 and A2 are the orbit positions

from which a point on the surface is imaged in s1 and s2, respectively. The angle θ is

the incidence angle for the point in s1. Intuitively, because the backscattered signal

varies with incidence angle, the incidence angles for s1 and s2 must be almost equal in

order for the interferogram phase to represent a meaningful phase difference. Thus,

in practice, we choose pairs of SLC’s for which the difference in incidence angle, δθ,

is very small, and we use the perpendicular baseline, B⊥, in lieu of δθ, since B⊥ is

much more easily measured. Perpendicular baseline is defined as the component of

the baseline perpendicular to the radar look direction (the direction from A1 to the

point on the surface).


(a)

(b)

Figure 3.3: Interferogram (a) and flattened Interferogram (b).


The data for this study are repeat-pass InSAR data, which means that the two

orbit paths were flown on different days. The temporal separation between the ac-

quisition times of s1 and s2 is called the temporal baseline.

3.2.3 InSAR Coherence

Definition of coherence

Whereas InSAR phase is the phase difference between two SAR images s1 and s2,

InSAR coherence is a normalized measure of the variance of this phase difference,

denoted ρ. It is defined as

ρ ≡ | < s1s∗2 > |√

< s1s∗1 >< s2s∗2 >(3.15)

Note that coherence is also sometimes called correlation. The angle brackets in Equa-

tion 3.15 denote expected value. Treating s1 and s2 as random processes, the expected

value represents an ensemble average. However, because we do not have many re-

alizations of this process, we will assume that the signals in this study are locally

stationary and ergodic, meaning that empirical averages over a small rectangular

window of pixels can be used to estimate the coherence. This is physically reason-

able because the statstical properties of the Greenland Ice Sheet vary quite slowly

spatially.

From Equation 3.15, we can show (using the Cauchy-Schwarz inequality), that

0 ≤ ρ ≤ 1 (3.16)

for all interferograms. We can interpret coherence as a measure of phase noise, or

local phase variability. In areas where the coherence is nearly 1, the phases of all

interferogram pixels within a small local window will all lie within a very narrow


(a)

(b)

Figure 3.4: InSAR geometry (a) is similar to Figure 3.1 but contains a second orbitpath. InSAR baseline and incidence angle is shown in (b). Note that the diagram isnot drawn to scale; the difference between incidence angles for A1 and A2 is exagger-ated for clarity.


Figure 3.5: Normalized coherence for the interferogram shown in Figure 3.3b.

distribution. Conversely, in areas where the coherence is nearly 0, the phases of

interferogram pixels within a small local window are almost uniformly distributed

between −π and π.

As an example of coherence estimation, Figure 3.5 shows a map of coherence

estimates for the interferogram shown in Figure 3.3.

Estimating coherence

To estimate coherence, we choose a local window, typically a rectangular window

in radar coordinates representing a square region on the ground. We then form an

estimate, ρ, of the coherence using

ρ =

∣∣∣∑Ni=1 s1(i)s∗2(i)

∣∣∣√∑Ni=1 |s1(i)|2

∑Ni=1 |s2(i)|2

(3.17)


Here, N is the total number of pixels in the local window, and the summations are

taken over the pixels in the window.

The number of independent resolution elements in the estimation window is called

the effective number of looks, L. Note that the coherence estimator is biased, but the

bias decreases as L increases. A closed-form expression for the expected value of the

coherence estimator is given by [Touzi and Lopes, 1996]

< ρ >=Γ(L)Γ(3/2)

Γ(L+ 1/2)· F2,3(3/2, L, L;L+ 1/2, 1; ρ2) · (1− ρ2)L (3.18)

Here F2,3 is a generalized hypergeometric function. Figure 3.6a shows a plot of < ρ >

versus ρ for various values of L. For this study, we have chosen N = 4 × 28 = 112,

corresponding to L ≈ 50 for a small bias, and also incorporate the bias into our

forward model for estimating accumulation rate. The proximity of the curves in

Figure 3.6a for L = 25, L = 50, and L = 120 indicate that the coherence estimator is

not very sensitive to L for L ≈ 50.

To understand the uncertainty in our coherence estimates, we turn to the Cramer-

Rao bound for the estimator, which is given by [Rice, 2007]

std(ρ) =1− < ρ >2

√2L

(3.19)

Touzi showed that this can be used as an approximate standard deviation of the

coherence estimator, with the approximation being most accurate at high coherence

values [Touzi et al., 1999]. We will use this in Chapter 6 to understand the uncertainty

in our inverse problem solution. Figure 3.6b shows a plot of std(ρ) for various values

of L.


(a)

(b)

Figure 3.6: Expected value (a) and standard deviation (b) of coherence estimator.


Figure 3.7: Geocoded interferogram.

3.2.4 Geocoding

Thus far, all the SAR and InSAR images we have shown are in radar coordinates.

In order to geolocate pixels and compare our results against geophysical in-situ data,

we can regrid the images into the polar stereographic xp, yp-plane before geophysical

modeling. This is called geocoding.

As an example, the result of geocoding the flattened interferogram shown in Fig-

ure 3.3b is given in Figure 3.7.


Figure 3.8: Conceptual illustration of radar wave scattering.

3.3 Electromagnetic Scattering Models

To understand how SAR and InSAR image pixel values can be used for remote sensing,

we examine the electromagnetic wave scattering processes that occur when a radar

pulse interacts with the Earth’s surface. Figure 3.8 depicts the main phenomena of

interest.

The incident wave is the wave transmitted by the radar instrument. It can be

considered a plane wave incident upon the Earth’s surface. The surface scatters a

portion of the wave energy. The surface scattering may be diffuse or specular, or

a combination of the two. The surface material properties determine the amount

and type of scatter. Note that the radar is only able to measure the backscatter, the

energy scattered back toward the radar.

The angle between the incident wave propagation direction and the normal to the

surface is the incidence angle, θ. Because this is oblique incidence, the wave refracts

as it penetrates the surface, and the propagation direction of the refracted wave is

given by Snell’s law. Portions of the refracted wave can be scattered by scatterers


beneath the surface. This is called volume scatter.

Because the resolution elements for ALOS PALSAR images (∼ 10 × 5 m) are

much larger than a wavelength (23.61 cm), each resolution element in an InSAR

image results from the superposition of backscatter from many surface or volume

scattering elements. We model this using integrals, or, numerically, using discrete

sums.

The following subsections examine the physical interpretations of each of the In-

SAR measurables: backscatter power, phase, and coherence.

3.3.1 Backscatter Power

The amplitude of a SAR pixel can be related (after appropriate calibration) to the

backscatter power, which is the total power of the wave scattered back toward the

radar. Sometimes, this is referred to as the radar brightness. This can also be con-

verted to normalized radar cross-section σ0. Various models, such as Bragg scattering

and the Hagfors model, have been developed for expressing backscatter power in terms

of surface roughness parameters. Rough surfaces and surfaces tilted toward the radar

tend to result in higher backscatter power. For smooth and radar-penetrable surfaces,

such as the Greenland Ice Sheet, the backscatter power depends to a large extent on

the amount of volume scatter.

Surface roughness varies over the Greenland Ice Sheet and depends on factors such

as wind, but previous studies have reported that the standard deviation of the surface

height is around 5 mm for the interior regions [Long and Drinkwater, 1994]. Since

this is much smaller than a wavelength at L-band, we will, for this study, assume that

the contribution of surface scattering can be neglected. Of course, the validity of this

assumption depends on location and weather conditions.

We can calculate σ0 for a volume scattering terrain by integrating the backscatter


over all depths:

σ0 =

∫ ∞0

σv(z)dz (3.20)

where σv(z) is the volumetric radar cross-section, or radar cross-section per unit

volume.

The penetration depth of the radar wave into the volume scattering medium can

be many wavelengths for dry snow because it is not a very lossy medium. However,

for wet snow, water, or other lossy materials, the penetration depth is much smaller.

We will consider models for σv(z) from a radiative transfer perspective and quantify

absorption in dry snow in Chapter 5.

3.3.2 InSAR Phase

Neglecting atmospheric effects, the phase of a SAR pixel can be expressed as

6 s1 = −2π

λ(2R1) + φ1,scat (3.21)

= −4π

λR1 + φ1,scat (3.22)

where R1 is the distance from the radar to the center of the pixel and φ1,scat is the

resultant phase from the coherent superposition of all the backscatter contributions.

The first term in Equation 3.22 results from a total two-way propagation distance of

2R1.

An interferogram is formed by multiplying s1 by the complex conjugate of another

SAR image s2, whose phase can be expressed as

6 s2 = −4π

λR2 + φ2,scat (3.23)


Therefore the interferogram phase is given by

6 s12 = 6 s1 − 6 s2 (3.24)

= −4π

λ(R1 −R2)︸︷︷︸

range difference

+ (φ1,scat − φ2,scat)︸︷︷︸incidence angle difference

(3.25)

This can be interpreted as the sum of one term due to range difference and another

term due to incidence angle difference. The range difference is due to topophgraphic

height and surface deformation, and results in topographic phase

φtopo ≈ −4π

λR0 sin θB⊥z (3.26)

and deformation phase

φmotion = −4π

λδR (3.27)

Here, R0 is the distance to the first range pixel, z is the topographic height above a

mean reference surface, and δR is any distance that the pixel center moved between

the s1 and s2 acquisition times. For glacial ice, the motion is due mostly to glacial

ice flow. Recall also that we remove φtopo when we “flatten” the interferogram using

the DEM.

The difference φscat ≡ φ1,scat − φ2,scat is best viewed as a stochastic process since

it depends on the random distribution of scatterers within a resolution element. In

some applications this is treated as noise. However, φscat is what we use to infer ice

sheet structure from InSAR data.

We denote phase due to thermal noise and all other signal processing error sources

by φnoise. Adding phase resulting from propagation through the neutral atmosphere,

φatmos, and phase resulting from propagation through the ionosphere, φiono, we can


write

φint = φtopo + φmotion + φatmos + φiono + φscat + φnoise (3.28)

Since φtopo is removed during flattening, and φmotion, φatmos, and φiono tend to be

slowly varying functions, most of the variance of φint within small local windows is

usually attributed to φscat and φnoise. It is this variance that coherence measures.

3.3.3 InSAR Coherence

Coherence can be modeled as the product of several contributions [Zebker and Vil-

lasenor, 1992]:

ρ = ρsurf · ρvol · ρtemp · ρtherm · ρother (3.29)

Each of these factors is between 0 and 1 and lowers the overall coherence from the

highest possible value of 1. Therefore the factors are sometimes referred to as decor-

relation sources. Surface correlation, denoted ρsurf , volume correlation, denoted ρvol,

and temporal correlation, denoted ρtemp are related to φscat. Thermal correlation,

denoted ρtherm is related to φnoise. Finally, other decorrelation sources are lumped

together and denoted ρother. We now examine each of these factors individually.

