InSAR time series analysis of the 2006
Slow Slip Event on the Guerrero
Subduction Zone, Mexico
MSc. Thesis
David Bekaert
December 26, 2010
Student:
David Bekaert
Email: D.P.S.Bekaert@student.tudelft.nl
Mobile: (+31) 64 24 59 440
Supervisor: Dr. Andrew Hooper
Email: A.J.Hooper@tudelft.nl
Telephone: (+31) 15 27 82574
Delft Institute of Earth Observation and Space Systems
Faculty of Aerospace Engineering
Delft University of Technology
All Rights Reserved
c
2010
By Bekaert David
Preface
The report presented here is written as part of my master graduation in Aerospace
Engineering at Delft University of Technology. I started my master in the field of
Earth and Planetary Observation with a focus towards Radar Remote Sensing. As
the first year progressed, my interest did grow towards the science studied with remote Sensing Techniques. Therefore I decided to put more emphasis on combining
geodetic techniques and modeling. During my search for an interesting and challenging thesis assignment I consulted dr. Andrew Hooper from the Department of
Earth Observation and Space Systems at our faculty. With his background in geology and radar I certainly was at the right spot. After exchanging ideas, I decided
to dedicate my thesis on the study of Slow Slip Events in Guerrero (Mexico) using
satellite radar as observation technique. This is not an easy task, moreover only a
few people have tried to study SSEs with radar.
Besides the graduation thesis, master students are obliged to fulfill an internship
for at least 3 months at a company or institute. By approaching Andy’s contacts,
we were able to arrange a position at the Jet Propulsion Laboratory of NASA.
Working at JPL was a dream coming to reality. However, the JPL Visiting Student
Researcher Program required the applicant to be self sufficient through grants and
other types of funding. Unfortunately I was not eligible for almost all of the Dutch
and Belgian grants because of my Belgian nationality whilst studying abroad in the
Netherlands. Being granted a teaching assistantship for the Multivariate Data Analysis course did gave me the first finances. However this would not be sufficient. The
internship was not possible without the financial support of the Flemish Institute for
Technological Research (VITO), the fund international internships and my family.
January 10th 2010, about one year after the application, I started my 5-month
lasting internship at JPL. Under the supervision of dr. Sang-Ho Yun I was able
to participate in a few projects, giving me the experience how a research institute
operates. Moreover I was given the opportunity to present some of my graduation
work. My JPL experience convinced me in pursuing a PhD degree.
As my master research progressed, I submitted two abstracts. The first abstract
was submitted for the 2010 Wegener conference in Istanbul. This is a rather small
conference, making it ideal for a first experience in presenting your results to community. My expenses where covered by the International Travel Award granted by
the International Association of Geodesy (IAG).
The second abstract was submitted for the 2010 fall assembly of the American Geophysical Union (AGU) in San Francisco, USA. This is the largest conference in the
i
Preface
community, with this year about 20 000 geophysicists from the whole world attending it. Around November I received the news, which I hoped for, I have been
accepted to present at AGU. Attendance was made possible by extra support from
my parents, TU Delft and the Delft University Fund.
I learned a lot during my master, it allowed me to get the first grips on radar remote
sensing and solid earth processes. I spent time listening to others and joined in the
discussions wherever possible. Along the way I got more familiar with the Linux
and Mac operation system, and software like Matlab, LATEX, DORIS, ROI PAC
and StaMPS. I am really proud of myself, when looking back on what I have accomplished during my master so far. Combining my masters with an honors track
program, doing an internship at a well recognized institute, teaching other students,
and as last being a master student while attending conferences and presenting your
research. All the arranging and preparation work was worth it.
I would like to thank, my parents, sisters and family for their support mentally and
financially. We are almost there! Special thanks to VITO, IAG, Delft University
Fund, Fund International Internships and Van Der Maas Fund for their financial
support. During the last two years, I have met a lot of people that I should thank.
All the colleagues, who were willing to listen to my questions and problems. Special
thanks to Erik Schmidt for proof reading and providing valuable feedback. As last,
I would like to thank my supervisor dr. Andy Hooper for his guidance and support.
I did really like the discussions that we had. As you know, sometimes a small question expanded into an afternoon of discussion. Im hoping I did not interfere in your
schedule too much.
David Bekaert
Delft, December 26, 2010
ii
Contents
Preface
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Contents
iv
Abstract
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1 Introduction
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2 Interferometric Synthetic Aperture
2.1 Radar . . . . . . . . . . . . . . . .
2.2 SAR . . . . . . . . . . . . . . . . .
2.2.1 Resolution . . . . . . . . . .
2.2.2 Radar Coordinates . . . . .
2.2.3 SAR Platforms . . . . . . .
2.3 InSAR . . . . . . . . . . . . . . . .
2.3.1 Decorrelation . . . . . . . .
2.3.2 Atmosphere . . . . . . . . .
2.4 Processing Software . . . . . . . .
Radar
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3 Time series InSAR
3.1 Persistent Scatterer technique
3.2 Small Baseline technique . . .
3.3 Combining PS and SB . . . .
3.4 StaMPS software . . . . . . .
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4 Guerrero Subduction Zone
4.1 Slow Slip Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Subduction interface . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Observed SSEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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5 PS-InSAR processing
5.1 Study feasibility . . . . . . . . . . . . . .
5.1.1 Available GPS data . . . . . . . .
5.1.2 Feasibility . . . . . . . . . . . . . .
5.2 SAR and PS-InSAR processing steps . . .
5.2.1 Focusing and cropping of raw SAR
5.2.2 Interferometry (DORIS) . . . . . .
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PAC)
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iii
Contents
5.2.3
PS selection (StaMPS) . . . . . . . . . . . . . . . . . . . . . .
6 Long wavelength tropospheric signal
6.1 Tropospheric delays from sounding data . . . . . . . . . . . . . . .
6.2 Convential estimation method . . . . . . . . . . . . . . . . . . . . .
6.3 Region decomposition and multiscale tropospheric delay estimation
6.4 Estimation results . . . . . . . . . . . . . . . . . . . . . . . . . . .
7 PS-InSAR analysis of the 2006 SSE
7.1 Extracting 2006 Slow Slip deformation
7.1.1 Functional model . . . . . . . .
7.1.2 Stochastic model . . . . . . . .
7.2 Extracted slow slip deformation signal
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8 Modeling the 2006 SSE
8.1 Slow slip model . . . . . . . . . . . . . . . . .
8.1.1 Fault slip . . . . . . . . . . . . . . . .
8.1.2 Subduction zone interface . . . . . . .
8.1.3 Surface deformations . . . . . . . . . .
8.2 Model constraints . . . . . . . . . . . . . . . .
8.2.1 Slip towards subduction zone only . .
8.2.2 GPS for constraining long wavelengths
8.2.3 Laplacian smoothness . . . . . . . . .
8.3 Model results and discussion . . . . . . . . . .
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9 Conclusions and recommendations
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A GPS time series Guerrero
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B Multiscale tropospheric delay estimation
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C 2006 SSE Small Baseline model
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iv
List of Figures
2.1
2.2
2.3
Radar coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . .
InSAR decorrelation . . . . . . . . . . . . . . . . . . . . . . . . . . .
Interferometric configuration . . . . . . . . . . . . . . . . . . . . . .
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Persistent Scatterer . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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4.1
4.2
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Slow Slip Events in the pacific subduction zones . . . . . . . . . . .
Continuous GPS stations in Guerrero and Oaxaca states . . . . . . .
Guerrero subduction interface . . . . . . . . . . . . . . . . . . . . . .
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5.1
5.2
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5.5
GPS displacements due to the 2006 SSE
Work and processing flow chart . . . . .
Single Master baseline plot of 2006 SSE
Single Master interferograms . . . . . .
PS-InSAR interferograms . . . . . . . .
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6.1
6.2
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Theoretical (integrated) water vapor curve . . . . . . . . . . . . . . .
Refractivity and tropospheric delay estimated from sounding data .
Global tropospheric stratification estimates for Guerrero . . . . . . .
Multiscale tropospheric delay estimation . . . . . . . . . . . . . . . .
Mean tropospheric delay . . . . . . . . . . . . . . . . . . . . . . . . .
Estimating the long wavelength tropospheric signal (Tropospheric
stratification delay) . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Example for combining region decomposition and multiscale tropospheric delay estimation . . . . . . . . . . . . . . . . . . . . . . . . .
Influence of the long wavelength tropospheric signal on the time series
estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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7.1
7.2
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PS-InSAR time series model of the 2006 SSE in Guerrero . .
Including slave atmosphere in the variance-covariance matrix
Estimated master atmosphere and DEM error . . . . . . . . .
Estimated secular motion and 2006 SSE deformation . . . . .
Comparing GPS and PS-InSAR slow slip deformation signal .
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8.1
8.2
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Modeled Guerrero subduction zone . . . . . . . . . . . . . . . . . . .
ENU unit vectors to LOS projection . . . . . . . . . . . . . . . . . .
Laplacian smoothness of patches on subduction interface . . . . . . .
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v
List of figures
8.4
Laplacian Smoothness trade off: misfit-roughness curve and maximum likelihood distribution . . . . . . . . . . . . . . . . . . . . . . .
8.5 Modeled maximum likelihood slip distribution of the 2006 SSE . . .
8.6 Standard deviations of the modeled maximum likelihood slip distribution of the 2006 SSE . . . . . . . . . . . . . . . . . . . . . . . . . .
8.7 Comparing estimated and modeled slow slip deformations . . . . . .
8.8 Comparing the 2006 slip distribution with the slip distribution of
[Correa-Mora et al., 2009] . . . . . . . . . . . . . . . . . . . . . . . .
8.9 Comparing the 2006 slip distribution with the slip distribution of
[Radiguet et al., 2010] . . . . . . . . . . . . . . . . . . . . . . . . . .
8.10 Comparing the 2006 slip distribution with the Ultra Slow velocity
Layer by [Song et al., 2009] . . . . . . . . . . . . . . . . . . . . . . .
vi
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A.1 Guerrero GPS time series of North component . . . . . . . . . . . .
A.2 Guerrero GPS time series of East component . . . . . . . . . . . . .
A.3 Guerrero GPS time series of up component . . . . . . . . . . . . . .
81
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B.1 Multiscale estimation of the stratification coefficient . . . . . . . . .
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C.1 Small Baseline time series model of the 2006 SSE in Guerrero . . . .
92
List of Tables
2.1
2.2
Characteristics ALOS PalSAR . . . . . . . . . . . . . . . . . . . . .
Characteristics Envisat ASAR and ERS SAR . . . . . . . . . . . . .
9
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4.1
Large periodic SSEs in Guerrero . . . . . . . . . . . . . . . . . . . .
24
5.1
5.2
5.3
GPS inter-seismic rates and 2006 SSE displacements . . . . . . . . .
Perpendicular, temporal and Doppler baselines . . . . . . . . . . . .
Persistent Scatterer processing parameters . . . . . . . . . . . . . . .
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vii
Abstract
The uppermost part of the Earth consists of a number of tectonic plates that move
at rates of a few centimeters per year. It is the motion between adjacent plates;
one plate moving with respect to another causes a build up of stress, which is the
driver for earthquakes. During earthquakes slip on faults occurs at high velocities.
However, this slipping can also occur at slower rates. In the last decade Slow Slip
Events (SSEs) have been discovered in many parts of the world, mostly at subduction zones where one plate descends beneath another. While the direct danger from
these SSEs is negligible, their occurrence alters the surrounding stress field, having
implications for the timing of subsequent, potentially damaging earthquakes.
The Guerrero subduction zone, located in the southern part of Mexico, has been
exposed to large earthquakes. Sill there is a region with a ”seismic gap”, where no
earthquakes have occurred since 1911. It is estimated that a rupture of this gap
results in a Mw 8.0 to 8.4 earthquake [Singh and Mortera, 1991]. In the past few
years it has become apparent that the subduction zone is also very active in terms
of SSEs. Since 1995, one large event has been observed every four years (1998, 2001,
2006 and 2010).
Up until now, most geodetic observations of the SSEs have been made using GPS
techniques only. Modeling, using these observations, indicated slip to be extending
into the seismogenic zone, and consequently reducing the stress field in the seismic gap. The slip solution is unfortunately not well constrained due to low spatial
density of the GPS stations. By using radar imaging, spatial resolution can be considerably increased, allowing subduction interface models to be better constrained.
Where radar and Synthetic Aperture Radar are all about measuring distance to a
target on the ground, the Interferometric SAR technique is about the change in
distance with respect to the target. Applying InSAR to Guerrero is challenging
due to high decorrelation noise near the coast and large net atmospheric delays.
However, by using a time series InSAR technique and by developing and applying
new methods for atmospheric correction, reasonable results for the slow slip surface
deformations are obtained that correlate well with the GPS observations.
Both GPS data and the obtained InSAR results were used for modeling of the SSE.
The GPS data constrain the long wavelength errors in the InSAR data, while the
InSAR data themselves constrain the distribution of slip at a higher resolution than
possible with GPS alone. Physically unreasonable slipping values are prevented by
ix
Abstract
allowing only slip towards the subduction zone and by adding smoothness criteria