Surface correlation

Surface correlation results from the different distances from the radar to individual

scatterers in a resolution element when viewed from different angles. In fact, a closed-

form expression for ρsurf can be given in terms of perpendicular baseline and other

geometric parameters:

ρsurf =

1−B⊥/Bcrit, |B⊥| < Bcrit

0, otherwise(3.30)


Here we define the critical baseline, Bcrit, as

Bcrit =R0λ| tan θ|

2δR(3.31)

A derivation of this formula is given in [Hoen, 2001]. Note that ρsurf decreases linearly

with increasing B⊥, until B⊥ reaches Bcrit. For B⊥ ≥ Bcrit, ρsurf = 0, which means

that no meaningful interferograms can be formed. Because only geometric parameters

are used in calculating ρsurf , it does not tell us anything about firn structure.

For the polarimetry mode in which most Greenland SAR data were acquired, the

critical baseline for ALOS PALSAR is about 3.3 km. This study uses perpendicular

baselines of less than 500 m, and we include ρsurf in our modeling.

Volume correlation

Whereas surface correlation is related only to geometric parameters, volume correla-

tion, ρvol, is related to both geometry and the structure of the subsurface scatterers.

It is thus useful for studying firn structure. Because energy is lost to backscatter and

attenuation, the radar wave interacts mainly with a layer near the surface of the ice.

In general, a thicker layer of ice contributing to the received backscatter results in

lower value of ρvol.

Volume decorrelation is given by

ρvol =

∣∣∣∣∫ σv(z)e−i2πkzzz∫σv(z)dz

∣∣∣∣ (3.32)

where

kzz =2√εδθr

λ sin θr(3.33)

In Equation 3.33, ε is the relative dielectric constant of the medium, θr is the incidence

angle of the refracted wave, and δθr is the incidence angle difference for the refracted


waves.

We can interpret the numerator of Equation 3.32 as a coherent sum of backscatter

contributions from all depths. The denominator represents the usual power normal-

ization factor. Readers are referred to [Hoen, 2001] for a derivation of the above. We

will use these formulae in the forward model in Chapter 5.

Temporal correlation

Temporal correlation is denoted ρtemp. This is caused by changes to the relative

positions of individual scatterers within a resolution element during the time between

the two SAR acquisitions used to form an interferogram.

For example, if the scatterer motion in the horizontal and vertical directions are

independent and follow Gaussian distributions with mean distances of σmotion,y and

σmotion,z respectively, then the temporal correlation is given analytically by [Zebker

and Villasenor, 1992]

ρtemp = exp

−1

2

(4π

λ

)2 (σ2motion,y sin2 θ + σ2

motion,z cos2 θ)

(3.34)

Larger values of σmotion,y and σmotion,z, generally associated with longer temporal

baselines, result in lower values of ρtemp.

In general, ρtemp decreases with increased temporal baseline, yet it is difficult to

quantify because we do not have in-situ data on the statistical distribution of the

scatterer motions. However, since the ice grains within the Greenland Ice Sheet are

generally not free to move about relative to one another, there is relatively little

temporal decorrelation in the 46-day temporal baseline that we use in this study.


Thermal correlation

Thermal correlation, ρtherm, results from the variance in φnoise due to receiver thermal

noise. Thus, for images with high SNR, the variance of φnoise approaches zero, and

ρtherm approaches unity. We can calculate ρtherm from SNR by

ρtherm =1

1 + 1/SNR(3.35)

Other factors

The factor ρother accounts for all other decorrelation sources. This may include missing

lines or other errors in the raw data, high phase gradients that are misinterpreted

as phase noise, or other processing errors. Inaccurate coregistration of s2 with s1

also causes decorrelation. In the next chapter, we will address decorrelation due to

ionospheric effects.

Chapter 4

Spatially Varying Ionosphere

4.1 Introduction

Whereas the previous chapter presented general principles of InSAR signal processing

and scattering models, this chapter presents a method we developed for mitigating

ionospheric effects in our InSAR data. Much of the research in this chapter has also

been published as [Chen and Zebker, 2014].

The ionosphere is a region of the Earth’s outer atmosphere consisting of partially

ionized plasma extending from about 100 km to 1000 km above the Earth’s surface

[Budden, 1985]. In the ionosphere, the number of electrons1 per unit volume is called

the electron concentration, denoted ne. The electron concentration varies with al-

titude, with several distinct peaks. A typical value for the largest peak is around

1012 m−3 at an altitude of about 350 km. The ionization is caused by radiation from

the sun, so the vertical electron concentration profile varies with the time of day,

and also with latitutde, time of year, and sunspot cycle. The ionosphere is very

dynamic, with disturbances and irregularities generated by various mechanisms at

1In this chapter, the term electron refers to free electrons that result from ionization of neutralatoms or molecules.

46

CHAPTER 4. SPATIALLY VARYING IONOSPHERE 47

distance scales ranging from centimeters to kilometers.

For this thesis, we are usually interested in the total electron content (TEC) along

a path L, which is defined as

TEC =

∫L

ne(l)dl (4.1)

The ALOS PALSAR instrument orbits at an altitude of about 700 km [Shimada

et al., 2009]. Thus, the pulses that it transmits propagate through much of the

Earth’s ionosphere. This is depicted conceptually in Figure 4.1. When a radar pulse

propagates through the ionosphere, its phase is advanced by an amount proportional

to the TEC along the propagation path.

Large spatial TEC variations within a synthetic aperture length would hinder az-

imuth focusing and degrade SAR resolution. (Similarly, GPS and microwave satellite

communications systems can also be adversely affected by the ionosphere.) For the

SAR scenes in this study, there is no noticeable loss of resolution, so we assume that

the TEC varies slowly enough that it can be considered a linear function of azimuth

position within any given synthetic aperture length.

In Section 4.2, we show how spatial variations in ionospheric TEC result in spa-

tially varying azimuth offsets between an interferometric pair of SAR images. Quite

often InSAR processing software does not account for these varying offsets, which

results in regions of low InSAR coherence. These have sometimes been observed by

other authors and have also been called “azimuth streaks.” Without correction, these

regions would cause misleading results in accumulation rate estimates.

Section 4.3 describes the algorithms we have developed to mitigate the effects of

spatially varying offsets due to ionospheric TEC variations. In Section 4.4, we present

results from two test sites. Finally, in Section 4.5 we analyze the effectiveness and

limitations of our method.


Figure 4.1: Conceptual illustration showing relative altitudes of ALOS PALSAR andthe E-layer and F-layer of the ionosphere. Note the atmospheric water vapor thatcontributes to φatmos, described in Section 3.3.2, is much closer to the ground thanthe ionosphere.


4.2 Theory: Ionospheric Effects in InSAR

In this section, we review the physics of electromagnetic wave propagation in the

plasma of the ionosphere. We then derive expressions for ionospheric InSAR phase

and azimuth offsets due to spatially varying TEC. Details on electromagnetic wave

propagation are given in [Inan and Inan, 2000; Ulaby, 2004]. A comprehensive treat-

ment of waves in the ionosphere is given in [Budden, 1985], and a good review of

plasma physics can be found in [Chen, 2006].

4.2.1 Microwave Propagation in the Ionosphere

As explained in Chapter 3, the radar transmits a pulse at carrier frequency fc, which

we model as a sinusoid because of its small fractional bandwidth. For this devel-

opment, we use a Cartesian coordinate system with the z-axis pointing in the wave

propagation direction. We denote the phasor x- and y- components of the electric

field by Ex and Ey respectively.2 The wave polarization is defined as

Ψ =EyEx

(4.2)

Plasma frequency and gyrofrequency

Dispersive propagation effects in the ionosphere depend on the plasma frequency

[Thompson et al., 2001, Sec. 13.3], which is given by

fp =1

2π

√nee2

ε0me

(4.3)

where ne is the electron concentration, e and me are the charge and mass of an

electron, and ε0 is the permittivity of free space. In the absence of a magnetic field,

2Our convention is that the x-component of the electric field is given by Ex = <Exe

i2πfct

, andsimilarly for the y-component.


the plasma frequency is the frequency below which electromagnetic waves cannot

propagate in the plasma. This frequency is around 2.84 MHz for a typical electron

concentration of 1011 m3. It is also customary to define the gyrofrequency (also known

as the cyclotron frequency) by

fB =1

2π

∣∣∣∣∣e ~Bme

∣∣∣∣∣ (4.4)

where ~B represents the Earths magnetic field. The gyrofrequency is the frequency

at which electrons spiral around magnetic field lines in the absence of any electro-

magnetic wave. This frequency is about 1.4 MHz for a typical magnetic field of

0.5× 10−4 T.

Polarizations

The two polarizations whose propagation are supported by plasma in a static magnetic

field are given by the two solutions to this quadratic equation [Budden, 1985, Sec.

4.3]:

Ψ2 − iΨ Y sin2 θ(1− i ν

2πf−X

)cos θ

+ 1 = 0 (4.5)

where ν is the collision rate of particles in the plasma, i is the imaginary unit, and

X =f 2p

f 2c

(4.6)

and

Y =fBfc

(4.7)

Note that for L-band systems, we have X 1 and Y 1. For the frequencies

used in microwave InSAR instruments, the ionosphere is well-modeled as a low-loss

plasma [Nicolet, 1953], and we can assume that ν 2πf . Then, we can approximate


Equation 4.5 by

Ψ2 − iΨY sin2 θ

cos θ+ 1 = 0 (4.8)

or

Ψ2 − iΨfBfc

sin θ tan θ + 1 = 0 (4.9)

For relatively large carrier frequencies, we find that

Ψ2 + 1 ≈ 0 (4.10)

or

Ψ ≈ ±i (4.11)

Physically, this means that right- and left- circular polarizations are supported. Note

that a linearly polarized wave, such as that used by ALOS PALSAR, can be considered

the superposition of two circularly polarized waves. However, we will show below that

the two constituent circularly polarized waves have slightly different refractive indices.