to the model, estimated using a Bayesian framework.
I show that the regions where slip is constrained by both the GPS and InSAR technique, the slip does not enter into the seismogenic zone. Regions in absence of the
InSAR data tends to model slip downdip and partly in the seismogenic zone.
x
Nomenclature
List of acronyms
ALOS
Advance Land Observation Satellite
ASAR
Advanced Synthetic Aperture Radar
ASTER
Advanced Space borne Thermal Emission and Reflection Radiometer (DEM)
DEM
Digital Elevation Model
DORIS
Delft Object-oriented Radar Interferometric Software
ENU
Components in East, North and Up direction
Envisat
Environmental satellite
ERS
European Remote-sensing Satellite
ESA
European Space Agency
FBD
Fine Beam Dual polarization acquisition mode of ALOS PalSAR
FBS
Fine Beam Single polarization acquisition mode of ALOS PalSAR
InSAR
Interferometric SAR
LD
Look Direction
LOS
Line Of Sight
NVT
Non Volcanic Tremor
PLR
Polarimetric acquisition mode of ALOS PalSAR instrument
PRF
Pulse Repetition Frequency [Hz]
PS
Persistent Scatterer
PS-InSAR
Persistent Scatterer Interferometric Synthetic Aperture Radar
ROI PAC
Repeat Orbit Interferometry PACkage
RSF
Range Sampling Frequency [Hz]
xi
Nomenclature
SAR
Synthetic Aperture Radar
SB
Small Baseline
SNR
Signal to Noise Ratio
SRTM
Shuttle Radar Topography Mission (DEM)
SSE
Slow Slip Event
TEC
Total Electron Content [TECU]
UNAM
National Autonomous University of Mexico
List of symbols
δatm
Atmospheric delay
[m]
∆a
Azimuth resolution
[m]
∆r
Range resolution
[m]
γ
Coherence
[-]
γgeom
Geometric or baseline coherence
[-]
γsat
Flight path angle
[◦ ]
γtemporal
Temporal coherence
[-]
γthermal
Thermal coherence
[-]
λ
Wavelength
[m]
τ
Pulse length
[s]
θ
Angle of incidence
[◦ ]
B
Baseline
b
Constant in linearized tropospheric stratified atmosphere [mm]
BR
Range bandwith
B⊥
Perpendicular Baseline
[m]
Bcrit
Critical perpendicular baseline
[m]
BDoppler
Doppler bandwidth
c
Speed of light
f
Frequency
G
Antenna gain
h
Satellite height
xii
[m]
[Hz]
[Hz]
[m/s]
[Hz]
[-]
[m]
K
Slope relating atmospheric phase to height
[mm/km]
L
Antenna length
P
Power
R
Satellite range
T
Temporal baseline
t
Time or two way travel time
Tcrit
Critical temporal baseline
[days]
vs/c
Satellite velocity
[m/s]
X
Ground range
fDC
Absolute Doppler centroid
Mw
Moment Magnitude
[m]
[dB]
[m]
[days]
[s]
[m]
[Hz]
[-]
xiii
Chapter 1
Introduction
We can’t solve problems by using
the same kind of thinking we used
when we created them.
ALBERT EINSTEIN
On September 19th 1985, Mexico City was hit by a 8.0 Mw earthquake (USGS).
An estimated 10 000 people were killed, 50 000 were injured, and 250 000 lost their
homes − a dark day in Mexican history. To try to avoid a repetition of these disastrous consequences, the Mexican Government made hazard monitoring a priority.
During the last decade, technological developments have led to the discovery of a
”new” type of earthquake, a silent one.
These silent earthquakes, also known as Slow Slip Events (SSEs), form the topic of
this thesis research. Up until now SSEs have been observed mainly by time series
analysis of continuous GPS stations embedded in the Earth’s crust. Using GPS
one is able to constrain the SSEs very well in the temporal domain. Unfortunately
GPS lacks good spatial resolution needed to model where exactly on the subduction
interface, and with what magnitude, slip took place. Moreover the operation and
maintenance of a continuous GPS network is expensive.
The problem of spatial resolution can be addressed using spaceborne Synthetic Aperture Radar Interferometry (InSAR). Where good correlation is maintained, displacement measurements can be obtained approximately every 20 m. Unlike GPS, InSAR
measurements are most sensitive in the vertical direction. Furthermore, SAR images
have been acquired since the launch of the ERS-1 satellite in 1992, allowing us to
probe silent earthquakes that occurred even before the installation of the GPS network. One drawback to InSAR measurements is the level of associated error, mainly
due to the varying delay in propagation through the atmosphere. By applying time
series InSAR techniques, one is able to model the effects of the atmosphere and
DEM errors.
The area under investigation, Guerrero, a province in the southern part of Mexico,
has been exposed to large thrust earthquakes. However there is a region with a
seismic gap, where no earthquakes have occurred since 1911. Scientists have es-
1
1. Introduction
timated that a rupture of this gap would result in an 8.0 to 8.4 Mw earthquake
[Singh and Mortera, 1991]. Moreover, the Guerrero subduction zone is subjected to
a SSE approximately once every four year. Considering the seismic hazard, it is important to know where and with what magnitude slip occurs on the interface. If the
slow slip extends into the seismogenic zone, it is expected to reduce the strain energy
and in that perspective reduce the hazard, while in case it would stop just below
the seismogenic zone it would result in additional loading [Rubinstein et al., 2010].
There are about 27 continuous GPS stations (property of different research groups),
altogether covering an area of 63 000 km2 (approximately one and a half times the
area of the Netherlands). Unfortunately, almost all stations are clustered, resulting
in large observations gaps. Stations inland form a profile perpendicular to the coast.
The InSAR study area should contain multiple GPS stations, in this way InSAR
and GPS can be combined during the modeling of the SSE.
SAR data acquired during the SSE itself is preferred but not a necessity for success.
To be able to successfully extract the slow slip deformation signal, it is required to
have at least three SAR acquisitions from the inter-seismic period before the SSE
and two acquisitions after, or vice-versa. Moreover, the applied time series method,
Stanford Method of Persistent Scatter, requires in total a minimum of 12 images
[Hooper et al., 2007].
Guerrero has been subjected to a SSE in 1995, 1998, 2001, 2006 and 2009/2010
(see table 4.1). Currently there is still discussion if the 2009/2010 SSE that started
around July 2009 has finished or not. Considered the available SAR data and the
requirements described above, satellite platforms ERS I/II/I+II, JERS, ALOS and
RadarSAT were rejected. ALOS covers the 2006 and 2009/2010 SSE, but changed
its viewing angle early 2007, making interferometry impossible for the 2006 SSE. A
full analysis of the current event was not yet possible since during the start of this
research the data was still being acquired. Envisat on the other hand covers the
2006 SSE and fulfills all requirements, having about 35 images for track 255. This
report is focused on the analysis and modeling of the 2006 SSE. The objectives are
formulated as:
1. Extract and visualize the 2006 slow slip deformation signal in Guerrero, Mexico, using time series SAR Interferometry.
2. Modeling the slip of the 2006 SSE using the obtained results.
Thesis outline
Chapter 2 gives the introduction to InSAR. Starting the subject from conventional
radar, it is expanded to SAR and ends with InSAR. Meanwhile information is provided about resolution, radar coordinates, SAR platforms and the limitations of the
InSAR technique. This is followed by chapter 3 addressing two main time series
InSAR techniques, Persistent Scatterer InSAR (PS-InSAR) and Small Baseline approach (SB).
2
Background information on the Guerrero subduction zone is provided in chapter 4.
This chapter starts with a general introduction on slow slip, followed by the Guerrero subduction interface and a description of the large SSEs that have occurred in
Guerrero. Chapter 5 gives an InSAR feasibility study for the 2006 SSE. Moreover,
starting with raw data, it pressents all the PS-InSAR processing steps. Due to the
Mexican tropical climate, Guerrero interferograms are subjected to large amounts of
net atmospheric delays. In chapter 6 I present a new method for estimating the long
wavelength tropospheric signal. Chapter 7 addresses the implemented time series
model of the 2006 SSE, together with its results. These results are then combined
with GPS data in modulation of the 2006 SSE. The methodology behind this modulation together with the results are presented in chapter 8. A work and processing
flowchart is provided in figure 5.2. As last, chapter 9 gives the conclusions and recommendations.
3
Chapter 2
Interferometric Synthetic Aperture Radar
InSAR is a nested acronym between Radio detection and ranging (radar), Synthetic
Aperture Radar (SAR), and interferometric SAR (InSAR). While the backbone
of the technique stated with measuring distance in conventional radar and SAR,
interferometry is about computing the change in distance. Sections below give a
brief overview starting with radar in section 2.1, SAR in section 2.2, interferometric
SAR or (InSAR) in section 2.3 and in the last section the processing software used
in this report for interferogram formation.
2.1
Radar
Perhaps you might not even notice it, but in one way or another we often use radar
in our daily lives. Consider for example weather prediction, traffic speed control
and aviation radars. Radar known as Radio detection and ranging, operates in the
microwave (1 mm to 1 m) regime. When using radar, one measures distance. The
range R between the radar antenna and the scatter object follows from the two way
travel time t and the propagation speed of electromagnetic waves, which is the speed
of light c.
ct
R=
(2.1)
2
The resolution along track or azimuth resolution ∆a and the resolution across track
or range resolution ∆r are often different resulting in rectangular pixels. The aspect
ratio gives the relation between the pixels in azimuth with respect to the pixels in
range such that by approximation square pixels are obtained. The range resolution
is given as the minimum distance to separate two objects from each other and is
determined by the pulse length τ . Keeping in mind the two way travel time, the
range resolution can be written as:
cτ
(2.2)
∆r =
2
The beamwidth, given by Fraunhofer Diffraction, determines the ground or azimuth
resolution. The azimuth resolution follows by multiplying the beamwidth with the
travelled range.
λ
∆a =
(2.3)
L
5
2. Interferometric Synthetic Aperture Radar
Where λ and L are respectively the wavelength and the antenna length. In case of
Envisat (C-band) with an antenna length of 10 m and pulse length of 36 µs this
results in 4.2 km azimuth and 5.5 km range resolution.
2.2
SAR
The first radars were incoherent radars, meaning that the phase information of the
emitted and received waveform were not stored. It was only later with the development of coherent radars that both amplitude and phase was stored for later
processing. Moreover by maintaining stable phase during operation, one was able
to create a long synthetic antenna. The latter is referred to as Synthetic Aperture
Radar (SAR).
2.2.1
Resolution
SAR is an attractive remote sensing technique due to its high resolution. The range
resolution (equation 2.2) is improved by applying chirp compression. This means
that one emits a signal with increasing frequency, resulting in a range bandwidth BR
that is inversely proportional with the pulse length. Envisat has a range bandwidth
of 16 MHz improving the range resolution to 9.4 m (about 20 m on the ground).
∆r =
c
2BR
(2.4)
The azimuth resolution (equation 2.3) on the other hand can be improved by applying the principle of the SAR. The Doppler frequency changes from positive when
flying towards the object to zero when the satellite is directly above the object and
becomes negative when flying away of the object. The bandwidth determined by the
range of Doppler frequencies is used in the computation of the azimuth resolution.
The azimuth resolution improves to 5 m for Envisat.
∆a =
vs/c
BDoppler
=
L
2
(2.5)
With vs/c the satellite velocity.
2.2.2
Radar Coordinates
During acquisition the data is sampled and stored in the so-called Radar Coordinate
(RC) format. This means that signals are stored in the same sequence as they were
received. The term early and late azimuth respectively indicates the first and last
received signal along track. In case of the range direction, this becomes near and far
range for the received signals close and far away of the satellites flight path. Often
it is preferred to visualize the SAR image in approximately North, East, South and
West orientation.
6
2.2. SAR
(a) Ascending aquisition (b) Descending aquisition
(c) Radar Coordinates
Figure 2.1: Relation between an acquisition made during an ascending/descending
track and the displayed Radar Coordinate format. NESW give the geographical
orientations, while EA, LA, NR and FR represent respectively Early Azimuth, Late
Azimuth, Near Range and Far Range.
Figure 2.1 gives a SAR image as acquired in an ascending, descending track and
stored in the RC. In the ascending plot, figure 2.1a, the first acquisition is in the
South. Keeping in mind the early and late azimuth, it is clear that the North and
South component are swapped.
In case of a descending orbit, figure 2.1b, the North is acquired first, corresponding
to the RC format. The near range on the other hand corresponds to the East while
it is displayed on the left side in RC. In other words in case of descending data the
East and West are flipped.
2.2.3
SAR Platforms
In the last decades many of SAR platforms have been launched and operated. Think
of missions like the European ERS I/II and the successor Envisat. Moreover there
is the Canadian RadarSat, the Japanese JERS and ALOS. Data from ALOS and
Envisat have been used during this thesis research. A detailed description of both
ALOS PalSAR and ASAR Envisat platforms is given below.
ALOS PalSAR
The Advance Land Observation Satellite (ALOS) operated by Japan carries onboard a Phase Array type L-band Synthetic Aperture radar known as the PalSAR
instrument. ALOS was launched in January 2006 into a sun-synchronous orbit with
orbital height of 691.56 km and a repeat cycle of 46 days. The PalSAR instrument
is right looking and can operate at four different modes: Fine Beam Single polarization (FBS), Fine Beam Dual polarization (FBD), fully-polarimetric (PLR) and as
last the ScanSAR mode. Table 2.1 gives an overview of the characteristics for each
mode. In case of FBS and FBD there is almost no difference. Both modes have
the same nadir-viewing angle and scan with the same swath. The chirp or range
7
2. Interferometric Synthetic Aperture Radar
bandwidth BR on the other hand is half as large for the FBD mode. This has as
consequence that the range resolution ∆r (equation 2.4) is twice as high for the FBS
acquisition mode and equals 5.35 m.
Data acquired before 2007 is not suitable for interferometry with later acquired data
due to a change in viewing angle near the end of 2006. Another advantage beside
the high resolution in range and azimuth is the higher critical baseline. The longer
wavelength of 24 cm together with the large range bandwidth increases the critical
perpendicular baseline (equation 2.10) to 13 km (FBS) over flat areas [Lu, 2007]
resulting in baseline lower decorrelation (equation 2.9). For FBD and PLR the
critical baseline is in the order of 6.5 and 3.4 km. The long wavelength makes the
signal less sensible for vegetation as well as for the wet troposphere delay. One of the
drawbacks of a long wavelength is the interaction with the ionosphere resulting ionospheric delays (equation 2.17). An estimation of the Faraday rotation is provided for
every acquisition, indicating the quality of the image affected by ionospheric effects
[Lu, 2007].
ASAR Envisat
Envisat, also known as the Environmental satellite from the European Space Agency,
carries onboard the Advanced SAR (ASAR) instrument. Envisat was launched in
March 2006 into a sun-synchronous orbit at an orbital height of 799.8 km with a
repeat period of 35 days. The ASAR instrument (right looking) operates in Cband and has five different operational modes, from which only the image mode is
considered here. This mode has seven predefined swaths (IS1 -IS7 ) with either HH or
VV polarization having incidence angles from 15 to 45.2◦ . The swath width ranges
from 100 to 56 km for respectively IS1 and IS7 having a spatial resolution of 30 m.
The IS2 swath corresponds to the acquisition mode of the ERS satellites allowing
cross-platform interferometry (table 2.2) [Holzner et al., ]. The range bandwidth
and the lower wavelength of 5.6 cm result in a critical baseline (see section 2.3.1) of
about 1100 m causing rapid spatial decorrelation, relative to ALOS, with increase of
perpendicular baseline. C-band radars are also more sensitive for the wet troposperic
delays, but less affected by ionospheric distortion.
2.3
InSAR
While radar and SAR was all about measuring distance, interferometric SAR or
InSAR is all about the change in distance. A SAR image consists out of a real and
complex part (coherent radar), which can be written as
y = |a|eiφ
(2.6)
With |a| the amplitude, φ the phase and i the imaginairy unit. An interferogram
ym ys is the phase difference between two SAR acquisitions. This phase difference
8
2.3. InSAR
Table 2.1: Characteristics of the ALOS PalSAR acquisition modes: Fine Beam
Single polarization (FBS), Fine Beam Dual polarization (FBD), Fully-polarimetric
(PLR) and ScanSAR. PalSAR senor operates with a central frequency of 1270 MHz.
Included are the parameters that differ between different modes. PRF and RSF
represent respectively the Pulse Repetition Rate and the Range Sampling Frequency.
Parameter
PRF [Hz]
RSF [MHz]
BChirp [MHz]
Polarisation
FBS
1500 - 2500
32
28
HH or VV
Off-nadir angle [◦ ]
Incidence angle [◦ ]
Swath Width [km]
Data rate [Mbps]
9.9-50.8
7.9-60.0
40-70
240
FBD
1500 - 2500
16
14
HH/HV
or VV/VH
9.9-50.8
7.9-60.0
40-70
240
PLR
2 x FBS PRF
16
14
HH/HV/VV/VH
ScanSAR
1500 - 2500
16
14
HH or VV
9.7-26.2
8-30
20-65
240
20.1-36.5
18.0-43.3
250-350
120 or 240
Table 2.2: Characteristics of Envisat ASAR sensor operating in IS2 mode and the
ERS SAR sensor. PRF and RSF represent respectively the Pulse Repetition Rate
and the Range Sampling Frequency.
Parameter
PRF [Hz]
RSF [MHz]
BChirp [MHz]
Polarisation
Incidence angle [◦ ]
Swath Width [km]
ASAR (IS2 )
1652.4
19.20
16
HH or VV
18.6-26.2
100
SAR
1679.9
18.96
15.5
VV
19.6-26.5
100
9
2. Interferometric Synthetic Aperture Radar
(a) B⊥ = 282 − 864 m
(b) Btemp = 420 − 1470 days
Figure 2.2: Interferograms decorrelate with increasing temporal and perpendicular baseline. The following interferograms (Envisat) of Guerrero, Mexico, indicate
this limitation of InSAR. Figure a) compares an interferogram with an increased
perpendicular baseline while keeping the temporal baseline approximately constant
(70 versus 35 days). The effect of an increased temporal baseline, while keeping approximately the same perpendicular baseline (85 versus 83 m) is visualized in figure
b).
can be computed by multiplying one image with the complex conjugate of the other
ym ys = |as |eiφs |am |e−iφm = |as ||am |ei(φs −φm )
(2.7)
The interferograms obtained after the inteferometric computation are wrapped between [-π,π]. A few examples are provided in figure 2.2. In these results, one color
cycle represents a fringe and corresponds to a change of 2π or, expressed in distance,
half the wavelength. Clearly some interferograms contain multiple fringes. The purpose of unwrapping is to integrate these fringes and reconstruct a continuous signal,
making interpretation much easier. Unwrapping is not trivial, different unwrapping
paths exist to get to a certain location which in the end should all end up with
the same unwrapped value. Atmospheric noise and noise due to decorrelation are
the limitations of radar interferometry, as is visualized in figure 2.2. Decorrelation
sources togehter with the atmopsheric effects are discussed respectively in section
2.3.1 and 2.3.2. More on InSAR processing steps is contained in section 5.2.
2.3.1
Decorrelation
At least two radar images are required for computing an interferogram. But how
can one discriminate between a good and a better combination in case there are
multiple images available? Coherence γ describes how well images fit together. It is
defined in such a way it has a value between 0 and 1, where 1 means that the images
10
2.3. InSAR
match perfectly. Different factors influence the coherence. The following sources of
decorrelation can be distinguished according to [Hanssen, 2001], [Gens, 2006] and
[Zebker and Villasenor, 1992]:
•
•
•
•
•
•
•
Geometric decorrelation
Non-overlapping Doppler decorrelation
Volume scattering
Temporal decorrelation
Atmospheric pertubations
Thermal decorrelation (SNR)
Processing induced decorrelation
[Zebker and Villasenor, 1992] has proven that the total coherence γtot , given by equation 2.8, can be computed by multiplying the individual decorrelation sources. Depending on how coherence is defined some of the sources are left out or additional
ones are included.
γtot = γgeom γtemporal γthermal
(2.8)
When computing or estimating the total coherence for a lot of image pairs, one
should pick the pairs with the highest coherence values. These are theoretically
the combinations the best fit for interferometry, having lesser noise, which makes
unwrapping easier.
Baseline or geometric decorrelation
When considering baseline B, one often refers to the perpendicular baseline B⊥ .
Baseline is defined as the distance between two satellite antennas, while the perpendicular baseline is the distance measurered perpendicular to the viewing direction
between the two sensors, figure 2.3. Baseline decorrelation is also referred to as
a change in viewing angle θ. Baseline decorrelation is linear and increases with
perpendicular baseline (figure 2.2).
γgeom =
Bcrit − B⊥
Bcrit
(2.9)
The baseline corresponding to a zero coherence value is referred to as critical baseline
Bcrit . This is the baseline corresponding to a spectral shift having the same size as
the bandwidth BR [Hanssen, 2001].
BR
Bcrit = λ
R tan (θ − ς)
(2.10)
c
Here ς, c, R and λ represent respectively the topographic slope, the speed of light,
the range and as last the wavelength of the SAR platform. Both ERS and Envisat
are C-band (5.6 cm) and have a critical baseline in the order of 1 km to 1.1 km.
Temporal decorrelation
This type of decorrelation is mainly determined by variations with time like vegetation growth or snow cover, as well as man-induced changes like farming and urbanization. Depending on the location around the world some areas will be affected
11
2. Interferometric Synthetic Aperture Radar
Figure 2.3: Interferometric combination: two satellites imaging over the same area,
showing the perpendicular baseline B⊥ , the baseline B, the satellite height h, the
satellite range R and the looking angle θ.
stronger. Surface scatter characteristics change as time passes by. The sensitivity to
change is determined by the wavelength. Larger wavelengths, like L-band (24 cm),
are less sensitive to vegetation growth and therefore coherence decorrelates slower
in time.
Temporal decorrelation is complex to model and depends on numerous variables.
For simplicity it is assumed that the decorrelation is linearly increasing with time
(figure 2.2). Zero coherence corresponds to the critical temporal baseline Tcrit . This
value is chosen depending on the area under investigation.
γtemp =
Tcrit − T
Tcrit
(2.11)
Thermal decorrelation
Thermal decorrelation can be fully described by the Signal to Noise Ratio. The
SNR is defined as the ratio of the received signal power Pr to the noise level Pn .
Thermal noise generated by the device itself is the main source contributing to the
SNR. The received power dependents on the surface scattering properties and can
be computed using the radar equation. The SNR is given in [Hanssen, 2001] as:
SNR =
Pr
Ptrans GAscat σ 0 A
=
Pn
(4πR2 )2 kTsys BR
(2.12)
Here Ascat describes the scatter area on the ground, which corresponds to the footprint area. The scattering properties of the ground are described in the σ 0 parameter. The remaining parameters are the antenna gain G (can be computed using
equation 2.13), the antenna area A, the travelled range R from the platform to
the surface, the Boltzmann constant k, the noise temperature Tsys and the range
12
2.3. InSAR
bandwidth BR .
G=
4πA
λ2
(2.13)
[Zebker and Villasenor, 1992] derived equation 2.14 for the thermal decorrelation,
assuming that for both acquisitions the SNR is identical. [Hanssen, 2001] derived
equation 2.15 to scope with two different SNRs.
γthermal =
γthermal = q
1
1 + SN R−1
1
−1
1 + SN R1
1 + SN R2−1
(2.14)
(2.15)
Thermal decorrelation can be neglected when comparing coherence between interferograms, when the images are acquired from the same satellite or from two satellites
with same payload characteristics. In this circumstance the system temperature will
be unchanged between acquisitions. In other words the thermal coherence remaines
constant.
2.3.2
Atmosphere
Electromagnetic signals are affected by the medium in which they travel. The propagation delay depends on the local atmospheric composition, the travelled path
as well as the frequency of the emitted signal. Atmospheric signals have a high
temporal variability, implying that averaging of multiple interferograms reduces the
atmospheric contribution.
The atmospheric signal of an interferogram δatm is composed of the difference between the atmospheric delay present in both SAR acquisitions. The atmospheric
delay follows from the integration of the refractivity N along all paths between the
antenna a to the resolution cell on the earths surface [Hanssen, 1998]. The atmoz,t1
spheric zenith delay δp,q
between resolution cells p and q at instant t1 can be written
as:
Z a
Z a
z,t1
−6
δp,q = 10 cosθ
N dz −
N dz
(2.16)
p
q
Here θ is the angle of incidence and N the refractivity, which is given as:
h e
Pd
e i
7 ne
N = k1
+ k2 + k3 2
+ [1.4W ]liquid + −4.028 10 2
T dry
T
T wet
f iono
(2.17)
Here Pd and e are respectively the partial pressure of dry air and water vapor. The
absolute temperature is given by T . The coefficients k1 , k2 and k3 are constants (k1
= 77.6 K hPa−1 , k2 = 71.6 K hPa−1 and k3 = 3.75 105 K2 hPa−1 ). The frequency
of the emitted signal is given by f , the electron density by ne and the liquid water
content by W .
13
2. Interferometric Synthetic Aperture Radar
The first two terms of the refractivity (equation 2.17) refers to the dry and wet
component of the atmosphere. The dry component is smooth and appears as a phase
trend of a couple of mm in the interferogram [Hanssen, 2001]. It is the variability
of the wet component which dominates the atmospheric signal in interferometry.
An increase in water vapor will cause the signal to travel slower, resulting in an
increased range or a phase delay.
The liquid component, third part of refractivity, is present in case the air is saturated.
Implying that it only occurs for presence of high water vapor density.
The last term is defined as the ionospheric term. Free electrons characterize the
ionosphere. The main driver for this ionization is solar radiation. The ionosphere
is a dispersive medium, meaning that the delays are frequency dependent. Besides
this, also the travelled path needs to be considered. The longer a signal propagates
through the ionosphere, the more it is affected. The Total Electron Content is
defined as the sum of all electrons within a box of 1 m2 along the travelled path. An
increase in TEC results in a phase advance, or in other words the observed range
has decreased.
2.4
Processing Software
The major space agencies deliver radar data in two formats. One can choose between Raw satellite data or already focused data. The latter is referred to as Single
Look Complex data (SLC).
The raw data used in this report has been focused using ROI PAC. The Repeat Orbit Interferometry PACkage is originally developed at the Jet Propulsion Laboratory
[Rosen et al., 2004] for ERS and Envisat radar data. Also ALOS and radarsat have
been incorporated for processing. ROI PAC can be used for focusing raw data and
constructing SLC images. Further it has its own toolbox for applying interferometry
on SLC images.
For interferometric computations the DORIS software was preferred above ROI PAC.
The Delft Object-oriented Radar Interferometric Software (DORIS) is the InSAR
processing software developed at Delft University of Technology [Kampes et al., 2003].
It can handle SLC images of the ERS, Envisat, JERS, RADARSAT and ALOS satellites. When starting from raw data focusing can be done using ROI PAC followed
by a roi2doris operation. This operation will put the SLC into the right DORIS
input structure. DORIS requires the SLC images to be focused in zero Doppler.
14
Chapter 3
Time series InSAR
Conventional InSAR is a valuable technique for observing crustal deformation, but
unfortunately it is affected by noise due atmosphere and decorrelation. Decorrelation becomes more apparent with increasing temporal baseline, temporal separation
between both SAR acquisitions, and increasing perpendicular baseline, which is the
distance measured perpendicular to the viewing direction between the two acquisitions.
The purpose of combining multiple images in a time series approach is to reduce
the errors and to better extract the deformation signal. Two main approaches exist
dealing with multiple images, the Persistent Scatterer (PS) and the Small Baseline
(SB) technique, addressed respectively in section 3.1 and 3.2. Section 3.4 describes
the time series InSAR software, StaMPS, which has been used to process the InSAR
data in this report.
3.1
Persistent Scatterer technique
The principle behind the Persistent Scatterer technique (PS) is the search for permanent dominant scatterers. Figure 3.1a shows a resolution element as seen from
an onboard radar sensor at two different acquisition times, i.e. the master and slave
acquisition. The reflected signal of a pixel is the coherent sum of all the scatters
within the resolution element. A change in viewing angle will change the value of
the coherent sum. Another acquisition will illuminate the surface under another
viewing configuration. Figure 3.1b simulates for different viewing geometries. One
can clearly see that the phase strongly varies with every other acquisition. In case
that a strong dominant scatterer is present, like e.g. a rock, tree trunk, buildings etc.
one will have a less varying phase. This simulation result is visualized in figure 3.1c.
This result indicates an important property, namely PS pixels do not decorrelate in
time.
Two methods of PS processing can be identified [Hooper, 2008], both searching for
the same PS pixels. The first method searches for the pixels having stable phase in
time [Ferretti et al., 2001][Kampes, 2006]. The initial step consists out of the selection of bright PS pixels based on amplitude variation. In this way pixels with a high
15
3. Time series InSAR
(a) Resolution element
(b) Distributed scatterer pixel