Dispersion relation

The dispersion relationship for plasmas in a static magnetic field is given by

n2 = 1− X(1−X)

1−X − 12Y 2 sin2 θ ±

14Y 4 sin4 θ + Y 2 cos2 θ(1−X)2

1/2(4.12)

where n is the refractive index and θ is the angle between the magnetic field and the

propagation direction [Budden, 1985, Sec. 4.6]. This is often known as the Appleton-

Hartree relation. In the denominator of Equation 4.12, a positive sign is used in front

of the bracketed expression for one polarization, and a negative sign for the other

polarization. We now seek a suitable approximation of Equation 4.12 for microwave


radars. Because we have X 1 and Y 1, we can make the approximation

n2 ≈ 1− X

1±

14Y 4 sin4 θ + Y 2 cos2 θ(1−X)2

1/2(4.13)

and thus

n2 ≈ 1− X

1± Y cos θ= 1−

f 2p

f 2c

(1± fB cos θ

fc

)−1

(4.14)

Finally, using the binomial approximation, we have

n2 ≈ 1−f 2p

f 2c

∓f 2p fB cos θ

f 3c

(4.15)

We can interpret the terms in Equation 4.15 physically as follows. For sufficiently

high carrier frequencies, we simply have n2 ≈ 1, which is the same as propagation in

free space. The second term on the right side of Equation 4.15 represents carrier phase

advance independent of wave polarization. The last term means that the right- and

left- circular polarizations propagate with slightly different phase velocities, which

causes Faraday rotation. In fact, ionospheric effects in SAR and InSAR systems can

be corrected by measuring Faraday rotation [Meyer and Nicoll, 2008], but this only

works in fully polarimetric systems.

The third term in Equation 4.15 is smaller than the second term by a factor of

fc/fB cos θ, which is approximately 103 for L-band systems. For the data in this

thesis, we find that the Faraday rotation associated with the third term is generally

much less than 10 and is difficult to measure accurately. We therefore focus on the

first two terms. Using the binomial approximation again and neglecting the magnetic

field terms, we can now write

n =

√1−

f 2p

f 2c

∓f 2p fB cos θ

f 3c

≈ 1− 1

2

f 2p

f 2c

(4.16)


This expression is commonly cited for L-band models of the ionosphere.

4.2.2 Ionospheric phase advance

We assume the length scale of spatial ne variations is much larger than one wavelength

of a radar pulse. We further assume that the ionosphere is static over the time frame

of a single SAR acquisition. Using these assumptions, we now examine the effect of

phase advance on InSAR phase, neglecting Faraday rotation.

Using the W.K.B. approximation [Elmore and Heald, 1969], we can write the

phasor of a plane wave propagating through the ionosphere as

E ≈ E0n−1/2 exp

(i2π

λ

∫L

ne(l)dl

)(4.17)

where E0 is the free-space amplitude, and λ is the free-space carrier wavelength. The

integral is taken along the wave’s propagation path. Note that the amplitude is not

changed, but the phase is given by the integration of refractive index.

We can give a closed-form expression for the phase:

φ =2π

λ

∫L

(1− 1

2

f 2p

f 2c

)dl (4.18)

=2π

λ

∫L

(1− 1

2

(1

2π

)2nee

2

ε0me

1

f 2c

)dl (4.19)

=2π

λL− 2π

λ

40.23

f 2c

∫L

ne(l)dl (4.20)

The first term in Equation 4.20 is the phase we would expect due to free-space

propagation. The second term represents the phase advance induced by the iono-

sphere. Thus, using the definition of TEC from Equation 4.1, the phase of a radar


signal due to the ionosphere is given by

φ = −2 · 2π

λ

40.23

f 2c

TEC = −4π

λ

40.23

f 2c

TEC (4.21)

where the extra factor of 2 arises from 2-way propagation. The constant

e2

8π2ε0me

= 40.23m3s−2 (4.22)

occurs frequently in ionospheric propagation formulas. Note that the phase advance

varies inversely with carrier frequency, so the effect is more noticeable at lower fre-

quencies (e.g. L-band radars) compared to higher frequencies (e.g. C-band radars).

4.2.3 Azimuth Offsets due to Ionospheric Inhomogeneities

We now relate the above results to azimuth shifts in SAR images. As explained in

Section 3.2.1, a SAR processor uses the Doppler frequency of targets on the ground

to focus the raw SAR data in the azimuth direction. Consider a point located at

azimuth position x0 = 0 whose range at closest approach is denoted R0.3 Then, using

the binomial approximation, we find that the range to the point, as a function of the

azimuth position x of the antenna, is given by

R(x) =√x2 +R2

0 (4.23)

= R0 +1

2

x2

R0

for R0 x (4.24)

Then the phase history of the point is given by

φ(x) = −4π

λR(x) = −4π

λR0 −

2πx2

λR0

(4.25)

3This derivation assumes an un-squinted geometry, meaning the antenna look direction is per-pendicular to the radar orbit path. The assumption is valid for all the data in this thesis.


Displacing this point from azimuth position x0 = 0 to x0 = ∆x changes its phase

history to

φ = −4π

λR0 −

2π

λR0

(x−∆x)2 (4.26)

= φ(x)− 2π

λR0

∆x2︸︷︷︸constant

+4π∆x

λR0

x (4.27)

Thus, the phase history of the displaced point is equivalent to the phase history of

the point at x0 = 0 with an additional constant phase (which does not affect azimuth

focusing) along with an added phase gradient of

∂φ

∂x=

4π∆x

λR0

(4.28)

Note that this is equivalent to a Doppler frequency of

fdop ≡1

2π

∂φ

∂t=

1

2π

∂φ

∂x

∂x

∂t(4.29)

=1

2πv

4π∆x

λR0

=2v∆x

λR0

(4.30)

Differentiating Equation 4.21, we can relate an azimuth gradient in the ionospheric

TEC to a resulting phase gradient in the received data from a point on the ground:

∂φ

∂x= −4π

λ

40.23

f 2c

∂TEC

∂x(4.31)

Setting the phase gradient in Equation 4.28 equal to the above, we find that

∆x = −40.23

f 2c

∂TEC

∂xR0 (4.32)

Thus an azimuth gradient in the TEC results in a range-dependent azimuth phase


gradient being added to the along-orbit phase histories of the pixels being imaged.

These phase gradients are equivalent to Doppler shifts. Since the SAR processor

equates a change in Doppler frequency to a change in azimuth position, this causes

offsets between the actual and imaged azimuth positions of the pixels.

Because of temporal variations in the ionosphere, these offsets are different in the

two SAR images of an interferometric pair and can vary widely over spatial scales

comparable to a synthetic aperture in length. As a result, when the offset between the

two images is described using a low-order polynomial function of range and azimuth

position, there are regions where the two images will not be correctly coregistered.

These regions often form “azimuth streaks” which can be particularly obvious in

coherence images.

Finally, note that the shift is proportional to the TEC gradient, and inversely

proportional to the square of the carrier frequency. For example, lowering the oper-

ating frequency from fc = 5.2967 GHz used by the ERS C-band SAR instrument to

fc = 1.2698 GHz used by the ALOS L-band Re PALSAR instrument would increase

∆x by a factor of 17.4. In conventional SAR and InSAR techniques, we do not have

independent measurements of these phase gradients, and thus we must correct for

their effects empirically. Gray and Mattar have presented numerical simulations of

this azimuth shift effect [Mattar and Gray, 2002], whereas Liu et al. have given a

stochastic approach [Liu et al., 2003].

4.3 Accurate Coregistration Method

In this section we present our method for correcting regions of low coherence caused

by spatial variations in the ionosphere. For this chapter, we denote the two images of

an interferometric pair by s1(x1, R1) and s2(x2, R2). Here x1 and R1 denote azimuth

and slant range position in s1. Similarly, x2 and R2 give azimuth and slant range


position in s2. The two images are not in the same coordinate system because they

are acquired from slightly different satellite orbits.

To form an InSAR image, we first estimate the offset between s1(x1, R1) and

s2(x2, R2). That is, we estimate the displacements ∆x and ∆R such that s1(x1, R1)

and s2(x1 + ∆x,R1 + ∆R) correspond to the same location on the ground. Note

that these offsets vary as a function of azimuth and range. We estimate the offsets

between the two SAR images using cross-correlation. We then resample s2 into the

x1, R1 coordinate system, and denote this by s2. The InSAR image, s12 is then given

by

s12(x1, R1) ≡ s1(x1, R1)s2(x1, R1)∗ (4.33)

In other chapters of this thesis, we assume that the resampling has been performed

accurately so that we do not need to differentiate between the x1, R1 and x2, R2

coordinate systems (cf. Equation 3.12).

Most existing InSAR software packages parameterize ∆x and ∆R as a low-order

polynomial function of range and azimuth. For example, a 4-term polynomial param-

eterization might be

∆x(x1, R1) = a0 + a1x1 + a2R1 + a3x1R1 (4.34)

∆R(x1, R1) = b0 + b1x1 + b2R1 + b3x1R1 (4.35)

Existing software typically uses amplitude cross-correlation to estimate ∆x and ∆R at

a few hundred locations in s1 and then estimates coefficients a0, · · · , a3 and b0, · · · , b3

using linear least-squares curve-fitting, which tends to be robust to outliers and in-

accurate offset estimates in noisy data.

As shown in Equation 4.32, azimuthal TEC gradients cause pixels to be displaced

in the azimuth direction. Indeed, these offsets have been detected even in single


SAR acquisitions [Nicoll and Meyer, 2008]. For the InSAR case, these azimuth dis-

placements are different in s1(x1, R1) and s2(x2, R2) because of temporal changes in

the ionosphere. This results in a spatially varying azimuth offset which cannot be

modeled by a low-order polynomial of the form used in Equation 4.35. As a result,

software which uses low-order polynomial parameterizations of ∆x and ∆R generate

interferograms with regions of decorrelation due to misregistration.

4.3.1 Accurate Coregistration Algorithm

To avoid the disadvantages of using a low-order polynomial parameterization of ∆x

and ∆R, we first choose an azimuth spacing Sx and a range spacing SR of typically

32 pixels (see Figure 4.2). We then estimate the azimuth and range offsets on a

regular grid of points in the SAR images with spacings Sx and SR. In other words,

we estimate

∆x(iSx, jSR), i = 1, · · · ,M, j = 1, · · · , N (4.36)

∆R(iSx, jSR), i = 1, · · · ,M, j = 1, · · · , N (4.37)

to form an M×N array of offset estimates, which we denote by ∆x(i, j) and ∆R(i, j).

Ideally, we might set the spacings Sx and SR equal to 1 in order to estimate the offset

at every pixel location in s1(x1, R1), but this generally requires prohibitively long

computation times, so we choose to use a larger spacing and then interpolate the

results.