(c) Persistent scatterer pixel
Figure 3.1: The principle of a Persistent Scatterer. Figure a) gives the relation
between a pixel and the corresponding illuminated resolution element on the ground
for two acquisitions t1 and t2 . The phase of the pixel is the coherent sum of all
scatterers within the resolution element. [Hooper, 2008] gives a simulation of the
phase in case of varying distributed scatterers b) and for the case a dominant scatter
is present as well c). Figure c) indicates that the phase remains stable, indicating
an important property that PS pixels do not decorrelate in time.
SNR are selected. Next these pixels are tested for phase stability using a functional
model. The method has proved to be suitable for urban areas, where man-made
buildings result in stable scatterers. However, the method is not well applicable
to natural areas, like volcanoes and post seismic deformation sites, having a more
complex functional deformation model. Often the functional model is not known in
advance, indicating a drawback of this method. Pixels deviating too much from the
stable deformation model will not be selected.
The second method [Hooper et al., 2004][van der Kooij et al., 2006] searches for pixels for which its phase correlates in space. Using this method no a priori information,
except smoothness, is needed about the deformation signal. The selection of the
pixels having coherent phase is based on the statistical relations between the amplitude dispersion and the phase stability. Moreover, the PS selection procedure has
changed, resulting in a better applicability in all areas (urban and vegetation rich
areas). [Hooper et al., 2007] developed a method called Stanford Method for Persistent Scatterer (StaMPS) which requires a minimal amount of 12 interferograms for
identification of reliable scatterers.
3.2
Small Baseline technique
The methodology behind the Small Baseline technique (SB) is as the name indicates
the combinations that minimize all the baselines. In other words the grouping of
SAR images that have a small perpendicular and temporal baseline. When the temporal difference between two images is low, the interferogram is more likely to have
16
3.3. Combining PS and SB
low temporal decorrelation. The temporal and baseline decorrelation can be approximated to increase linear with time (see section 2.3.1 and 2.3.1). [Berardino et al., 2002]
and [Schmidt and Burgmann, 2003] apply the SB method. Using the available images they make, based on baseline threshold, as many SAR combinations as they
can. Depending on the area under investigation also thresholds for temporal and
Doppler baselines can be incorporated. Due to the short baselines, chance of good
coherence is higher, making possible to extract the deformation signal.
3.3
Combining PS and SB
Hooper [2008] developed a method which combines the PS and SB techniques. One
is no longer limiting to an extreme case of ground scattering. In case of PS pixels
the phase is dominated by a single scatterer, while for SB there is no dominant
scatterer and the phase is varying slowly with time. By combining both PS and SB,
one is able to use the advantages of both, making it fit for a large range of ground
scatterer characteristics.
3.4
StaMPS software
StaMPS (Stanford Method of Persistent Scatter) is developed by [Hooper et al., 2007].
Through the years this software was further developed at the University of Iceland
and Delft University of Technology. The StaMPS software can be used for Persistent Scatter (single master approach), Small Baselines and a combined time series
method. StaMPS uses the interferograms as computed using DORIS. If needed raw
data is focused to zero Doppler first using ROI PAC. All the software is open source
and free for non-commercial applications.
17
Chapter 4
Guerrero Subduction Zone
Guerrero is a province in the southern part of Mexico adjacent to the Pacific Ocean.
Here the Cocos plate subducts beneath the North America plate ∼60 to 80 km off
the coastline. In the past, large thrust earthquakes have occurred along the interface between the subducting and overriding plates. However, there is a region with a
seismic gap where no earthquakes have occurred since the 1911 7.6 Mw earthquake.
It is estimated that a rupture of the gap would result in a 8.0 to 8.4 Mw earthquake
[Singh and Mortera, 1991]. In the past few years it has become apparent that the
subduction zone is also very active in terms of Slow Slip Events (SSEs). Approximately in every four years one large event (>6 Mw ) has been observed since 1995
(see table 4.1). Considering the seismic hazard, it is important to know where and
with what magnitude slip occurs on the interface.
The most important reason to investigate silent earthquakes at Guerrero is to assess
the effect that these events have on the seismic gap region. Does the perturbation
to the stress field caused by these SSEs lead to a delay or a hastening of the next
large earthquake? Could one even be a trigger of a large earthquake, in which case,
it would have predictive power? Bearing in mind the proximity of Mexico City and
the destruction that wreaked there by the 1985 earthquake, any knowledge gleaned
about the timing of the next big one could prove invaluable.
Section 4.1 introduces the different types of slow slip together with their geographical
location. The Guerrero subduction interface and the large SSEs that have occurred
in Guerrero are respectively addressed in section 4.2 and 4.3.
4.1
Slow Slip Events
The Earth’s lithosphere is segmented into a number of tectonic plates, riding on the
asthenosphere (the upper mantle of the Earth). These plates move with respect to
each other at rates of a few cm a year. The oceanic-continental convergence region,
where the plate with ocean crust subdives the other plate, better known as a subduction zone, forms the area of interest. It is at these subduction zones, but not
limited to, where most SSEs have been recently observed by GPS (figure 4.1).
19
4. Guerrero Subduction Zone
Figure 4.1: Slow Slip Events in the pacific subduction zones. Here only slip
at subduction zones is indicated. The numbers in the figure represent slow slip
locations, additional information like the slow slip magnitude, slip size, time period
and location are contained in [Schwarts and Rokosky, 2007].
During subduction, the oceanic crust takes the overlaying continental crust with
it in the direction of motion, resulting in deformations, strains. However at some
parts of the interface the plates are locked or coupled. It are these regions where
during an earthquake both plates will start to slip until the friction force is able to
compensate for the slip displacement. The slip rate at which earthquakes happen
can be in the order of meters per second, often resulting in devastating surface waves
[Schwarts and Rokosky, 2007]. Recent GPS analyses also indicated slowly slipping
rates to occur. These earthquakes are in literature often addressed as slow or silent
earthquakes.
One can expect that SSEs need something to trigger them. One mechanism suggests
that the slip is initiated once a critical stress threshold is exceeded. This phenomenon
is already observed at the Cascadia region, where accumulation of stress has been
monitored in the form of SSEs. A second mechanism would be a sudden increase
in stress caused by a small to moderate earthquake triggering the slow slip to start.
Another possible mechanism is where harmonic seismic tremor, long lasting rhythmic signal, accompanies the SSE.
However, with the latter mechanism strong discussion remains between the relationship of harmonic seismic tremor, more specific Non-Volcanic Tremor (NVT), and
the occurrence of SSEs. NVT, occurs in the 1 to 10 Hz frequency range and where
frequencies higher than 2 Hz are dominated by high ambient or environmental noise
[Husker et al., 2010] and has only reccently been observed.
At subduction zones it occurs that both SSEs and NVT are strongly related in time
as in space. [Ide et al., 2008] suggested that SSEs are the macroscopic sum of many
20
4.1. Slow Slip Events
small tremors. This model however, fails to explain the case when a SSE is not accompanied by NVT. Two other models were suggested by [Rubinstein et al., 2010].
In the first model, tremor is only generated in specific cases depending on the frictional properties causing a rupture or slipping, resulting in radiation frequencies
larger than 1 Hz. In the second model it is suggested that SSEs and NVT are temporal and spatially related, but where the amplitude of tremor is depending on the
local friction characteristics, making at undetectable in certain areas.
Based on the mechanism that trigger slow slip, different types of slow slip can be
identified. According to [Schwarts and Rokosky, 2007] three types of slow slip can
be considered, called preslip, inter-seismic slip and afterslip.
• Preslip is a type of slip indicating the opponent threat of an earthquake. Preslip events are seldom monitored. In the past, indications for possible preslip
were the change in leveling, tide gauges and well data. With the expansion
of dense GPS ground stations one will be able to detect preslip with a higher
probability.
• Inter-seismic slip is the type which is not clearly related to an earthquake.
Moreover this type of slip can act over long time periods, running from days
to months. The magnitudes accompanying these slip events can be harmless
but also reach large earthquake like magnitudes (6 to 8 Mw or even more).
Almost all subduction zones equipped with a dense GPS network were able to
monitor this type of slip.
• Afterslip is the slip which has been detected the most, since it occurs at the
earthquake site. Often the amount of slip caused by the preceding earthquake
is in the same order as the afterslip. [Schwarts and Rokosky, 2007] gives some
characteristics which have to be satisfied to be called afterslip. Firstly the
location should be on the coseismic fault plane. Secondly, the slip duration is
in the order of days to months, where the decay has a high initial rate and
decreases with a logarithmic decay. As last the size of the slip should exceed
50% of the average coseismic slip.
Whereas the slip during earthquakes is over in minutes, SSEs can occur over periods
ranging from a few days to months, and in extreme cases to a year. It is not yet
understood why the slip during these events occurs at such a snail’s pace, but in
subduction zones at least, it is hypothesized that pulses of water, released by waterrich minerals deeper within the subduction zone, increase the pore pressure as they
pass through. The high pore pressure acts against the pressure from the overlying
rock, which clamps the two sides of the fault together, allowing slip to occur at
lower stresses. Although this slow slip does not generate large seismic waves, it does
cause an instantaneous elastic response in the surrounding rock, which results in
deformation of the Earths surface.
The Moment magnitude (Mw ) of a SSE is identical to the scale used for earthquakes.
The magnitude scale is based on the potential energy released and is calculated from
the slip area, the amount of slip and the elastic properties of the rock. These days
the Mw scale is used more often than the famous Richter scale, calculated from
21
4. Guerrero Subduction Zone
seismic wave amplitudes. Both scales are logarithmic, i.e. a 7 Mw is 10 times as
large as 6 Mw , and have approximately similar values, but the Mw scale does not
saturate for large earthquakes while Richter scale does. Silent earthquakes measured
up to now have magnitudes ranging up to 7.5 Mw , large enough to cause significant
damage if the energy were released in a normal earthquake.
4.2
Subduction interface
The Cocos plate subducts the North America plate about 60 to 80 km off the coast.
Based on the seismotectonic features the subduction zone appears to be segmented
in five regions: Jalisco-Colima, Michoacan, Guerrero, Oaxaca and Chiapas region,
with increasing secular rates towards the Southeast. These regions are bounded
by the Rivera fracture zone, the East pacific rise, the Orozco and O’Gorman fracture zones, and the Tehuantepec ridge [Yoshioka et al., 2004]. Figure 4.2 gives a
geographic visualization of southern Mexico and more specific of the Guerrero and
Oaxaca regions.
During the years the Guerrero subduction interface evolved significantly [Franco et al., 2005],
[Payero et al., 2008], [Perez-Campos et al., 2008], [Kim et al., 2010]. The Mexican
subduction interface is unusual. The oceanic crust in absence of the asthenosphere
subducts the North America plate with a dip angle of 15◦ and becomes sub-horizontal
at a depth of 40 km, which is about 80 km from the coast. The interface remains horizontal till about 280 km from the coast, after which it steeply dips, below the TransMexican Volcanic Belt, at an angle of 75◦ . This interface, by [Kim et al., 2010], is
obtained from analysis of the receiver functions. Figure 4.3 gives the visualization of
the Guerrero subduction interface. It represents the interface along the continuous
GPS station profile.
4.3
Observed SSEs
During the past decade a lot of GPS data has been analyzed, indicating the occurrence of SSEs. One large SSE (table 4.1) has been observed approximately every
four years. The area affected by the associated deformation of these SSEs is rather
large, with the highest deformation near the coast, reaching as far as Mexico City
(300 km inland). Smaller SSEs have been observed as well by [Lowry et al., 2005],
[Lowry, 2006] and [Vergnolle et al., 2010]. These smaller SSEs are indicated by numbered events in the GPS time series contained in appendix A. Current discussion
remains about whether these small SSEs are periodic or not. Below more information is given about the past large periodic SSEs.
• SSE September 1995 : This event is observed mainly through modeling. No
continuous GPS stations were operational. Fortunately there were some GPS
campaigns that can be used to validate the fitted models. [Larson et al., 2004]
has modeled this event and finds the largest displacements (∼3 cm) to occur
22
4.3. Observed SSEs
Figure 4.2:
Continuous GPS stations in Guerrero and Oaxaca states
[Franco et al., 2005]. These stations are used as calibration of the InSAR data during modeling of the SSE. Contours indicate the depth of the Cocos plate under the
North America plate. The subduction rates increase for West to East. Continuous
GPS ground stations are indicated by triangles.
Figure 4.3: Simplified visualization of the Guerrero subduction interface, defined
by [Kim et al., 2010], with the cross-section at the CAYA station and perpendicular
to the trench (figure 4.2). The Cocos plate subducts 60 to 80 km outside the coast.
The cross-section of the interface can be split into three segments having different
dip angles. Initial the plate subducts at an angle of 15◦ after which it becomes sub
horizontal and in the last stage continues dipping at a steep dip angle of 75◦ .
23
4. Guerrero Subduction Zone
Table 4.1: Large periodic SSEs in Guerrero. The different columns address the
start, magnitude, duration and size of the different SSEs. The latest SSE that
started around July 2009 is not contained in this table. During the writing of this
report, it was not clear if the 2009/2010 SSE has stopped at all GPS sites.
Start SSE
[year]
September 1995
January 1998
October 2001
April 2006
Mw
∼7.1
>6.5
∼7.5
∼7.5
Time span
[days]
∼180
>183
>195
∼300
Slip
[cm]
∼1.5
∼10
∼6
near the coast (along the CAYA and ACAP stations). The slip started near
September 1995 and lasted for about 6 months till February 1996, corresponding to a 7.1 Mw event.
• SSE 1998 : [Lowry et al., 2001] This SSE has been discovered first. Only one
continuous GPS station CAYA was operational at that time, spanning the SSE.
[Larson et al., 2004] indicated that also survey measurements and continuous
GPS data from the far away POSW Popocatepetl volcano site can be used.
From figures A.3, A.1 and A.2, showing the thee components of the GPS
time series, it is clear that the largest deformations are observed at the CAYA
station. Still to few observations at other locations are made to indicate the
slow slip signal.
Making use of modeling is another option. [Larson et al., 2004] finds results
that indicate that the largest displacements acted near the coast, extending
west to east of the CAYA station. The displacements decease when going
more inland. The event lasted for about 6 months from January 1998 till
June. During this event deformations of around 1 cm resulted in a >6.5 Mw
magnitude.
• SSE 2001 : [Kostoglodov et al., 2003] proposed two models for assessing this
SSE. The first model is based on slow slip over the transition zone while for the
second model the slip is extended over the seismogenic zone. [Iglesias et al., 2004]
investigated which of the two models is best by inverting the available data using physical constraints to map the slow slip. Based on their best model (physical reasonable constraints) they find that slow slip occurred over the transition
zone at 100 - 170 km from the trench or 20 to 90 km inland. The average slip
is about 22.5 cm acting over a 4 months period. [Kostoglodov et al., 2003]
suggest that slip acts over a period of 6 months starting from October 2001
till April 2002. The difference can be due to the fact that the latter also takes
the transition accompanying the slip event into account. The estimates for the
size of the SSE are almost the same for both 7.5 Mw .
Recent refined GPS analyses of [Vergnolle et al., 2010] indicate that the 2001
SSE continued for a longer period, and had a second phase. Depending on
the location of the stations the SSE lasted for 4 to 15 months. The largest
24
4.3. Observed SSEs
deformations are measured at the CAYA, ACAP, ZIHP and IGUA stations.
Overall the horizontal displacements for all stations are in Southwest direction,
having a size of 4 cm near the coast and at the IGUA station, and decreasing
further inland. The vertical deformation is about 4 cm upward near the coast,
3 cm subsidence at the IGUA station and decreasing further inland.
• SSE April 2006 : [Vergnolle et al., 2010] finds large resemblance between the
2001 and 2006 SSE. Vertical motion is 6 cm upward near the coast at CAYA
and COYU station, decreasing inland, resulting in subsidence of 4 cm at MESC
and again decreasing to zero further inland. The area of deformation extends
300 km along the coast till Mexico City. Horizontal displacement for all GPS
stations are in the Southwest direction, with the largest deformation of 6 cm
at CAYA and COYU stations. The start of the SSE was spread over two
months, with the first deformations observed at the CAYA station. Slip occurred from April till late December 2006 [Larson et al., 2007], corresponding to a 7.5 Mw event. [Vergnolle et al., 2010] indicated that by analyses of
the vertical GPS components (figure A.3) the SSE ended near late February
2007 rather than end of 2006 as suggested by [Correa-Mora et al., 2009] and
[Larson et al., 2007].
25
Chapter 5
PS-InSAR processing
Section 5.1 presents the feasibility of the InSAR study, based on available GPS data.
This is followed by section 5.2 giving an overview of the processing steps required
for the PS selection and unwrapping of the PS interferograms.
5.1
Study feasibility
Based on analysis of the available SAR data, see chapter 1, the descending track
288 of Envisat was chosen. Multiple acquisitions were made during the SSE, which
unfortunately do not extend to the area of interest. Eight raw SAR images were
ordered initially. Through collaboration with our colleagues at the University of
Grenoble the amount increased to 19 images in total. All the images range from the
Mexican coast to Mexico City. Figure 5.1 shows the illuminated ground area and
the location of the continuous GPS stations in Guerrero.
5.1.1
Available GPS data
Most of the continuous GPS stations in Guerrero are located near the coast, and the
ones away from the coast form a profile approximately perpendicular to the coast
(see figure 5.1). The measured slow slip surface deformations are small in magnitude
and extend over a large area. Table 5.1 gives the results of the GPS analyses over the
time series as contained in appendix A. The GPS network contains, besides stations
in Guerrero, also stations in the adjacent southern state Oaxaca. The original GPS
data is property of the National Autonomous University of Mexico (UNAM). The
results of the GPS analysis are combined in chapter 8 with the results of the PSInSAR analysis for modeling slip on the interface.
Horizontal deformations
From figure 5.1 one can deduce that the horizontal deformations are in Southwest
direction. They have a magnitude of 5 cm near the coast and reduce to zero near
Mexico City (300 km inland). The descending track of Envisat is perpendicular to
27
5. PS-InSAR processing
Figure 5.1: Map of Guerrero showing the horizontal and vertical GPS displacements of the 2006 SSE, corrected for the inter-seismic motion [Larson et al., 2007].
Uncertainties (1 σ) are approximately three times larger for the vertical compared to
the horizontal motion. InSAR is most sensitive for deformation in the Line Of Sight
(LOS). The largest contribution to the LOS due to the 2006 slow slip deformation
signal is comming from the vertical. The rectangular box indicates the illuminated
area of the SAR data. The grey shaded area shows the assumed non-deforming region, which is confirmed by the GPS displacements. Mexico City (MC) area however
is excluded from this region due to large subsidence rates.
the coast. The horizontal deformations as well as the horizontal inter-seismic rates
are almost parallel to the direction of flight and hardly affect the LOS deformations.
Vertical deformations
The vertical deformations are approximately 1 to 4 cm upward near the coast. They
reduce as you go inland, resulting in subsidence and diminish to zero near Mexico
City (see figure 5.1).
5.1.2
Feasibility
When considering horizontal motion, radar is only sensitive to the deformation in
Line Of Sight direction (LOS). Most SAR satellites are right looking. In case of
Envisat, the nominal viewing angle equals 23◦ . Keeping this all in mind, part of the
vertical deformation of the SSE as well as the vertical inter-seismic rates is mapped
in the LOS direction, being detectable with InSAR.
28
5.2. SAR and PS-InSAR processing steps
Table 5.1: Inter-seismic rates and displacements of the GPS stations during the
2006 SSE as obtained after GPS analysis [Vergnolle et al., 2010] for the three components North, East and Up. Last column of every component gives the error of
the SSE size at 95% confidence. SSE displacement is computed as the displacement,
between two zero slope crossings, after subtracting the secular motion.
Station
HUAT
PINO
CPDP
ACAP
ACYA
COYU
CAYA
ZIHP
DOAR
MEZC
OAXA
IGUA
YAIG
UNIP
DEMA
5.2
Rates
[cm/y]
2.21
1.64
1.94
1.79
2.05
1.70
1.69
1.85
2.04
1.71
1.38
1.29
0.78
-0.29
0.34
North
SSE
[cm]
-1.20
-2.12
-3.31
-3.77
-3.99
-5.27
-5.49
-0.40
-4.36
-4.55
-1.06
-3.15
-1.59
-0.22
-0.44
σ95%
[cm]
0.15
0.11
0.14
0.12
0.11
0.16
0.21
0.19
0.15
0.15
0.07
0.15
0.14
0.10
0.13
Rates
[cm/y]
1.19
1.46
1.30
1.28
1.37
1.09
1.13
1.68
1.15
0.95
0.98
0.64
0.31
0.12
0.02
East
SSE
[cm]
-0.12
-1.43
-0.57
-0.79
-0.99
-1.74
-2.71
-1.22
-1.71
-2.34
-1.05
-1.50
-0.66
-0.08
-0.04
σ95%
[cm]
0.15
0.19
0.25
0.21
0.22
0.19
0.18
0.21
0.24
0.19
0.11
0.13
0.14
0.15
0.17
Rates
[cm/y]
-1.59
-1.30
-1.24
-1.38
-1.22
-1.50
-1.50
-0.65
-1.11
0.50
0.72
0.44
0.20
-0.04
-0.21
Up
SSE
[cm]
1.24
2.27
2.03
2.29
1.90
5.68
5.67
0.66
4.63
-2.63
-0.93
-2.01
-0.37
-0.08
0.13
σ95%
[cm]
0.33
0.34
0.30
0.38
0.41
0.34
0.33
0.41
0.42
0.37
0.24
0.50
0.33
0.21
0.46
SAR and PS-InSAR processing steps
The end goal is to model the slip on the interface between the two plates. All required processing steps are visualized as a processing flowchart in figure 5.2. This
section focuses on the processing up to the selection of PS points. The processing
starts with the focusing and cropping of raw SAR data, followed by the interferometric computations and lastly the PS selection. These processing steps correspond
to the ROI PAC, DORIS and StaMPS blocks respectively in the flowchart.
5.2.1
Focusing and cropping of raw SAR data (ROI PAC)
The pre-processing stage consists of focusing and cropping the raw data. This is
visualized in the upper part (ROI PAC) of the flowchart, given in figure 5.2. Raw
data is focused in Zero Doppler, which is required for later interferometric processing using DORIS. The crop is chosen such the maximum area is contained in every
image. This resulted in a crop ranging from the coast to, as far as, Mexico City.
This is indicated by the rectangle shown in figure 5.1.
29
5. PS-InSAR processing
The PS-technique requires that all interferograms be with respect to the same image,
referred to as the master. The master, 16 December 2005, is chosen by minimizing
the temporal and perpendicular baseline. Table 5.2 gives the perpendicular and
temporal baselines together with the absolute Doppler centroids of each of the interferograms. Variations of the absolute Doppler centroid are small and do not affect
the choice of the master. Figure 5.3 shows the single master network for the descending track 288 of Envisat.
5.2.2
Interferometry (DORIS)
After pre-processing, one can continue with the interferometic processing, as is visualized in the DORIS stage of the flowchart. Computing interferograms requires
first of all the coregistration and resampling of the slave to the master image.
Coregistration
Coregistration is done in a coarse and a fine coregistration step. The coarse coregistration step uses orbit information as starting point. The slave image is subdivided
into large windows and amplitude correlated with the master image. For every window the translation with the highest correlation is kept. By comparing the different
translations in range and azimuth of the different windows, a coarse estimate is obtained.
The fine coregistration step uses the result of the coarse coregistration as starting
point. The principle of fine coregistration is identical to the second step of the coarse
coregistration. Now more and smaller windows are used. Translation estimates at
sub-pixel precision are obtained after oversampling and amplitude correlation between master and slave in the region where the maximum correlation at the pixel
level was obtained. As last, these estimates are used in estimation of the 2d coregistration polynomial.
Large baselines introduce a decrease of coherence and consequently seem to be more
noisy making correlation to become more difficult and unreliable. To cope with this
problem StaMPS has an option to coregister a slave image to other closer slaves images, decreasing the perpendicular baseline. During coregistration a baseline threshold of 150 m has been used.
Resampling and interferometric computations
Next, the slave is resample to the grid of the master image, after which the interferometric phase can be computed using equation 2.7.
The interferometric phase φ is the sum of different phase contributions like the reference surface, topography, deformation, atmosphere and noise. Both the reference
surface and the topography can be removed. The reference surface has been introduced by a different position of the satellites during acquisition and can be corrected
30
5.2. SAR and PS-InSAR processing steps
Figure 5.2: Work and processing flow chart. A distinction is made between processing using the ROI PAC (focusing), the DORIS (interferograms) and the StaMPS
(PS points) software. Remaining processing blocks are dedicated on estimating the
long wavelength tropospheric signal, extracting the slow slip deformation signal and
as last the modeling of the slip due to the 2006 SSE. Rectangular boxes indicate
products or intermediate results, while the rounded boxes give the key operations.
Grey shaded rectangular boxes indicate input sources. The intermediate processing
steps are indicated on the processing line on the left.
31
5. PS-InSAR processing
Figure 5.3: Single Master baseline plot of the data used for the 2006 SSE. All
interferograms are with respect to the master of 16 December 2005. The 2006 SSE
ranges from April till late December 2006.
for using the position information with respect to the reference ellipsoid (WGS84).
The topographic contribution can be corrected for using a Digital Elevation Model
(DEM) from ASTER [EROSDC, 2003] or SRTM [Farr et al., 2007].
ASTER covers the globe from 83◦ N till 83◦ S with a spatial resolution of 1 arcsec
or 30 m. SRTM on the other hand mapped the earth from 60◦ N till 60◦ S with a 3
arsec or 90 m resolution. For the American continent there is in addition a 1 arcsec
or 30 m resolution SRTM DEM available. The main reason to opt for SRTM is the
higher precision. For Guerrero this means a spatial resolution of 90 m. This can
introduce topographic features during interferometry. These are however estimated
in the time series model.
Interferograms
Figure 5.4 shows the interferograms after subtraction of the reference surface and
topography. Quite some interferograms have considerably lower coherence. It is
mainly the large perpendicular baseline (>400 m) that is causing the interferograms
to decorrelate. Higher temporal baselines on the other hand (300 days) causes interferograms to decorrelate at even smaller perpendicular baselines. From figure 5.4 it
is clear that urban areas remain longer coherent. Large deformations are expected
near Mexico city (upper part of the interferograms), subsidence rates as high as 38
cm/year have been observed by GPS and InSAR [Cabral-Cano et al., 2008]. The
numerous fringes apparent in the Mexico City region confirm this. The lower, vegetated, part of every interferogram seems to have low coherence, making unwrapping
32
5.2. SAR and PS-InSAR processing steps
Table 5.2: Perpendicular (B⊥), temporal (T) and Doppler (fDC ) baselines of the
interferograms as displayed in figure 5.4. fDC gives the absolute Doppler centroid.
All interferograms are relative to the master of 16 December 2005.
Date
26 Nov 2004
31 Dec 2004
4 Feb 2005
11 Mar 2005
15 Apr 2005
20 May 2005
24 Jun 2005
29 Jul 2005
2 Sep 2005
B⊥
[m]
210
134
-120
283
561
123
700
305
652
T
[days]
-385
-350
-315
-280
-245
-210
-175
-140
-105
fDC
[Hz]
-155.93
-156.90
-158.50
-149.69
-153.45
-158.66
-157.27
-157.07
-157.79
Date
7 Oct 2005
11 Nov 2005
20 Jan 2006
24 Feb 2006
5 Jan 2007
9 Feb 2007
16 Mar 2007
9 May 2008
11 Sep 2009
B⊥
[m]
-110
690
-864
-282
430
86
349
138
352
T
[days]
-70
-35
35
70
385
420
455
875
1365
fDC
[Hz]
-129.55
-133.59
-129.66
-131.85
-138.93
-134.73
-129.26
-137.38
-148.73
in that region difficult.
5.2.3
PS selection (StaMPS)
Persistent Scatterer processing using StaMPS forms the third processing block of
the flowchart shown in figure 5.2. The tweaking of selection parameters is done