For the Greenland data, we estimate the offsets using amplitude cross-correlation

because we have found that complex cross-correlation is not effective in areas of high

fringe-rate, such as at the margins of the ice stream in our Greenland data. For

the Iceland data, we found that estimating offsets by complex cross-correlation gave

fewer outliers. In both our complex and amplitude cross-correlation algorithms, we


Figure 4.2: InSAR coregistration coordinate systems and geometry. The red points ins1 and the corresponding red points in s2 represent the same locations on the ground.


up-sample the data by a factor of 32, thus giving offset estimates with accuracies of

up to 1/32 = 0.031 pixels or about 0.11 m.

To facilitate the computation, we have implemented a parallelized version of our

offset estimation code. Using 8 computation threads, we typically compute offset

estimates with spacings Sx = SR = 32, for a total of about 32, 000 offset estimates

in a single frame of ALOS data, in under 1 minute. Finer offset spacing results in

correspondingly longer computational times.

When s1(x1, R1) and s2(x2, R2) can be appropriately modeled as partially corre-

lated stationary circular Gaussian signals, the amplitude cross-correlation method for

empirically estimating the offset between them generates unbiased estimates whose

standard deviation increases as the correlation between s1(x1, R1) and s2(x2, R2) de-

creases [Bamler and Eineder, 2005]. In some areas, such as the interior regions of

the Greenland ice sheets, we find large areas where interferometric correlation is high

(greater than 0.5) and the stationarity assumption is not violated. Here we observe

that the estimated offset fields are smooth and accurate. In other regions, our im-

ages often have areas in which correlation is very low, or in which the stationarity

assumption is violated due to spatial terrain variations. In these regions, we often

find that the offset estimates are very noisy and contain many outliers. Therefore,

the estimates ∆x(i, j) and ∆R(i, j) cannot be directly used to resample s2 into the

x1, R1 coordinate system and form the interferogram.

When the number of outliers in the offset measurements is not too large, we find

that it is very effective to apply a median filter followed by a low-pass filter to the

arrays ∆x(i, j) and ∆R(i, j). These filtered estimates of ∆x and ∆R, which we denote

∆xfilt(i, j) and ∆Rfilt(i, j), can be used for resampling and interferogram formation.

For cases where a large percentage of the estimates in ∆x(i, j) and ∆R(i, j) are

inaccurate, we also find that a fairly good representation of the actual offset fields

can be obtained by low-pass filtering or using a smoothing spline (see Section 4.4.2).


To resample s2 into the x1, R1 coordinate system, we must first interpolate ∆xfilt(i, j)

and ∆Rfilt(i, j) to obtain estimates of offsets at all range and azimuth locations. This

can be challenging, because the interpolation must not introduce errors of more than

a few tenths of a pixel in order to avoid decorrelation, as shown by Equation 4.42

(see below). We find that in many cases, using a truncated sinc interpolation kernel

introduces ringing artifacts of unacceptably large amplitude. Also, ∆x and ∆R are

generally not band-limited. We find that bilinear interpolation avoids these ringing

artifacts and produces smooth, accurate maps of range and azimuth offsets.

Finally, we use the interpolated offset estimates to resample s2 into the x1, R1

coordinate system. Because s2 is bandlimited (since it is produced using SAR matched

filters), we interpolate it using a two-dimensional truncated sinc kernel, typically

of size 8 × 8 pixels. It has been shown that resampling using this kernel has a

negligible effect on the interferometric coherence [Hanssen and Bamler, 1999]. We

have implemented parallelized code for this operation, so that we can resample a

frame of ALOS data in about 1 minute using 8 computation threads.

4.3.2 Ionospheric Phase Estimate

We can use the theory developed above to estimate the contribution of the ionosphere

to the interferometric phase. By combining Equation 4.21 and Equation 4.32, we find

that

φiono =4π

λR0

∫∆x(x,R)dx (4.38)

Using our empirical estimates of ∆x(x,R), we are thus able to estimate the iono-

spheric contribution to the InSAR phase by numerical integration using the trape-

zoidal rule:

φiono(iSx, jSR) ≈ 4π

λR0

·j∑

n=1

(∆xfilt(n+ 1, j) + ∆xfilt(n, j)) ·Sx∆x

s(4.39)


where ∆x is the azimuth pixel size. We then use bilinear interpolation to find φiono at

other values of x and R. Finally, an ionospheric phase screen signal can be calculated

using

siono = exp(iφiono) (4.40)

We thus estimate that the true InSAR phase is given by s12(x1, R1) · s∗iono, so we can

remove an estimate of the ionospheric phase from the interferogram without phase

unwrapping.

4.4 Results

To test our method, we use ALOS PALSAR data from two test scenes. The first is

a scene in the dry snow zone of the Greenland ice sheet; the second is a scene over

central Iceland. The SLCs and maps showing the locations of the two test scenes

are shown in Figure 4.3. The SAR image of the interior of the Greenland ice sheet

looks largely featureless due to the lack of topographical relief on the ice sheet. We

do, however, see interesting phase patterns in the interferogram due to glacier ice

motion, as shown in Section 4.4.1.

4.4.1 Greenland test scene results

SAR images of the Greenland ice sheets are well-modeled as partially correlated sta-

tionary circular Gaussian signals, so the Greenland ice sheets form an ideal testing

ground for our algorithm. We form focused SAR images from data acquired in March

and April 2007, with a perpendicular baseline of about 150 meters, a temporal sepa-

ration of 46 days, a range resolution of 9.52 m and an azimuth resolution of 4.45 m.

The left side of Figure 4.4 shows the interferogram generated using a low-order

polynomial function to represent the offsets. The most prominent artifact is indicated


(a) Greenland SLC (b) Iceland SLC

(c) Greenland map (d) Iceland map

Figure 4.3: Test scenes for our method. The SLC images for the Greenland andIceland test scenes are shown in (a) and (b), respectively. Similarly, maps showingthe locations and topography of the Greenland and Iceland test scenes are shown in(c) and (d), respectively. The scene boundaries are indicated by the black rectangles.


Figure 4.4: Original and corrected interferograms for the Greenland test scene. Theclosely-spaced fringes near the top of the interferograms are due to ice motion at theedge of the Northeast Greenland Ice Stream (NEGIS). The most prominent artifact,or “azimuth streak”, is indicated by the red outline. The noisier, darker band withinthis outline is an area of lower coherence because of ionospheric inhomogeneity. Inthe corrected interferogram, the coherence in this area is restored.


Figure 4.5: Original and corrected coherence images for the Greenland test scene.The streaks of low coherence in the original coherence image (indicated by blue onthe color scale) are removed by our method in the corrected coherence image.


by the red outline. In contrast, the right side of Figure 4.4 shows the interferogram

generated using our accurate coregistration method. Figure 4.5 shows the corre-

sponding coherence images. These images demonstrate that interferometric fringes

are much more clearly visible and coherence is improved using our accurate coregis-

tration method.

Figure 4.6 shows the azimuth and range offsets that we used in the resampling

process. The azimuth offsets vary from about −2 to +2 pixels, and this spatial

variation is correlated with the locations of the azimuth streaks. On the other hand,

there is negligible variation in the range offsets.

To further quantify the improvement in coherence, Figure 4.7 shows histograms of

the coherence values in this image, using both the low-order polynomial offset fit and

our accurate coregistration method. Whereas the low-order polynomial fit results in

many pixels with coherence below 0.5, almost all the pixels have coherence greater

than 0.5 using our method. Thus this method improves our ability to recover useful

geophysical information from Greenland InSAR data that are affected by ionospheric

inhomogeneities.

In Figure 4.8 we have plotted the estimated azimuth offset, ionospheric TEC, and

ionospheric contribution to InSAR phase for the near-range pixels in the Greenland

scene. Our analysis shows that the ionosphere contributes about 4 radians of phase

over the interferogram. This is approximately 1 fringe of interferometric phase. This

is not worse than fringes often observed due to tropospheric inhomogeneities or orbit

errors. Figure 4.9 shows an image of the estimated ionospheric phase over the entire

InSAR scene. Removing the estimated ionospheric phase using our method improves

the accuracy of ice velocity estimates derived from InSAR data. This is especially

important because ice velocity measurements are an important source of data on

Greenland’s outlet glacier ablation rates. (Efforts to compare these TEC results with

those derived from GPS measurements are given in [Chen and Zebker, 2012].)


Figure 4.6: Azimuth and range offsets for Greenland scene. The spatial variationin the azimuth offset is correlated with the locations of the azimuth streaks. In theregion with the most prominent azimuth streak artifact, at azimuth location around30 km, the azimuth offset is around -2.5 pixels.


Figure 4.7: Coherence histograms for the original and corrected Greenland interfero-grams.

4.4.2 Iceland processing results

Compared to inner Greenland, the terrain of Iceland shows significant spatial vari-

ability. We use an interferometric pair of SAR images from September and October

2007. Again the temporal separation is 46 days. The perpendicular baseline for this

pair is approximately 500 meters.

The left side of Figure 4.10 shows the interferogram formed using the traditional

low-order polynomial fit method. There is a salient artifact, indicated by the red out-

line. Note that these images were acquired by the PALSAR instrument in fine-beam

dual polarization (FBD) mode, so there are approximately twice as many range bins

as in the Greenland images. The right side of Figure 4.10 shows the interferogram

formed using our accurate coregistration method. Figure 4.11 shows the correspond-

ing coherence images.


Figure 4.8: Azimuth offset, TEC, and estimated ionospheric phase for near-rangepixels in Greenland test scene. In the middle plot, the TEC is the differential TEC,the difference between TEC on the first and second acquisition dates. Note that1TECU = 1016electrons/m2.


Figure 4.9: Estimated ionospheric phase in Greenland scene. The ionospheric phaseamounts to about one fringe, which would be equivalent to about 12 cm of defor-mation or ice motion. For many applications, this would represent an unacceptableamount of error if uncorrected.


Figure 4.10: Original and corrected interferograms for Iceland scene. The most promi-nent artifact, or “azimuth streak”, is indicated by the red outline. The noisier, darkerband within this outline is an area of lower coherence because of ionospheric inho-mogeneity. In the corrected interferogram, the coherence in this area is partiallyrestored.


Figure 4.11: Original and corrected coherence images for Iceland scene.


Figure 4.12: Azimuth and range offsets for Iceland scene.


Figure 4.12 shows the azimuth and range offsets measured using complex cross-

correlation. Because of the large number of outliers in these estimates, we need

to process the offset estimates carefully before using them to resample s2 into the

x1, R1 coordinate system and form the interferogram. In this scene we have chosen

to keep those offset estimates which result in a correlation of over 0.68. We then

use a linear interpolator to fill in the locations at which we do not obtain accurate

offset estimates. We have found that fitting a 2-D spline and low-pass filtering the

estimates give similar results, with the spline approach perhaps slightly better. The

resulting estimates, which we show in Figure 4.13, are then used to resample s2 and

form the interferogram.

4.5 Discussion

4.5.1 Quantifying the Improvement in Coherence

To better understand how spatial ionospheric variations lower InSAR coherence, recall

from Section 3.3.3 that coherence can be modeled as the product of several factors:


When coherence is affected by inaccurate coregistration, ρother is given by

ρother = sinc(µ/δ) (4.42)

where µ is misregistration and δ is the resolution [Bamler and Eineder, 2005].

Figure 4.14 shows coherence in the Greenland interferogram as a function of mis-

registration. We also draw the model that best fits that data, assuming that the

product ρsurfρvolρtempρtherm is a constant, which we call the background coherence.


(a) Smoothed offset

(b) Comparison

Figure 4.13: Smoothed azimuth offsets for the Iceland interferogram (a), and com-parison of smoothing by filtering and fitting a 2-D spline, for the 3200th range bin(b).


Figure 4.14: Coherence as a function of azimuth coregistration error for Greenlandscene.

The best fit model has a background coherence of 0.693 and an azimuth resolution

of 1.65 pixels (compared to a theoretical range resolution of 1.4 pixels). The good

fit between the model and the data verify that the loss of coherence in the original

interferogram is due to local coregistration errors.

Figure 4.15 shows the coherence, averaged over the first 25 range bins, as a func-

tion of azimuth position in the Greenland interferogram. The large dips in coherence

corresponding to the “azimuth streaks” are removed by using our method, and co-

herence improved by as much as 0.5 in some areas.


Figure 4.15: Coherence improvement for near-range pixels in Greenland test scene.

4.5.2 Limitations and Conclusions

Our results show that spatial variations in ionospheric TEC do not necessarily hinder

measurement of InSAR coherence. When the TEC can be considered a linear function

of azimuth within any given synthetic aperture length, we can form focused SAR

images, but we find that there are spatially varying azimuth offsets of often more

than one single-look pixel. Accounting for these offsets ensures that interferometric

coherence is not significantly affected by ionospheric effects.

This method is limited primarily by our ability to estimate the true range and

azimuth offsets as a function of position. The cross-correlation algorithm for offset

estimation is prone to inaccurate offsets and outliers when used in a region where

the SAR images cannot be considered stationary partially correlated circular Gaus-

sian signals. Also, we cannot distinguish between true azimuthal displacement and

ionosphere-induced azimuth shifts. Therefore, this method is most accurate in regions


where displacement is primarily in the range direction, or where azimuth displace-

ment is much smaller than ionosphere-induced azimuth shifts. In the Greenland test

scene, phase unwrapping shows that the total slant-range (line-of-sight) displacement

is about 3.2 meters in the fastest-moving part of the ice stream (upper-right corner

of the interferogram). This is significantly smaller than the range resolution, which

explains why we cannot clearly see the ice stream in the range offset image.

Another limitation of this method is the accuracy of numerical integration. By

evaluating the integral in Equation 4.38 numerically (using a Riemann sum), we

potentially introduce a numerical error that can grow without bound as azimuthal

distance increases. In many cases, this would result in a small phase ramp in the final

interferogram, but noisy data could cause much worse inaccuracies. Using Equa-

tion 4.38 we can write an upper-bound on the ionospheric phase error, which we

denote by |εφ iono|, as follows:

|εφ iono| ≤4π

λR0

∫scene length

max(|ε∆x|)dx (4.43)

Here, ε∆x is the offset estimate error. For our Greenland data, the empirical

standard deviation of the offset estimates is about 0.1 pixels. If we assume that

there are no outliers in the estimates, it is reasonable to take max(|ε∆x|) ≈ 0.2 pixels,

which gives a worst-case error bound of |εφ iono| ≤ 3.9 rad over an 80 km long InSAR

scene. That is equivalent to about 7 cm of line-of-sight deformation, which makes this

method clearly unsuitable for measuring subtle crustal deformation signals. However,

since this is a worst-case error, our method is still clearly suitable for scenes with

relatively large deformation signals, such as the Greenland ice sheet.

Finally, in this method of estimating ionospheric phase, the null space is the set

of all functions of the form

φnull = g(R) (4.44)


In other words, this method is not sensitive to the variation of ionospheric phase

in the range direction. For the data in this study, we have found empirically that

the azimuth offset, and therefore the ionospheric phase, is primarily a function of

azimuth. However, this need not be the case in general, and depends on geometric

factors such as the alignment of the satellite orbit, the magnetic field direction, and

the variations in the ionosphere at the time of measurement.

Estimating ionospheric phase using Faraday rotation or a split-spectrum technique

may help overcome some of the difficulties in the azimuth offset method. However,

these methods have their limitations as well. Faraday rotation measurements are only

possible in polarimetric InSAR instruments, whereas our method works for all InSAR

data. Split-spectrum techniques can also be used to correct ionospheric effects in In-

SAR data [Rosen et al., 2011]. However, in InSAR data with limited range bandwidth

(e.g. 14 MHz for ALOS PALSAR polarimetry mode), split-spectrum techniques offer

limited signal-to-noise ratio, which must be compensated by taking a large number

of looks or low-pass filtering, which limits the spatial resolution. In contrast, the

azimuth offset method presented here can correct interferograms at a finer spatial

scale, with very good results in some areas, especially in the Greenland ice sheet.

To summarize, we have shown how spatial variations in the ionosphere can cause

spatially varying azimuth offsets between master and slave images of an interferomet-

ric pair of SAR images. We have shown that these varying offsets account for the

streaks of decorrelation observed in some InSAR images, and we have demonstrated

that the correlation can be improved by as much as 0.5 by accurately measuring the

azimuth offsets. We have demonstrated our method on two ALOS PALSAR datasets,

and shown that the result works very well over the Greenland ice sheet, because the

scattering properties of the terrain there are approximately wide sense stationary. We

have quantified the improvement in correlation, and also shown how to estimate the

ionospheric phase signature.

Chapter 5

Dry-Snow Zone Firn Modeling

5.1 Introduction

We now turn to the problem of estimating accumulation rates from observed InSAR

data, which we formulate as a mathematical inverse problem. This chapter concerns

the forward problem, showing how we can predict the InSAR data resulting from any

given accumulation rate. The next chapter deals with the inverse problem.

In the inverse problem formulation, we denote a vector of data values by ~d. Thus,

using a radar brightness σ0 and volume coherence ρvol as our data, we could write

~d =

σ0

ρvol

(5.1)

The model parameter of interest in this case is accumulation rate A. For a single

location, the vector of model parameters is simply

~m = [A] (5.2)

80

CHAPTER 5. DRY-SNOW ZONE FIRN MODELING 81

The observed data and model parameters are related by

~d = G~p(~m) (5.3)

where the function G~p(·) represents the underlying physics. For our experiment, this

function depends on a vector of geophysical and imaging system parameters

~p =

h

T

B⊥

θ

(5.4)

where h and T are the location’s elevation above sea level and mean annual tempera-

ture, respectively, and B⊥ and θ are the InSAR perpendicular baseline and incidence

angle (see Chapter 3).

This formulation is illustrated conceptually in Figure 5.1. To calculate ~d from an

accumulation rate, we first use a geophysical model to calculate the firn structure

resulting from that accumulation rate. This is discussed in Section 5.2. Then we use

an electromagnetic scattering model to calculate the InSAR data resulting from that

firn structure. This is discussed in Section 5.3.

As an illustrative example of how the forward model works, we show numerical

results from a site called B-26, whose location is shown in Figure 5.2. We chose this

site because in-situ firn structure measurements are available. We will show that our

forward model of firn structure fits in-situ data for this site, and that, for simulated

data, a simple inversion recovers the accumulation rate.


Figure 5.1: Conceptual illustration of forward and inverse problem formulation. Givenan accumulation rate along with relevant auxiliary data, we can simulate the resultingInSAR data by first calculating the firn structure (as explained in Section 5.2) andthen using an electromagnetic scattering model (as explained in Section 5.3).


Figure 5.2: The study location designated B-26 is indicated by the red star. B-26was one of the study sites for the North Greenland Traverse (NGT) conducted in1993-1995. The green boundary shows the edge of the dry-snow zone.


5.2 Geophysical firn model

In the geophysical portion of the forward model, we compute the firn grain size rg and

density1 γ, which are both functions of depth z. In this chapter we let z represent

depth below the surface of the ice sheet (z is positive and increases going downward).

The grain size and firn density depend on accumulation rate as well as surface

elevation and mean annual temperature. Recall that we know the surface elevation

from the DEM (see Chapter 2). For mean annual temperature, denoted T , we use a

linear model developed by Ohmura [Ohmura, 1987]:

T = 48.38− 0.007924h− 0.7512Φ (5.5)

Here h is surface elevation in meters above sea level, and Φ is latitude in degrees.

Intuitively, the negative coefficients in front of h and Φ mean that mean annual tem-

perature decreases with increasing elevation and increasing latitude. This model was

developed by linear regression using temperature data from in-situ weather stations.

5.2.1 Firn Density

As glacial firn is buried under successive years’ accumulation, its density increases.

Thus the density of glacial firn increases with depth. This densification, driven pri-

marily by the weight of the ice, happens in several stages and via several mechanisms.

This is a complicated process that has been described and quantified in many ways

(see, for example, [Cuffey and Paterson, 2010, Sec. 2.5] and [Herron and Langway,

1980]).

Freshly fallen snow, whose density is typically 300kg/m3 to 400kg/m3, first under-

goes densification as its grains settle and pack more closely. This proceeds until the

1The term density in this chapter always refers to mass density.


density reaches around 550 kg/m3. In the next stage, snow grains deform and grow

as ice molecules rearrange themselves, sublimate, and refreeze. This continues until

the air spaces between grains of snow close off, forming disconnected air bubbles. At

this point the density is around 830 kg/m3. Finally, the firn continues to increase in

density as air bubbles get compressed. Eventually, the firn density approaches the

density of pure ice, 917 kg/m3.

The densification process depends on accumulation rate because the accumulation

rate determines the rate at which the pressure from the weight of ice above a given

layer (overburden) increases. In addition, the densification process also depends on

temperature, since molecular scale ice crystal deformation, sublimation, and refreez-

ing are all thermodynamic processes. For our model, we assume that accumulation

rate and temperature are the only two variables needed to compute a density profile.