based on visual inspection of the wrapped and unwrapped interferograms. Table 5.3
summarizes the main processing parameters tweaked during the processing of the
Guerrero region.
PS candidate selection
During the initial processing step, the set of pixels were divided into a number
of patches that partly overlap. The main reason for the patches is to reduce the
required processing memory. The search for PS pixels is the search for pixels having
a dominant scatterer in time, meaning that the phase should remain stable (see
section 3.1). An initial set of PS candidates can be selected based on amplitude
dispersion. The latter is defined as the standard deviation of the amplitude divided
by its mean [Ferretti et al., 2001]. An amplitude dispersion value of 0.4 resulted in
a broad range of pixels, most of them not PS. The phase noise for the PS candidate
pixels is estimated using an iterative approach [Hooper et al., 2007].
PS selection and resampling
PS pixels were selected based on the pixel phase characteristics and by allowing
maximum 20 random phase pixels per km2 . In the next processing step additional
weeding is performed in order to reduce noisy pixels. A pixel was rejected if the
33
5. PS-InSAR processing
26 Nov 2004 31 Dec 2004
4 Feb 2005
24 Jun 2005
29 Jul 2005
2 Sep 2005
24 Feb 2006
5 Jan 2007
9 Feb 2007
11 Mar 2005 15 Apr 2005 20 May 2005
7 Oct 2005
11 Nov 2005 20 Jan 2006
16 Mar 2007 9 May 2008
11 Sep 2009
Figure 5.4: Single Master interferograms of the descending track 288 of Envisat.
Interferograms are displayed in radar coordinates, meaning that approximately left
and right are swapped, compared to the PS-InSAR interferograms as displayed in
figure 5.5. One color cycle corresponds to a LOS change of 2.8 cm. All images are
relative to the master of 16 December 2005. The slow slip started around April 2006
and lasted till late December 2006.
34
5.2. SAR and PS-InSAR processing steps
(a) Wrapped PS interferograms
(b) Wrapped PS interferograms corrected for initial estimate of SCLA
error and master atmosphere
35
5. PS-InSAR processing
(c) Unwrapped PS interferograms corrected for initial estimate of SCLA
error and master atmosphere
Figure 5.5: PS-InSAR interferograms, obtained after processing with StaMPS
using the parameters as specified in table 5.3, shown in the geographic reference
frame. All interferograms are relative to the master of 16 December 2005. Figure
a) shows the wrapped interferograms obtained after PS selection, which can be
compared to figure 5.4 (radar coordinates) showing conventional InSAR. Figure
b) gives the wrapped interferograms after correction of SCLA error and master
atmosphere as estimated by StaMPS. Both figure a) and b) have the same color
scale, where one cycle corresponds to a change of 2.8 cm. Figure c) shows the
unwrapped interferograms of figure b), where the star indicates the reference area.
minimum phase noise standard deviation of all arcs of that pixel was larger than 1
radian. Since the SSE deformation signal spans a large area it is opted to resample
the PS points during merging of the patches to a coarser grid of 1000 m. This in
the first place reduces the noise and secondly additional weeding can be performed.
During resampling, the new PS values as well as their variance is being estimated.
The final set of PS pixels was obtained by allowing a maximum merging standard
deviation of 0.45 radians. Figure 5.5a gives the wrapped PS interferograms, consisting out of 27100 PS pixels. The subsidence signal near Mexico City is being
underestimated, less fringes compared to figure 5.5a, due to coarser grid and aliasing of the signal.
36
5.2. SAR and PS-InSAR processing steps
Table 5.3: Parameters used in StaMPS processing of the Guerrero region.
Parameter
Amplitude dispersion
Density random
Weed max noise
Weed σ
Weed time window
Merge grid size
Merge σ
unwrap goldstein α
Unwrap number of goldstein windows
Unwrap grid size
Unwrap time window
Value
0.4
20 pixels/km2
Inf
1 rad
1460 days
1000 m
0.45 rad
0.8
16
1000 m
730 days
Unwraping
StaMPS uses an iteration during the unwrapping in order to improve the unwrapping
solution. Due to large baselines, there are still some interferograms that have noisy
regions. Since unwrapping is an integration of the different fringes, unwrapping errors could be propagated into other regions of an interferogram. After unwrapping
an initial estimate is computed for the master atmosphere and the Spatially Correlated Look Angle (SCLA) error, based on the well unwrapped interferograms. This
means that badly unwrapped interferograms need to be ignored, and identified by
the user, in the estimation processes. Next the estimate for master atmosphere as
well as the SCLA error are used to correct all wrapped interferograms. This result
is visualized in figure 5.5b. The remaining signal consist out of a contribution from
the deformation signal due to the SSE, secular motion, slave atmosphere, noise and
possible orbital ramps. From figure 5.5c it is clear that the spatially correlated noise
as well as the amount of fringes has decreased, moreover the signal remains longer
coherent in the coastal region.
Due to reduction of noise, unwrapping of the corrected interferograms is more likely
to be succesful. As last step in the StaMPS processing, the initial estimate of the
SCLA error and master atmosphere is added back to the unwrapped signal. Figure 5.5c shows the unwrapped interferograms of corrected wrapped interferograms
(figure 5.5b). Visual inspection of the unwrapped and comparing to the wrapped
interferograms does not reveal large unwrapping errors.
37
Chapter 6
Long wavelength tropospheric signal
When observing the unwrapped interferograms, shown in figure 5.5c), it is clear that
some of the signal correlates well with the topography, shown in figure 6.4a). This
chapter is dedicated to removing the net (difference between two SAR acquisitions)
propagation delays which correlate with topography, referred to as long wavelength
tropospheric signal. Conventional methods, where phase is correlated with topography, are biased in case of the Guerrero dataset, due to correlation of the deformation
signal with the long wavelength of the topography. For this reason I developed a
new method, section 6.3, and applied it to the Guerrero interferograms, section 6.4.
6.1
Tropospheric delays from sounding data
Signals that travel longer before being scattered back are more affected by atmospheric delays. Consequently variation in topography will introduce a different propagation delay. Figure 6.1a) shows the exponential decrease of water vapor ewv with
increasing height h at two different SAR acquisition instants t1 and t2 . As a radar
signal travels through the atmosphere, the propagation delay is proportional to the
integration
of the water vapor for the full travelled height range. Integrated water
R
vapor ewv curves are shown in figure 6.1b).
Pressure P , temperature T and humidity R data, obtained from the University of
Wyoming Department of Atmospheric Science and acquired by the United States
weather service, were used to compute the refractivity N . Since the tropospheric
stratified delay is mainly determined by the water vapor content, the liquid and
ionospheric term of the refractivity were neglected in equation 2.17. The tropospheric delay follows by integration of the refractivity with height and projection to
the LOS. Figure 6.2a) and b) give the refractivity and tropospheric delay for sounding stations Acapulco (dashed lines) and Mexico City Airport (solid lines) with the
colors representing the SAR acquisition dates. Soundings were acquired at noon
while the SAR data were acquired at 4.34 pm.
An interferogram, however, is subjected to the difference in integrated water vapor
between two acquisitions. Unfortunately not all SAR acquisition dates were contained in the data pool of both stations. Due to absence of the master date for the
39
6. Long wavelength tropospheric signal
(a) Water vapor content
(b) Integrated water vapor content
Figure 6.1: Figure a) visualizes an exponential decay curve of water vapor with
height at two SAR acquisitions times. Figure b) gives the integrated water vapor
curves. The interferogram is subjected to the difference of both integrated water
vapor curves.
Acapulco sounding station, 24 November 2004 was chosen as an arbitrary master in
order to compute the example of the relative tropospheric delay given in figure 6.2c).
6.2
Convential estimation method
[Cavalie et al., 2007] approximates the delay curve as a linear function between
heigth and phase. The linearization assumption is only valid for the lower part
of the atmopshere.
E{∆φ} = Kh + b
(6.1)
With E{·} the expectation operator, K the slope relating topography h to phase φ
and b a constant. One set of coefficients is estimated for each interferogram using the
full spectrum of topography and phase. While the constant b is biased by the long
wavelenght signals, it has no implications for the estimation of the long wavelength
tropospheric signal. The constant only results in a shift of the full interferogram
and for this reason can even be neglected.
The estimated long wavelength tropospheric signal is biased when the slope is biased.
This occurs when deformation relates to topography, as is the case for Guerrero. Figure 6.3 shows the estimated stratification coefficients K when applying the convential
estimation (blue) compared to the stratification estimated from a wavelength band,
8 to 64 km, insensitive to deformation (green). The solid lines indicate the evolution
of the stratification coefficient in time. However the stratification is undersampled
in the temporal domain, it can be deduced that the stratification estimates of the
full spectrum, convential estimation method, show a correlation between the topography and the inter-seismic deformation, which increases with time.
40
6.2. Convential estimation method
(a) Refractivity with height
(b) Tropospheric delaybetween 0 and 9600 m
(c) Relative tropospheric delay
Figure 6.2: Refractivity and tropospheric delay estimated from sounding data collected at Mexico City Airport (solid lines) and Acapulco (dashed lines) visualized in
respectively figure a) and b). Refractivity and tropospheric delay are composed from
the hydrostatic and wet component only. Colors representing the SAR acquisitions
dates are contained in the legend. Unfortunately sounding data is not acquired for
all SAR dates. Figure c) gives the relative tropospheric delay with respect to the
master. Due to absence of the master SAR acquisition date (16 December 2005) for
the Acapulco dataset, 26 November 2004 was defined as an arbitrary master for the
sounding data. Sounding data, pressure P, temperature T and humidity R, were
obtained from the University of Wyoming Department of Atmospheric Science and
acquired by the United States weather service.
41
6. Long wavelength tropospheric signal
Figure 6.3: Global tropospheric stratification estimates. These are obtained from
the full spectrum or full wavelength range (blue) and band filtered wavelengths
between 8 to 64 km (green). The lines give a linear approximation, indicating the
change of stratification coefficient with time. Full wavelength range estimates are
not reliable due to correlation of the slow slip and inter-seismic deformation signals
with the long wavelength of the topography. As time passes by, the signal appears
to be better correlated with topography, as can concluded from the full spectrum
line. However, the temporal variation of the stratification is undersampled, one can
clearly see a reduction for the stratification when the long wavelengths are neglected
in the estimation. Time is relative with respect to the master date of 16 December
2005. Purple shaded areas indicate the rain season, which occurs from May through
September or October.
6.3
Region decomposition and multiscale tropospheric
delay estimation
Here I propose a new method, called region decomposition and multiscale tropospheric delay estimation, that is not biased when deformation correlates with topography, and which allows for spatial variability of the net tropospheric delays.
Region decomposition
The assumption of constant refractivity in each height layer, is limited to specific
cases. In Guerrero, a large mountain range of 2000-2400 m height blocks the atmospheric circulation close to the coast, as can be seen in the topography profile
shown in figure 6.4a). This can imply a different weather conditions on both sides
of the mountain. A coastal region will have a different stratification due to moisture
compared to a region located more inland. Here I propose to divide the area in
42
6.3. Region decomposition and multiscale tropospheric delay estimation
multiple patches. Consequently when the patches are small enough the assumption
of constant refractivity in each height layer becomes valid again within each patch.
The coefficients known at the center of the patches needs to be extrapolated to the
location of the PS points. The closest patch to a PS point is most likely going to
represent the correct atmospheric stratification. However, it could occur that the
coefficients are not reliably estimated for that patch. In case of a bad patch, the
neighboring patches should have a higher weighting factor. This is acceptable since
atmospheric signal has a smooth behavior and consequently the variation between
the different patches should be small. This is accounted for by applying weighting
that is based on the distance from the patches to the PS point, and by weighting
the correctness of the estimated coefficients in each patch. Patches having a standard deviation higher than 3 mm/km for the estimated slope, are rejected from the
estimation and regarded as outliers. The constant b, representing the shift of each
patch with respect to the reference, can no longer be neglected.
The weights wdist , based on distance, are computed using a Gaussian window with
zero mean and a standard deviation equal to range Rgauss of the filter.
wdisti =
1
√ e
Rgauss 2π
−d2
i
2
2Rgauss
(6.2)
Here di represents the distance from the PS point to patch i. 500 bootstrap simulations [Efron and Tibshirani, 1986] are applied in order to evaluate the stability
of the linear fit, giving a standard deviation for each coefficient. The inverse of the
standard deviation σ for patch i gives the weight wbooti , indicating the correctness
of the coefficient.
wbooti =
1
σi
(6.3)
The total weight for patch i follows by a normalization of the two weights multiplied.
wdisti wbooti
wi = Pn
i=1 wdisti wbooti
(6.4)
With n the number of patches. Each patch has two weights, one for each coefficient.
The extrapolated coefficients at the PS location are given as:
KP S =
bP S =
n
X
i=1
n
X
wi Ki
(6.5)
wi bi
i=1
The estimation of the long wavelength tropospheric signal is based on the variation
of the number of patches. The filter range is chosen to be half of the patch size and,
in this way, is determined by the number of patches.
43
6. Long wavelength tropospheric signal
Multiscale tropospheric delay estimation
Deformations at the surface can be caused by numerous confounding processes,
like ongoing subduction, creep and slow slip, acting at certain wavelength scales
[Lin et al., 2010]. In Guerrero, the 2006 slow slip deformations and the inter-seismic
deformation correlate well with the long wavelength of the topography. For this reason convential estimation of the stratification coefficient K is biased. Tropospheric