Because many physical models of densification depend on many unknown quan-

tities or else do not fit in-situ observations perfectly, we have chosen to use a semi-

empirical exponential model developed by Linow instead. In this model, the firn

density γ is related to depth z by the exponential relationship

γ(z) = a0 exp(a1z) + a2 (5.6)

where the coeffiecients depend on mean annual temperature and accumulation rate

by

a0 = −557.93 + 1.27T + 0.6621A (5.7)

a1 = −0.04193− 0.00054T + 0.0000257A (5.8)

a2 = 856.92− 2.71T − 0.0417A (5.9)

for accumulation rate A in cm/yr w.e., temperature T in degrees Celcius, and density


Figure 5.3: Comparison of in-situ and modeled firn density at B-26.

γ in kg/m3. Linow determined these coefficients by linear regression from 11 dry-

snow zone ice cores from Greenland and Antarctica [Linow, 2011]. While it would be

relatively straightforward to use other models in our analysis, this model has the key

advantange that it closely matches recent in-situ measurements.

Figure 5.3 shows the model and in-situ data for density versus depth at the B-26

site. The in-situ data are from [Schwager, 2000]. The model matches the available

in-situ data well, and also approaches the density of pure ice, 917 kg/m3, at large

depths.

5.2.2 Firn grain radius

As firn density increases, its average grain radius also increases. Firn grains grow as ice

molecules move from smaller grains to larger grains to reduce the overall free energy of

the system by reducing the total surface area. Firn grain radius is important because


larger grains generally result in more backscatter, which increases radar brightness.

Grain growth can be described by the Arrhenius equation, a relationship often

used by chemists for the temperature dependence of reaction rates [Drinkwater et al.,

2001]. This equation gives grain radius rg as a function of time t by

rg(t)2 = k0 exp(−E/RT )t+ r2

g0 (5.10)

where k0 is a scaling constant, E is the activation energy of the grain-growth process,

R is the ideal gas constant, T is absolute temperature, and rg0 is the initial grain size.

For grain-growth in polar firn, k0 is approximately 6.75 × 107 mm2/yr [Drinkwater

et al., 2001]. An empirical relationship between temperature, accumulation rate, and

initial grain size is given by

rg0 = 0.781 + 0.0085T − 0.00279A (5.11)

where T is in degrees Celcius, A is in cm/yr w.e., and rg0 is in mm [Linow et al.,

2012]. We have also found that an activation energy of E = 44.1 kJ/mol results in

the minimum mean-square error fit between the model and the in-situ data at the

B-26 site. This matches the values found by other authors, such as E = 42.4 kJ/mol

given in [Cuffey and Paterson, 2010], fairly closely.

Finally, to find grain size as a function of depth, we can relate firn age to firn

depth by

t(z) =

∫ z

0

γ(z′)dz′

A · γice(5.12)

where γice is the density of pure ice. Thus, for an array of depths z, we can calculate

the firn age t at each depth, and the corresponding grain radius rg using Equation 5.10.

A comparison between the in-situ data and the model results is shown in Figure 5.4.

Because of the limited availability of in-situ measurements of grain radius in


Figure 5.4: Comparison of in-situ and modeled grain radius at B-26. The in-situ dataare from [Linow et al., 2012].

Greenland, especially for depths greater than 10 m, firn grain radius is one of the

major sources of uncertainty in our forward model. We do find that the grain radius

values used in this study are also similar to those reported in other studies of glacial

firn, such as [Alley et al., 1986]. Further investigation of the variation of grain size

with depth and other factors is left for future work.

5.3 Electromagnetic scattering model

In the previous section we explained how we can calculate two key components of

firn structure, density and grain radius, using accumulation rate along with mean

annual temperature and surface elevation. We now turn to the second component

of the forward model: computation of simulated InSAR data using electromagnetic

scattering equations (refer again to Figure 5.1).


The interaction of electromagnetic waves with polar firn is described by Maxwell’s

equations. However, due to the complexity of the scattering interaction, it is difficult

to solve Maxwell’s equations for the scenario in this study using either analytical

or numerical approaches. We instead use a model based on the radiative transfer

approach [Chandrasekhar, 1960; Ishimaru, 1978; Kendra, 1995]. This approach is

based on a consideration of the transport of energy as the radar beam propagates

through the scattering medium.

The radiative transfer approach is based on the equation of transfer :

∂I

∂s= −κeI + Je +

∫4π

P (~s, ~s′)IdΩ′ (5.13)

Here I is the intensity (in W ·m−2 · sr−1 · Hz−1) in the s-direction, and ds is a dif-

ferential distance in the propagation direction. The change in intensity, ∂I∂s

, is given

by the sum of three terms. The first term represents extinction, which is caused by

absorption and scattering. It is quantified by the extinction coefficient κe, which is

the sum of the absorption coefficient κa and the scattering coefficient κs:

κe = κa + κs (5.14)

The negative sign in front of this term indicates that absorption and scattering both

reduce the intensity of the radar beam. The second term, Je, represents thermal

radiation emitted by the scattering medium, which is governed by Planck’s law. This

term is negligible for active radar systems, with which this study is concerned. Finally,

the last term in Equation 5.13 represents contributions from intensities propagating

in all other directions ~s′ scattering into the s-direction. Energy entering the beam

in this way is due to multiple scattering, which we will generally neglect by taking

P (~s, ~s′) ≈ 0.


5.3.1 Scattering model

Scattering and Absorption Coefficients

To model scattering from snow grains, we use the Rayleigh scattering formulation.

This assumes that the snow grains are small compared to the wavelength, which is

accurate in this case (since λ = 23.61 cm). It also assumes that snow grains are

spherical, which is approximately true for most polar firn. For a volume consisting

of spheres in air, the classical expression for scattering coefficient, assuming Rayleigh

scattering, is

κs = 2γ

γice

(2π

λ

)4

r3g

(εi − 1

εi + 2

)2

(5.15)

where εi = 1.9 is the (real part of the) relative permittivity of ice [Ishimaru, 1978]. As

before, γ is the firn density, γice is the density of pure ice, and rg is the grain radius.

Note that λ is the free-space wavelength.

The absorption coefficient for firn can be modeled by

κa =2π

λ

0.00033(0.52γ + 0.62γ2)√εi

(5.16)

where λ is in meters and γ is in g/cm3. This is again based on empirical measurements

of dry snow [Matzler, 1987].

Note that this relatively small absorption coefficient means that dry snow acts like

a dielectric. This model does not apply to wet snow (or even moist snow), because the

liquid water in those materials make them much more lossy media. Also, salts or other

ionic impurities can make ice much more lossy, but we assume the concentrations of

these are relatively low in the dry-snow zone.


Radar Cross-section

We can now use the absorption and scattering coefficients to calculate σ0 and ρvol

using relationships derived by Hoen [Hoen, 2001]. We will not repeat the derivations

here, but we give the equations used in our model with explanations of their physical

meaning.

We can calculate σ0 for a volume scattering situation by integrating the backscat-

ter over all depths:

σ0 =

∫ ∞0

σv(z)dz (5.17)

where σv is the volumetric radar cross-section. Intuitively, this means that the total

backscatter power is equal to the (incoherent) sum of backscatter powers from all

depths. In practice, we calculate this integral by forming a discrete, truncated sum

over thin layers of thickness h:

σ0 ≈Nmax∑n=0

σv(n) (5.18)

The σv for each layer is given in terms of scattering and extinction coefficients by

σv(n) =3κs(n)h

2exp

(−2h

n∑m=1

κe(m)

cos(θr(m))

)(5.19)

where θr is the incidence angle of the refracted wave, given by Snell’s law as

θr = sin−1

(sin θ√εi

)(5.20)


Volume correlation

In addition to σ0, we can also calculate volume decorrelation ρvol by

ρvol =

∣∣∣∣∣∣∑

n σv(n) exp(i4πλh∑n

m=1

√εδθ(m)

sin θr(m))∑

n σv(n)

∣∣∣∣∣∣ (5.21)

The numerator of this expression can be interpreted as a phasor sum (or coherent sum)

of the radar returns at each depth. The denominator is analogous to the normalization

factor used in the definition of coherence (see Equation 3.15). In Equation 5.21, the

incidence angle difference between the two SAR images, δθ, is related to perpendicular

baseline by

δθ =B⊥R0

cos θ√εi cos θr

(5.22)

We have shown that accumulation rate and other geophysical parameters deter-

mine firn structure, which determines ρvol. In order to use the measured InSAR

coherence ρ to estimate accumulation rate, we must isolate ρvol from the product


When signal processing is done accurately, we have ρother ≈ 1, and for SAR scenes

with high SNR, we also have ρtherm ≈ 1. For this study, we have chosen the InSAR

scenes with the shortest available temporal baseline, so that we can take ρtemp ≈ 1.

This means that using our analytic expression for ρsurf (see Equation 3.30), we can

approximately calculate ρvol from ρ.

5.3.2 Numerical results and discussion

We now illustrate the electromagnetic scattering model by showing results for the

B-26 test site. Figure 5.5 shows plots of modeled volumetric cross-section, σv(z) as


Figure 5.5: Modeled volumetric radar cross-section, σv(z), for the B-26 site, for L-band (top) and C-band (bottom) radars. The radar cross-section, σ0, is the integralof the curves shown.

a function of depth z at the B-26 site. The top plot is for the ALOS PALSAR L-

band radar. For comparison, the bottom plot shows the results for a C-band radar.

The peak of the σv(z) curve occurs at a larger depth for the L-band radar than for

the C-band radar, indicating that the L-band radar wave penetrates deeper beneath

the surface of the ice. The total radar cross-section is given by the integral of these

σv(z) curves, and is about −4.3 dB for the C-band case and −9.8 dB for the L-band

case. These numbers are similar to those found in experimental studies such as those

presented in [Ulaby et al., 1982].

Next, we examine the modeled volume correlation, ρvol. At a given site, such as

the B-26 test site, the modeled ρvol depends on both the radar wavelength and the

InSAR perpendicular baseline. Figure 5.6 shows ρvol as a function of perpendicular

baseline for the B-26 site, for both the ALOS PALSAR L-band radar, and for a

C-band radar with similar orbit geometry.


Figure 5.6: Modeled volume correlation, ρvol, for the B-26 test site.

The simulations show that volume correlation drops to nearly zero at perpendic-

ular baselines of around 200 m for both the L-band and C-band systems. In the

L-band case, the radar return comes from a greater range of depths, which tends to

lower ρvol. This, however, is offset by the fact that the larger wavelength causes ρvol

to decrease more slowly with B⊥. The net result is that the curves for L-band and

C-band systems look similar. These simulation results constrain the InSAR baselines

that are useful for accumulation rate studies.

It is illustrative to examine how the simulated data vary with accumulation rate.

Figure 5.7 shows simulated σ0 and ρvol data for a realistic range of accumulation

rates for the Greenland Ice Sheet. We used the wavelength and geometry of the

ALOS PALSAR system, with a perpendicular baseline of B⊥ = 100 m. The data are

monotonic functions of accumulation rate for all temperatures, which indicates that

it is possible to invert the curves and estimate accumulation rate from InSAR data.