stratification delays act at all wavelength scales and can be more reliably estimated
by correlateing band filtered phase and topography. The idea behind the multiscale
tropospheric delay estimation and a more sophisticated estimation procedure for the
stratification slope is described in [Lin et al., 2010].
Figure 6.4a) gives an example of a simulation of phase delay as consequence of
a stratified troposphere with a slope of 13 mm/km and a constant of 18.71 mm.
On top of the simulated phase, white noise was added with a standard deviation
of 2 mm. Figure 6.4b) shows the relation between of the phase and topography
for the full interferogram and High Pass (HP), Low Pass (LP) and Band Pass (BP)
filtered phase and topography. From the figure it is clear that the slope K is reliably
estimated in each case. The constant however is filtered out and cannot be estimated
when using the multiscale approach.
Combining region decomposition and multiscale tropospheric delay estimation
While the slope is estimated well in the multiscale approach, the constant b, required for the region decomposition, cannot be determined reliably since it is being
filtered out. However, a relation between the slope and the constant is found by
using sounding data, shown in figure 6.2.
Since the refractivity is only integrated for the 0 to 9600 m height range, it excludes
the delay introduced by the atmosphere at higher altitudes. At 9600 m the refractivity is 98 ppm with a maximum variation of ± 1.5 ppm. Here, the delay above 9600
m is assumed to be constant for each acquisition, and for this reason is neglected
when computing the delay.
Assuming a linear relation between the net atmospheric delay and the height, makes
it possible to compute a zero-crossing height where the net delay has reduced to zero.
This is however difficult to determine from the net delay. The sounding data give a
too much localized profile, resulting in variations for the computed net delay. The
zero-crossing height, however, can also be estimated from the mean tropospheric
delay as shown in figure 6.5a). Mathematically, the constant can then be defined
as the value required to have a zero tropospheric delay at the zero-crossing height
h0 (a), which is computed from a linearization of the mean delay at height a.
b = −Kh0 (a)
(6.6)
The assumption of the zero-crossing height is only valid for that specific linearization
44
6.3. Region decomposition and multiscale tropospheric delay estimation
(a) Simulated topography related atmosphere
(b) Relationship between phase and topography in different wavelength bands
Figure 6.4: Simulation example of multiscale tropospheric delay estimation. Figure
a) shows a profile of the topography for Guerrero. Phase is simulating using a
principle of a stratified atmosphere with a slope of 13 mm/km and a constant of
18.71 mm. White noise with a standard deviation of 2 mm was added on top of
this modeled signal. Figure b) shows the relation between phase and topography
for the full wavelength range, a Low Pass (LP) filter, four Band Pass (BP) filters
and as last a High Pass (HP) filter. The slope is well approximated in each case.
The constant on the other hand is filtered out and cannot be reliable estimated
when using the multiscale approach. Standard deviations are estimated from 400
bootstrap iterations. The principle behind multiscale tropospheric corrections and
a more sophisticated estimation is given by [Lin et al., 2010].
45
6. Long wavelength tropospheric signal
(a) Mean tropospheric delay
(b) Zero crossing heights
Figure 6.5: Mean tropospheric delay estimated from the Mexico City Airport and
Acapulco tropospheric delays (0-9600 m height range), computed from sounding
data, as visualized in figure 6.2b). Delay introduced by heights larger than 9600 m
(98 ppm) is assumed to be constant for each acquisition. Dashed line shows a linear
fit estimated from the height range 0-1 km. Intersection of the linear fit with the
height axis give the height at which the topographic delay has reduced to zero, at
a refractivity of 98 ppm, under the assumption of a linearized delay at the lower
heights. Figure b) shows the zero height crossings (98 ppm) for a range of height
intervals (solid line) and the linear regression fit (dashed line).
height. When applying the Taylor expansion (till first degree) for the lower height
range, one finds for each height the corresponding zero-crossing height as is visualized
in figure 6.5b). Using the mean height and estimated slope K of each patch together
with the linear approximation of the zero-crossing height gives the constant of each
patch.
6.4
Estimation results
DEM errors scale with perpendicular baseline and not with topography. Still they
cause a bias in the estimation of the stratification coefficient K. For this reason the
unwrapped phase, from now on referred to as phase, is corrected for DEM errors
prior to the estimation of the long wavelength tropospheric signal. For convenience,
the phase is corrected for the master atmosphere as well. The initial estimates for
the DEM errors and master atmosphere are computed as described in section 7.1.
Figure 6.6a) shows the phase (u) after correction for an initial estimate of the DEM
errors (d) and master atmosphere (m).
Slow slip deformation signals and topography correlated well in the long wavelength.
For this reason, the phase and topography are high passed filtered in order to remove
the wavelengths larger 64 km. The remaining signal is low pass filtered, to reduce
46
6.4. Estimation results
noise, passing only wavelengths larger than 8 km. This band (8-64 km) is chosen
based on consisitent estimates for the stratification coefficients in the different subbands, given in appendix B.
From the topography profile, top of figure 6.6, it can be seen that mountains, close
to the coast, form an obstacle for the atmospheric circulation. This can result in a
different wheather condition on each side of the mountain. To be able to account
for this variation the area is divided into 75 square patches.
Depending on the local weather conditions, one SAR acquisition will be more subjected to atmospheric delays than to an other. By visual inspection of the 7 October
2005 and 17 March 2006 interferograms, shown in figure 5.5c), it is clear that a strong
net stratification delay exist. In other words topographic features, figure 6.4a), are
present in the interferograms. Figure 6.7 shows the filtered topography and phase
(u − dm) of the 7 October 2005 interferogram, top row, together with an example
for the estimation of the stratification coefficient of a patch, second row. Strong correlation between filtered topography and phase remains. The top right figure shows
the estimated net stratification delay of the filtered data, which is reduced to a multiplication between the filtered height and the estimated stratification coefficient.
This estimation approximates very well the filtered phase, indicating that the filtered signal consist mostly out of net stratification delays, with a lesser contribution
from the slave atmosphere. The bottom row gives the estimated long wavelength
tropospheric signal, which follows from the sum of the multiplication of the heights
with the stratification coefficient and the constant.
Figure 6.6b) and c) show respectively the estimated long wavelength tropospheric
signal (a) for each interferogram and the phase after correcting for it (u − dma). As
was to be expected, strong net stratification is estimated for the 7 October 2005 and
17 March 2006 interferograms. By visual inspection of the corrected phase one can
conclude that the estimation worked properly. Overall the interferograms become
more consistent with eachother. Unfortunately it is not always clear if new small
features are now introduced due to a wrong estimate or are the result of tropospheric
stratification. Overall one can conclude that the topographic features are strongly
reduced in the corrected interferograms.
An other way to assess the correctness of the results is to inspect the estimates for
the master atmosphere, DEM errors, inter-seismic velocity or secular motion and
the slow slip deformation using the estimated long wavelength tropospheric signal
as source data. If the estimation is done properly, the estimated signals should
correlate well with topography. Figure 6.8 show these estimates and confirms this
statement. More information about how these estimates were computed is addressed
in section 7.1.
47
6. Long wavelength tropospheric signal
(a) Phase corrected for DEM errors and master atmosphere (u − dm)
(b) Long wavelength tropospheric signal (a)
48
6.4. Estimation results
(c) Phase corrected for DEM errors, master atmosphere and long wavelength tropospheric signal (u − dma)
Figure 6.6: Estimation of the long wavelength tropospheric signal, known as the
tropospheric stratification delay, for each interferogram. Figure a) shows the unwrapped phase (u) corrected for DEM erros (d) and master atmosphere (m), which
is used as starting point in the estimation. Figure b) gives the estimated tropospheric
stratification delay (a) for each interferogram. The estimation has been performed
by combining the region decomposition and the multiscale approach, as is described
in section 6.2. To do so, phase and heights were band pass filtered for wavelenghts
between 8-64 km and spatially divided into 75 approximately square patches. The
star indicates the reference area.
49
6. Long wavelength tropospheric signal
Figure 6.7: Example of the 7 October 2005 interferogram for combining region
decomposition and multiscale tropospheric delay estimation. The first two figures
show the band filtered (8-64 km) topography and phase (u − dm). The area is
divided into 75 approximately, red, square patches. The middle figure shows a
scatter plot between filtered phase and topography for the highlighted patch. The
slope or stratification coefficient for each patch is estimated from Least Squares
while the standard deviation is obtained by applying 500 bootstrap iterations
[Efron and Tibshirani, 1986]. In case the estimated slope of a patch is not well
constrained, it is down weighted based on its estimated standard deviation. The
top right figure shows the estimated net stratification delay of the filtered data,
which reduces to a multiplication between the filtered height hf ilt and the estimated
stratification coefficient K. The bottom figures show from left to right the original topography (full spectrum), the product of these heights with K and the final
estimate for the long wavelength tropospheric signal of the 7 October 2005 interferogram.
50
(f) SSE step size (a)
(e) Secular motion (a)
(g) Master atmosphere (a)
(c) Master atmosphere (u)
(h) DEM error (a)
(d) DEM error (u)
Figure 6.8: Influence of the long wavelength tropospheric signal on the time series estimates. First and second row give the
estimates for the secular motion, SSE deformation, master atmosphere and DEM for the unwrapped phase (u) and the long wavelength
tropospheric singal (a) respectively.
(b) SSE step size (u)
(a) Secular motion (u)
6.4. Estimation results
51
Chapter 7
PS-InSAR analysis of the 2006 SSE
Section 7.1 describes the time series analysis applied in order to extract the slow
slip deformation signal. This is followed by section 7.2 pressenting the results of the
analysis.
7.1
Extracting 2006 Slow Slip deformation signal
When considering the GPS time series of the 2006 SSE, see appendix A, one can
identify three stages: prior, during and after the SSE. Prior and after the event, the
only contribution to the surface deformations is coming from ongoing subduction.
During the event however, slip on the interface causes extra surface deformations.
While GPS has a high temporal resolution, the InSAR data have a considerably
lower temporal acquisition rate. None of the SAR data in our dataset are acquired
during the SSE itself (March 2006 till end of December 2006), and consequently no
transition stage is observed in InSAR time series. The entire event is visible in the
interferograms spanning the event.
My time series model consists out of a functional model, describing the relation
between observations and unknowns parameters (secular motion or inter-seismic velocity, slow slip deformations, DEM error and master atmosphere), and a stochastic
model, where unknowns (noise and slave atmosphere) are described by a stochastic
process in the form of probabilities.
7.1.1
Functional model
To constrain modeling it is assumed that the secular rate before and after the event
is identical. This is plausible when considering the GPS time series (see appendix
A). Figure 7.1 gives a simplified visualization of the time series for a random PS
point. Since no acquisitions were made during the SSE itself, the time series shows
a discrete jump at tSSE corresponding to the slow slip deformation. In time, the
phase φ behaves linearly as:
(
vsec t + b1 , t < tSSE
φ=
(7.1)
vsec t + b2 , t > tSSE
53
7. PS-InSAR analysis of the 2006 SSE
Figure 7.1: PS-InSAR time series model of the 2006 SSE. The time series for a
single PS point is visualized. Circles indicate the slave acquisitions, while the master
is indicated by the rectangle. The model assumes that the secular rate before and
after the event is identical. The shaded area indicates the SSE. Since no acquisitions
are made during the event itself, the SSE displacement can be modeled as a discrete
step.
Here vsec [rad/year], t [year], b1 and b2 [rad] represent respectively the secular motion
or inter-seismic velocity, time, the bias before and after the event. Conversion from
radians to meter is done by multiplying with a scaling factor of with -λ/4π. The
size of the SSE φSSE is given by the difference between the two biases:
φSSE = b2 − b1
(7.2)
An interferogram is the change in phase between the master and slave acquisition,
acquired at respectively tm and ts . Assuming only secular motion and deformation
due to the SSE the change in phase becomes:
(
vsec ∆t,
tm , ts < tSSE
∆φ =
(7.3)
vsec ∆t + φSSE , tm < tSSE < ts
This, however, is not complete. The signal of the unwrapped interferograms that is
obtained after processing consists out of a superposition of different contributions.
Besides the deformation signal from the SSE and deformation due to ongoing subduction, there will be DEM errors, net atmospheric delays, noise and possibly orbital
and unwrapping errors.
The residual topographic phase φtopo , the flat earth and DEM are subtracted during
interferogram formation, scales with perpendicular baseline Bperp .
φtopo = ctopo B perp
(7.4)
With ctopo a proportionality constant that can be translated to DEM error ∆h as:
∆h =
54
−λRsinθ
ctopo
4π
(7.5)
7.1. Extracting 2006 Slow Slip deformation signal
With the wavelength λ, the range R and the angle of incidence θ.
Depending on the local circumstances, one acquisition will be subjected to more
atmospheric distortion compared to another. Two components of atmospheric errors can be identified. These are an atmospheric component that is present in every
interferogram, called master atmosphere am , and another one that varies for each
interferogram, indicated by the term slave atmosphere as .
The slave atmosphere is modelled stochastically by updating the variance-covariance
matrix, see section 7.1.2. Keeping this all in mind, two additional terms, ctopo and
am , are added to the model. The deterministic model can be written in matrix
format as:
∆φ = Ax + e