Also, the slope of these plots reveals the sensitivity of our accumulation rate mea-

surements. For some situations, such as accumulation rates of around 35 cm/yr w.e.


(a) Radar cross-section, σ0

(b) Volume correlation, ρvol

Figure 5.7: Simulated InSAR σ0 and ρvol for ALOS PALSAR at a realistic range ofaccumulation rates for the Greenland Ice Sheet.


Figure 5.8: Objective function for estimating accumulation rate at B-26 site usingsimulated data.

and temperatures of about −35C, the σ0 is relatively small, which results in low

SNR, and the volume correlation varies very slowly with accumulation rate. This

means that small uncertainties in ρvol can create large uncertainties in accumulation

rate. Thus this type of simulation helps us understand the relevant challenges for our

approach to accumulation rate estimation.

Finally, to verify that our software is consistent, we confirm that we can recover

the true accumulation rate at the B-26 test site from our simulated data, dsim. This

can be done by defining an objective function

f(A) = ‖G(A)− dsim‖2 (5.24)

and solving the optimization problem

A = argmin f(A) (5.25)


Then A is the estimated accumulation rate, which should equal the true accumulation

rate for the ideal case of using simulated data. Figure 5.8 shows a plot of the objective

function. The minimum occurs at the true accumulation rate (A = 17 cm/yr w.e.),

as we expect. Also, note that the objective function is convex, which means it has no

local minima and the solution is unique. In the next chapter, we examine the results

of performing this type of inversion with ALOS PALSAR data.

Chapter 6

Accumulation Rate Case Study

Results

6.1 Introduction

In this chapter we examine SAR and InSAR data using the model presented in the

previous chapter.

We present two case studies. The first uses available SAR mosaic images of the

Greenland Ice Sheet. We extend previous work on C-band data from ERS-1/2 to an

L-band mosaic formed from L-band ALOS PALSAR data. The second case study uses

L-band InSAR data from a long transect that traverses the entire dry-snow zone, and

gives us the opportunity to conduct a large-scale comparison between InSAR data

and in-situ accumulation rate measurements.

We examine the relationship between simulated radar data, derived by applying

the forward model to in-situ accumulation rate measurements, and experimentally

observed radar data. Thus we can use our forward model to explain the observed

radar data. We also explain our regularized inversion approach, and invert our model

to estimate accumulation rate from InSAR data. We show and interpret our inversion

98

CHAPTER 6. ACCUMULATION RATE CASE STUDY RESULTS 99

results, along with approximate uncertainty intervals.

6.2 Case Study: Radar Mosaic

As a first case study, we examine mosaic images covering most of Greenland made

using both ERS and ALOS PALSAR.

Figure 6.1a shows a SAR image of Greenland made by forming a mosaic of many

SAR scenes acquired by the C-band SAR instrument on ERS-1/2. The data are

from [Fahnestock et al., 1997]. The data cover nearly the entire ice sheet. We have

also denoted the approximate dry-snow zone boundary on the image using the green

outline. The SAR mosaic image shows that the dry-snow zone generally has lower σ0

than the rest of the ice sheet, and the σ0 in the dry-snow zone is generally lower in

regions of higher accumulation rate (cf. Figure 2.7).

Figure 6.1b shows a similar mosaic made using the L-band ALOS PALSAR instru-

ment. The data are provided by the Japan Aerospace Exploration Agency (JAXA).1

We can examine the results of our forward model by comparing the σ0 data values

predicted by the model with the actual σ0 data measured by both the C-band and

L-band instruments. That is, we use in-situ accumulation rate measurements, Ainsitu

to compute predicted data values, dpre = G~p(Ainsitu), and we compare these against

the actual measured data, dmeas.

We have performed this comparison for all points within the dry-snow zone at

which in-situ measurements are available, excluding those within 125 km of the dry-

snow zone boundary. Points close to the boundary are excluded because the gradual

transition between the dry-snow and percolation zones means that the forward model

is less accurate near the boundary. We have also excluded points at which no SAR

data are available due to gaps in the mosaic. The points used in the comparison are

1Accessed at http://www.eorc.jaxa.jp/ALOS/en/kc mosaic/kc 500 arcticpole.htm, 2012.


(a) ERS SAR mosaic (b) ALOS SAR mosaic

Figure 6.1: Mosaic of SAR images covering most of Greenland using ERS (a) andALOS PALSAR (b). The green outline denotes the approximate dry-snow zoneboundary. The blue points indicate locations of in-situ accumulation rate measure-ments, excluding those within 125 km of the dry-snow zone boundary.


(a) ERS SAR mosaic (b) ALOS SAR mosaic

Figure 6.2: Scatter plots of SAR σ0 predicted by our forward model versus measured(though un-calibrated) σ0 from ERS (a) and ALOS PALSAR (b). The black line isthe best-fit line in the least-squares sense.

shown by the blue points in Figure 6.1.

The results of this comparison are shown in Figure 6.2. The plot in Figure 6.2a

is a scatter plot of σ0 values predicted by our forward model versus measured (but

un-calibrated) σ0 values from ERS-1/2. Figure 6.2b shows a similar scatter plot

using data from ALOS PALSAR. The correlation coefficient between predicted and

measured data values is 0.84 for the C-band data, and 0.56 for the L-band data. The

C-band results are very similar to those previously published in [Munk et al., 2003].

Here we have extended the results to include L-band data.

The results of this case study indicate that the forward model generally explains

the variations in dry-snow zone radar brightness, but they also show that estimating

accumulation rate using only σ0 is less accurate for L-band data than for C-band

data. Therefore this case study shows the need for development of additional InSAR

techniques beyond state-of-the-art SAR σ0 techniques when using L-band data for

accumulation rate studies. Fortunately, L-band InSAR data generally has better

temporal correlation, allowing us to use ρvol for accumulation rate studies over a

larger range of temporal baselines.


6.3 Regularized Inversion

We have already shown in Section 5.3.2 how we can use our forward model to estimate

accumulation rate from InSAR data by solving an inverse problem. Before applying

this to our transect case study (see Section 6.4), we extend the inverse problem

formulation to a regularized inversion, which we use to estimate accumulation rate

from noisy radar data without overfitting.

6.3.1 Problem Formulation

As before, we want to solve for accumulation rate A(~r), which we now consider to be a

function of position ~r. Equivalently, the set of model parameters that we are solving

for is a vector ~m consisting of accumulation rates at (adjacent) points of interest,

A(~ri), for N locations ~r1 · · ·~rN .

We add the term α‖L~m‖2 to our previous objective function (Equation 5.24) in

order to penalize solutions that overfit the data. Here, L is derivative operator, and

the parameter α controls the smoothness of the solution. The estimated accumulation

rate is thus the solution to the optimization problem

A(~r) = arg minA(~r)

[‖~d−G~p(~m)‖2 + α‖L~m‖2

](6.1)

This is a non-linear least-squares problem, which we solve using the Gauss-Newton

method, with partial derivatives in the Jacobian matrix estimated empirically using

finite differences.

6.3.2 Uncertainty Analysis

Here we develop a simple framework for approximating the uncertainty associated

with estimating accumulation rate from InSAR data. Recall from Chapter 3 that the


standard deviation of estimated coherence, ˆρvol, is approximately given by

std(ρ) =1− 〈ρ〉2√

2L(6.2)

As a first order approximation for cases in which ρvol is the dominant factor in the

coherence, we can apply this approximation to ρvol as

std( ˆρvol) ≈1− ˆρvol

2

√2L

(6.3)

This approximation works best for higher correlation values, but we take it as a first-

order approximation for lower correlation values as well. For this section we also

assume that the effective number of looks, L, is sufficiently large that estimation bias

is negligible, so that we can take 〈ρ〉 ≈ ρ.

If we linearize the model G~p(~m) around a particular estimated accumulation rate

A, we can write, approximately

std( ˆρvol) =

∣∣∣∣∂ ˆρvol∂A

∣∣∣∣ std(A) (6.4)

or equivalently

std(A) =

∣∣∣∣∂ ˆρvol∂A

∣∣∣∣−1

std( ˆρvol) (6.5)

This last equation gives an approximation of the uncertainty in estimated A using

InSAR correlation data. While this model neglects non-linearities and the covariance

between accumulation rate estimates at adjacent points introduced by regularization,

it shows the main factors upon which uncertainty in estimated accumulation rate

depend: uncertainty in ρvol, and the sensitivity of ρvol to accumulation rate, quantified

by ∂ ˆρvol∂A

.


Date (MM/DD/YYYY) Orbit # Frame #04/13/2007 06490 1400 to 163005/29/2007 07161 1400 to 1630

Table 6.1: ALOS PALSAR data used in this study. The temporal baseline is 46 days.

6.4 Case Study: Dry-Snow Zone Transect

6.4.1 Data and InSAR processing

In this case study we examine L-band InSAR data that covers a long transect travers-

ing the dry-snow zone of the Greenland Ice Sheet. The location of the transect is

shown in Figure 6.3. This particular study location was chosen because it contains

data that lies close to many in-situ data points, which facilitates a comparison be-

tween InSAR and in-situ accumulation rate estimates. The transect covers a wide

range of accumulation rates, and consists of 24 frames of ALOS PALSAR data.

For convenience, we also define a u − v coordinate system with coordinate u

measuring distance along the transect, using the southwest corner of the transect as

the origin. This is approximately parallel to the azimuth direction. The v axis is

orthogonal to the u axis, and is approximately parallel to the ground range direction.

In this coordinate system, the dry-snow zone extends from approximately u = 27.7 km

to 1297 km. Figure 6.4 shows the InSAR perpendicular baseline as a function of

distance along the transect. Finally, Table 6.1 lists the SAR acquisitions used for this

study.

After forming interferograms from the 24 pairs of SAR acquisitions, we remove

topography phase and digitally stitch together the resulting flattened interferograms

to form the transect. The resulting combined interferogram is shown in Figure 6.5.

The phase in the interferogram represents residual topography due to inaccuracies

in the DEM as well as phase due to ice motion. Note that in the dry-snow zone there

is an absence of high phase gradients that would hinder coherence estimation.


Figure 6.3: Map showing the location of the transect used for this case study. Ap-proximate dry-snow zone boundary and in-situ data point locations are also shown.


Figure 6.4: Perpendicular baseline for InSAR data as a function of distance along thetransect.

Figure 6.5: Interferogram for transect case study. Note the different distance scalesfor the horizontal and vertical axes.


(a) Photo (b) Interferogram

Figure 6.6: Photograph (a) and interferogram (b) showing Summit Camp, Green-land. (Photograph is from http://www.flickr.com/photos/coastaleddy/6961187802/,accessed 2012.)

Although the inner region of the ice sheet looks relatively featureless, as we have

seen before, this particular transect does cover Summit Camp, a research station near

the ice sheet’s highest point. The skiway at Summit Camp is relatively easy to see

in the InSAR data, as shown in Figure 6.6.

6.4.2 Forward Model Results

Before using the InSAR data to estimate accumulation rate, we examine the forward

model results, assuming that the interpolated in-situ accumulation rates along the

transect are accurate.

First we compare the measured σ0 and the simulated σ0 predicted by the forward

model. This is shown in Figure 6.7. In these plots, we plot the values at the InSAR

beam center as a function of distance along the transect in order to facilitate quan-

titative comparison. We have empirically calibrated the σ0 data so that they match

the model results around the middle of the dry-snow zone.

The results indicate that both the measured and predicted σ0 show litte variation

over the dry-snow zone. There is, however, an increase in measured σ0 at the ends of

the transect, which is probably due to brighter radar returns in the transition region

between the dry-snow and percolation zones.


Figure 6.7: Simulated σ0 predicted by forward model (red curve) and measured σ0

(blue curve) as a function of distance along the transect.

Figure 6.8: Measured ρvol (green curve) and ρvol predicted by forward model (tealcurve) as a function of distance along the transect. The calculated values of ρsurf areshown by the red curve. We obtain ρvol by dividing measured ρ (blue curve) by ρsurf .Because ρsurf is close to unity, the curves for ρ and ρvol nearly overlie each other. Thepurple interval shows the simulated uncertainty in ρvol.


Figure 6.9: Estimated penetration depths along the transect.

Next, we compare predicted and measured coherence values along the transect,

shown in Figure 6.8. The simulated ρvol predicted by applying the forward model

to in-situ accumulation rate along the transect is shown by the teal curve. For com-

parison, the measured ˆρvol is shown by the green curve. For reference, we have also

shown the ρsurf calculated from Equation 3.30. Because the perpendicular baseline

is much smaller than the critical baseline, surface correlation is very close to unity

for this data. The purple interval shows the simulated uncertainty in ˆρvol due to the

standard deviation of the correlation estimator. These results show that our model

correctly predicts the trends in the coherence data.

We can estimate the penetration depth dpene of the L-band radar waves using the

ρvol data using a formula given in [Hoen, 2001]:

ρvol =1√

1 +(

2π√εdpeneB⊥

R0λ tan θ

)2(6.6)


Figure 6.10: Accumulation rates estimated from σ0 data. The red curve shows the es-timated accumulation rate, while the purple interval shows the estimated±1-standarddeviation uncertainty interval.

The results of applying this formula are shown in Figure 6.9. Note that temporal cor-

relation and any other decorrelation effects that are not modeled cause over-estimation

of penetration depth. In general, however, the results indicate that penetration depths

in the dry-snow zone can be about 100 m to 250 m, which agrees well with the model

results shown in Figure 5.5.

6.4.3 Inversion Results

Now we show the results of inverting our model to estimate accumulation rates from

our InSAR data. Figure 6.10 shows the result of performing this inversion using σ0

data. For comparison, we also show the interpolated in-situ accumulation rate data

along the transect.

Here, the estimated accumulation rate has approximately the correct mean, partly

as a result of our empirical calibration. As we have seen, there is less variation in


Figure 6.11: Accumulation rates estimated from ρvol data. The red curve shows theestimated accumulation rate. The purple interval shows the estimated ±1-standarddeviation uncertainty interval. For comparison, the in-situ accumulation rate is alsoshown in blue.

L-band σ0 over the dry-snow zone, which tends to make it more difficult to estimate

accumulation rate using L-band σ0 data. For reference, we have also shown the ±1

standard deviation uncertainty interval for our estimates. In this case, we have used

the local empirical standard deviation of σ0 as std(σ0), and

std(A) =

∣∣∣∣∂σ0

∂A

∣∣∣∣−1

std(σ0) (6.7)

which is analogous to Equation 6.5.

We have also performed the inversion using ˆρvol, as shown in Figure 6.11. Again,

the estimated accumulation rate is shown, along with the estimated ±1-standard

deviation uncertainty intervals. For comparison, the in-situ accumulation rates are

also shown.


We note that uncertainties for this estimation are larger for regions where coher-

ence is lower. Shorter spatial baselines would thus help to improve the precision of

these estimates, though no ALOS PALSAR data with shorter spatial baselines is cur-

rently available. Also, better constraints on temporal correlation or shorter temporal

baselines would help to more accurately separate out ρvol from ρ.

Several observations concerning these inversion results are germane. First, the

results indicate that additional work is needed to develop a model and inversion

technique that is able to reproduce the in-situ results more precisely. We have shown

here that using InSAR coherence produces better results than simpler techniques

based on radar amplitude measurements alone. Further development using additional

data or model parameters could yield still more precise results.

Second, although our inversion results do not exactly match in-situ results, our

forward model tends to match the experimental data fairly well. This is likely because

the inversion accuracy is limited by the geometrical imaging parameters, which are

not optimized for accumulation rate studies. InSAR instruments are usually designed

to be suitable for a variety of applications, but data from a mission with parameters

optimized for ice sheet studies could yield better inversion results.

Finally, the L-band radar waves used in this study generally penetrate deeper

into the firn of the dry-snow zone than most of the ice cores used in the in-situ

accumulation rate measurements. It is therefore also possible that, when averaged

over a longer time, the true accumulation rate is closer to those that we have estimated

than to the in-situ values shown here; we do not currently have enough in-situ data

on historical accumulation rates with sufficient spatial resolution to test this.

Chapter 7

Conclusions

7.1 Summary

Data from the ALOS PALSAR L-band InSAR instrument gave us the opportunity

to examine the dry-snow zone of the Greenland Ice Sheet from a remote sensing

perspective. We developed signal processing methods and software for forming L-band

InSAR images of the Greenland Ice Sheet. We showed that azimuth offsets between

pairs of SAR acquisitions can be used to reduce decorrelation caused by ionospheric

effects in L-band InSAR data and to estimate the ionospheric phase signature in

an interferogram. We also developed software to geolocate InSAR pixels in a polar

stereographic coordinate system, to remove topography phase using an up-to-date

DEM, and to form geocoded InSAR images of long transects of the Greenland Ice

Sheet.

We then used InSAR data for the scientific application of estimating accumulation

rate in the dry-snow zone. We showed that a volume-scatter model can be used to

explain our L-band radar brightness and InSAR coherence data. We presented a large-

scale comparison between L-band InSAR measurements and in-situ accumulation rate

measurements along a transect that traverses the entire dry-snow zone. Our results

113

CHAPTER 7. CONCLUSIONS 114

indicate that estimates of accumulation rate based on L-band InSAR coherence have

the potential to perform better than estimates based on radar backscatter power

alone.

Specifically, the key contributions presented in this thesis are as follows:

1. We have implemented software to form geocoded InSAR images of long tran-

sects of the Greenland Ice Sheet by stitching together frames processed using a

motion-compensation InSAR processor. To avoid geometric distortion at high

latitudes, we geolocate pixels in a polar stereogrpahic projection.

2. We have shown that ionospheric propagation effects can interfere with accurate

measurements of InSAR phase and coherence in L-band studies of the Greenland

Ice Sheet. As explained in Chapter 4, we have developed a signal processing

technique for compensating for these effects.

3. We demonstrated that a volume-scatter model can be used to explain ALOS L-

band InSAR measurements of Greenland. We derive the first spaceborne InSAR

coherence measurements and estimates of accumulation rate from L-band SAR

data.

4. We presented an error analysis for the measurement of accumulation rate from

InSAR coherence data. We also show that estimates of accumulation rate based

on L-band InSAR coherence measurements perform better than estimates based

solely on radar backscatter brightness.

5. We implemented a large-scale comparison between in-situ accumulation rate

measurements and L-band InSAR measurements. The comparison is done along

a transect that traverses the entire dry-snow zone of the Greenland Ice Sheet.


7.2 Future Work

Science and engineering work is really never finished, as improvements and further

developments will certainly continue to contribute to our understanding of the Green-

land Ice Sheet. In particular, here are several interesting directions for further study:

1. Further development of methods for identifying and correcting ionospheric ef-

fects in L-band InSAR data would further improve our ability to use L-band

InSAR data to study geophysical phenomena. Range split-spectrum processing,

polarimetric Faraday rotation measurements, and other techniques could yield

additional ionospheric signals that could be used in conjunction with our az-

imuth offset method. L-band InSAR data from systems with a dual-frequency

design or other features may yield further opportunities for improvement. Us-

ing ionospheric measurements from other instruments (such as GPS) to correct

ionospheric effects in InSAR data also remains an active research area.

2. Refinement of the forward model for firn structure and electromagnetic scatter-

ing would help to improve the accuracy of accumulation rate estimates derived

from remote sensing data. In particular, additional in-situ data on firn grain

radius and density would help to refine the geophysical model. Enhanced elec-

tromagnetic scattering models that account for multiple scattering or other

higher order effects could prove helpful.

3. Additional remote sensing data would help to further improve the accumulation

rate estimation. For example, L-band InSAR data acquired with zero spatial

baseline and a variety of temporal baselines could help us better understand

how temporal correlation varies with temporal baseline over different regions

of the ice sheet. Also, inversions that incorporate data from a variety of SAR

instruments with varying wavelengths and incidence angles could prove to be


more robust.

4. Unusually widespread melting occurred at or near the surface of the Greenland

Ice Sheet on July 12, 2012. Observations from several satellite sensors show that

melt occurred across 98.6% of the ice sheet surface, coinciding with unusually

warm air over Greenland. The melt event created a melt layer which, over large

areas, subsequently refroze into an ice crust several centimeters thick [Nghiem

et al., 2012]. Historical ice records show that this type of event occurs less than

once a century. Although this melt event occurred after the data for this study

were acquired, it does present interesting challenges and opportunities for future

work. L-band InSAR instruments are less sensitive to anomalous melt layers

because of larger penetration depths. This is could be an advantage to using L-

band data for estimating accumulation rates for data acquired soon after 2012.

On the other hand, detecting and modeling the melt layer from the 2012 melt

event using remote sensing data could yield new insights into the geophysics of

the Greenland Ice Sheet.

As remote sensing technologies continue to improve and climate change increas-

ingly impacts our lives, we hope that the work reported in this thesis represents

one useful step in the continued effort to understand the complex geophysics of the

Greenland Ice Sheet.

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