∆t1 1 0 Bperp1
∆φ1
 .
.. ..
.. 
 .. 
 ..


. .
. 
 . 

 vsec



..
.. 
∆φt 
∆tk−1 . 0


. 
k−1 

  am 
E{
 ∆φt } = 


.
.
.. 1
..  φSSE 
k 

 ∆tk
 .. 


 ..
.. ..
..  ctopo
 . 
 .
. .
. 
∆φn
∆tn 1 1 Bperpn
(7.6)
With E{·} the expectation operator, ∆φ the deformation or observation vector, A
the design matrix of the model, x the unknowns and e the noise.
7.1.2
Stochastic model
Noise is pressent in the observations, which results in residuals or errors e. The
stochastic model is written as:
D{∆φ} = Qnoise
(7.7)
With D{·} the dispersion operator and Qnoise the variance-covariance matrix, computed from the noise of the interferograms. The noise is assumed to be random and
uncorrelated in time, resulting in a diagonal variance-covariance matrix, visualized
in figure 7.2a), in which case the estimation reduces to a Weighted Least Squares,
with the weights corresponding to the noise of the wrapped interferograms. The high
variance values corresponds to the more noisy wrapped interferograms as shown in
figure 5.5a).
As stated before, slave atmosphere is modelled stochastically. Filtering out the
slave atmosphere would result in smoothing of the interferograms, with a chance of
removing part of the long wavelength deformation signal as well.
Rather than applying a high pass and low pass spatial filter in respectively time and
space, it is opted for including the slave atmosphere as a stochastic process through
updating the variance-covariance matrix. This is done by locating a non-deforming
area and by computing the variance. Prior to this, the phase is corrected for an
55
7. PS-InSAR analysis of the 2006 SSE
(a) Initial variance-covariance matrix (b) Updated variance-covariance matrix
Figure 7.2: Figure a) shows the initial variance-covariance matrix computed from
the noise of the wrapped interferograms shown in figure 5.5a). Figure b) shows
the updated variance-covariance matrix, which beside the noise also includes slave
atmosphere. The updated variance-covariance matrix is obtained by computing
the variance in the non-deforming area (figure 5.1) for each interferogram. Both
variance-covariance matrices are assumed uncorrelated in time.
initial estimate of the DEM error and master atmosphere. This leaves the phase
in the non-deforming area to be a superposition of slave atmosphere, noise and
possibly orbital and unwrapping errors. The unwrapping errors are included in the
modeling by using the noise of the interferograms as weights. Orbital errors εor on
the other hand are approximated by a Least Squares (LS) estimation and subtracting
a bilinear ramp. The corrected phase φ∗ remaining in the non-deforming area comes
from possible unwrapping errors, noise and the slave atmosphere.
φ∗ = φ − am − φtopo − εor
(7.8)
Assuming that the noise of the wrappend and unwrapped interferograms is identical, the new variance-covariance matrix Qnoise is updated to include the slave
atmosphere by computing the variance of the non-deforming area for each corrected
interferogram. It is assumed that the variance is constant in all directions, meaning
that the variance-covariance matrix is changing in time and not in space, as was the
same before.
Near Mexico City deformations due to the SSE diminished to almost zero. No SSE
deformations are expected above this region, making it valid as the non-deforming
area. However it is known that the Mexico City region itself subsides rapidly and
for this reason it is excluded from the non-deforming area, visualized as the grey
shaded area figure 5.1. Computing the variance of this non-deforming for each
corrected interferogram results in the updated variance-covariance matrix shown in
figure 7.2b).
56
7.2. Extracted slow slip deformation signal
7.2
Extracted slow slip deformation signal
By applying the time series analysis as described in the section above, I obtained
an estimate for the master atmosphere, DEM errors, secular motion or inter-seismic
velocity and the slow slip step. The results for the master atmosphere and DEM
errors are shown in figure 6.8 and 7.3. As expected, the estimates are not strongly
influenced by the atmospheric correction.
The secular motion and slow slip step, visualized in figure 6.8 and 7.4, on the other
hand are improved by correcting for the long wavelength tropospheric signal. Figure
7.4 shows that a small topography-related signal remains present in the north around
the Popocatepetl volcano. The estimated secular motion and slow slip deformation
signal have, just like the GPS estimates, long wavelengths and are approximately
parallel with the trench. Since orbital errors are not corrected for in the time series
model, the difference between the InSAR and GPS technique are expected to have
a ramp-like behaviour.
During the time series analysis, 700 bootstrap estimations are applied in order to
get an idea about the stability of the time series fit. The second column of figure
7.4 shows the standard deviations for the secular motion and slow slip step. The
standard deviations are the smallest in the reference region and increase with increasing distance with respect to the reference area. This is caused by the remaining
atmospheric signal. Near the coast the standard deviations increase rapidly. This is
due to the atmospheric variations, which in the estimation of the long wavelength
tropospheric signals cannot be estimated well. The long wavelength tropospheric
signal is better estimated in regions with large height variation, since for those regions the standard deviations remain small.
Figure 7.5 shows a cross-section of the estimated slow slip deformations using the
PS-InSAR technique, resampled to a 5500 m grid, and compares it to the slow slip
estimates obtained from GPS analysis [Vergnolle et al., 2010]. All point and stations
within a distance of 15 km are included in the cross-sectional plot, showing a good
fit between the PS-InSAR and GPS technique. This indicates that orbital errors
appear to be small.
57
7. PS-InSAR analysis of the 2006 SSE
(a) Master atmosphere (u − a)
(b) DEM error (u − a)
Figure 7.3: Master atmosphere and DEM error estimated from time series analyses
after correcting phase for the long wavelenght topographic signal (u − a).
58
7.2. Extracted slow slip deformation signal
(a) Secular motion (u − a)
(b) Secular motion σ (u − a)
(c) SSE step size (u − a)
(d) SSE step size σ (u − a)
Figure 7.4: Secular motion (first row) and 2006 SSE step (second row), estimated
from time series analyses after correcting phase for the long wavelenght topographic
signal (u−a). The second column gives the estimated bootstrap standard deviations
(700 estimations). The standard deviations are the smallest near the reference area
and increase, due to remaining atmospheric signal, with increasing distance. Near
the coast the standard deviations increase rapidly. This is due to the atmospheric
variations, which in the estimation of the long wavelength tropospheric signals cannot be estimated well.
59
7. PS-InSAR analysis of the 2006 SSE
Figure 7.5: Comparing GPS and PS-InSAR slow slip deformation signal (resampled to a 5.5 km grid). Both techniques maintain the same definition for the slow slip
deformation. Black triangles indicate the location of the continuous GPS stations.
The figure on the right shows the projection of all PS points and GPS stations within
15 km distance from the profile as drawn in the left figure (black line). The estimated InSAR slow slip deformation signal has not been corrected for orbital errors.
From the cross-section it is clear that there is a good match between the time series
InSAR (grey) and GPS estimates (red). Uncertanties for GPS (red) and InSAR
(black) are drawn for 2σ. Differences between both techniques near the coast can
be explained by larger uncertainties for the InSAR estimates. InSAR uncertainties
are obtained by computing the variance of 700 bootstrap estimations.
60
Chapter 8
Modeling the 2006 SSE
The question remains, where and with what magnitude slip occurred on the interface. This chapter describes the applied modeling and results of the slipping
due to the 2006 SSE. Section 8.1 describes the model using InSAR data only. The
surface displacements, computed from rectangular dislocations [Okada, 1985], assume an homogeneous elastic half space. In this way, we are able to model the
surface displacements with low residual compared to the set of observations. However the applied model does not provide physically reasonable slip. For this reason
an additional constraint on the slip direction, the long wavelength signal and the
smoothness of the slip solution are added to the model. This is all described in
section 8.2. Finally, section 8.3 presents the results and conclusions.
8.1
Slow slip model
After the time series InSAR analysis (chapter 7), we continue with modeling of the
SSE. However, before setting up the model, one needs to consider what unknowns
are going to be solved for. The goal is to extract the location and magnitude of the
fault slip.
8.1.1
Fault slip
Three components of fault slip can be defined, being strike slip, dip slip and opening.
Strike and dip slip are both the slip occurring respectively parallel to the strike and
dip direction. Strike is defined to coincide with the fault line whereas dip is defined
on the subduction interface while being perpendicular to the strike direction. Opening is, as the names indicates, the separation between the overriding plates and is
defined perpendicular to the subduction interface.
The observation model with fault slip as an unknown can be written as:
d = AS + e
E{d} = AS
(8.1)
With d the displacement or observation vector, A the design matrix of the model,
61
8. Modeling the 2006 SSE
Figure 8.1: Modeled Guerrero subduction zone. The interface consists out of two
first stages as given in figure 4.3. The third stage is neglected since slow slip is not
expected to extend in this region. The modeled interface consists out of 20 patches
in strike and 14 in dip direction.
S the slip vector, e the residuals or errors and E{·} the expectation.
8.1.2
Subduction zone interface
The location of the slip is constrained by the subduction interface. As described
in detail in section 4.2, the subduction interface consisted out of three stages. In
the first stage, the continental plate dips at about 60 to 80 km outside the coast at
an angle of 15◦ , reaching a depth of 40 km. Here the interface enters the second
stage and becomes sub-horizontal. The third stage, starting approximately 200 km
from the coast, is neglected in the modeling of the interface, since slow slip is not
expected to occur in this region. The modeled interface, as shown in figure 8.1 is
divided into 20 patches in strike and 14 in dip direction, giving approximately 20
km square patches (trench 75 km outside the coast).
8.1.3
Surface deformations
The surface deformation Gij at location i as consequence of a unit slip on patch j of
the subducting interface is given by [Okada, 1985], assuming a homogeneous elastic
half space (poisson ration of 0.25). From now on, Gij will be referred to as Green’s
coefficient. Every patch results in three Green’s coefficients defined in East, North
and Up (ENU) direction. The total surface deformation in northern direction dN
i
at position i corresponds to a linear combination of the slip displacement of every
62
8.1. Slow slip model
(a)
(b)
Figure 8.2: Projection of the East, North and Up (ENU) unit vectors (i,j,k) to the
Line of Sight (LOS) of a descending SAR platform assuming a flat Earth. Figure a)
and b) give respectively the front and top view of the platform.
patch contributing to the deformation at i.
N
N
N
dN
i = Gi1 S1 + Gi2 S2 + · · · + Gin0 Sn0
n
X
dN
=
GN
i
ij Sj
(8.2)
j=1
With Sj the slip of patch j, n0 the total amount of patches and N the indication
of the North. Since the estimates for the SSE deformation, obtained from the time
series InSAR analysis, are given in Line Of Sight (LOS) direction, the Green’s coefficients need to be transferred to LOS as well.
The transformation from the LOS to the East North Up (ENU) coordinate system
is not unique. One would at least need three InSAR measurements of a different
direction or heading to be able to solve for the three components ENU. In other
words a combination from ascending and descending data would still result in an
undetermined system.
The transformation from ENU to LOS is on the other hand is trivial and requires
only the satellite heading and the incidence angle to be known. The heading γ
is defined as the angle between the North and the flying direction. This angle is
clearly different for ascending and descending tracks. The incidence angle θ corresponds to the angle between the incoming signal on the ground and the local vertical.
Figure 8.2 shows the projection form the ENU unit vectors i, j and k to the LOS
direction in case of descending orbit. Figure 8.2a) gives the top view of the platform.
The Looking Direction (LD), as well as the LOS are defined positive away from the
63
8. Modeling the 2006 SSE
platform. Projection of the N and E components to LD yields:
LDi = −cos(γ)i
LDj = sin(γ)j
(8.3)
LD = −cos(γ)i + sin(γ)j
Figure 8.2b) gives the plane perpendicular to the flight direction. From this plane
it is easy to make the projection of the U and the LD components to the LOS.
LOSk = −cos(θ)k
LOSLD = sin(θ)LS
(8.4)
LOS = −cos(θ)k + sin(θ)LD
By combining the two equations, the transformation can be written as:
LOS = −sin(θ)cos(γ)i + sin(θ)sin(γ)j − cos(θ)k
(8.5)
Equation 8.5 gives the transformation from ENU to LOS assuming a flat earth. For
the conversion it is important that variation of the angle of incidence and heading
are implemented as well. In case of ERS/Envisat the curvature of the earth will
result in an additional 2 to 3◦ for the incidence angle for the near and far range
respectively. The heading on the other hand is latitude dependant.
8.2
Model constraints
The estimation at this stage gives unreasonable slipping results. Different slip solutions exist, given a good approximation of the surface displacements. To reduce the
ill-posed nature of the model additional constraints are required.
8.2.1
Slip towards subduction zone only
It is physically resonable to assume that most of the slip occurs in the direction of
subduction. Besides strike slip, opening is assumed to be small and is neglected.
Assuming only dip slip, the model of observations can be written as:
InSAR
DS
E{dInSAR
DS
LOS } = GLOS
 InSAR 
 DS InSAR
dLOS1
GLOS11
···
 .. 

.
..
.
E{ . } = 
.
.
dInSAR
LOSn
InSAR
GDS
LOSn1
···
InSAR
GDS
LOS1n0
InSAR
GDS
LOSnn0


DS1
  .. 
 . 
(8.6)
DSn0
Dip slip, only in the direction is subduction, is implemented by a non-negative
Least Squares approximation, where the observations as well as the design matrix
are scaled with the weight matrix W, which is the Cholesky decomposition of the
variance covariance matrix inverse Q−1 .
WT W = Q−1
64
(8.7)
8.2. Model constraints
8.2.2
GPS for constraining long wavelengths
The estimated SSE step from the time series InSAR analysis may be affected by orbit
errors and ionospheric delays, since these are not corrected or estimated for. For this
reason, continuous GPS (table 5.1) data were used to constrain the long wavelength
components of SSE. It is assumed that the long wavelength can be corrected for by
including a planar correction, ax + by + c,for the InSAR data in the model. The
Green’s coefficients at the GPS sites do not have to be transformed to LOS, since
the estimated SSE step is provided in ENU components. Keeping this all in mind
the model can be written as:

 DS InSAR
 InSAR 

GLOS
x y 1 DS
dLOS
GP
S

 DS
S 
 dGP

GN
0 0 0 
N
 a 
} = 
E{
(8.8)


GP
S
GP
S
 d

 GDS
0 0 0  b 
E
E
S
GP S
c
dGP
U
GDS
0 0 0
U
The design matrix consist out of n + 3m rows, with m the amount of continuous
GPS stations, and n0 + 3 columns. The covariance matrix Q used for this model is
an uncorrelated combination of both observation techniques used. For the observations that were the product of the InSAR analysis, the variance-covariance follows
from the bootstrapping estimation, which was performed during analysis of the SSE
(see section 7.2). The GPS variance-covariance matrix was constructed using the
standard deviations as provided in table 5.1. The GPS stations were assumed to be
uncorrelated.
QInSAR
0
Q=
(8.9)
0
QGP S
By using Best Linear Unbiased Estimation it is possible to get an estimate for the
unknown dip slip for every patch as well as the coefficients a, b and c of the plane.
During the estimation, the covariance matrix needs to be inverted. For this reason
the amount of InSAR observations are resample to a grid of 5500 m, consisting out
of about 1300 points. Correcting the long wavelength does not result in a loss of
information.
8.2.3
Laplacian smoothness
Multiple slip solutions exist that approximate the set of observations very well. However, these slip solutions often have unreasonable amounts of slip on the interface.
Adding the requirement that slip should vary smoothly makes the model more realistic and avoids the model to fit the measurement noise, like residual atmospheric
delays that are still present in the set of observations.
I implemented the smoothness criteria by minimizing the sum of the second derivative of the slip for neighboring patches, known as Laplacian smoothing. Figure 8.3
shows an interface assuming ns patches in strike and nd in dip direction. In case of
patch i, the Laplacian can be written as:
1
1
min
(Si−1 − 2Si + Si+1 ) + 2 (Si−ns − 2Si + Si+ns )
(8.10)
ws2
wd
65
8. Modeling the 2006 SSE
Figure 8.3: Laplacian smoothness of patches on subduction interface. The example shows ns and nd patches in respectively the strike and dip direction. Every
patch has an index, which increases bottom to top, from left to right. Two special
cases are indicated with grey shaded colors. These are the patches where the Laplacian only remains in one direction. The corner patches do not have an individual
Laplacian. Still smoothness is maintained because all patches are part of adjacent
Laplacian.
The first and second term of equation 8.10 describe respectively the second derivative of the slip S between the adjacent patches in strike and dip direction, with ws
and wd the width in each direction. Patches on the edges of the interface will have
a reduced equation. In case of edge patches parallel to the strike direction only the
first term of equation 8.10 remains, which is the derivative in strike. Visa versa for
the second term in case of edge patches parallel to the dip direction. The four corner
patches do not have an individual Laplacian. Their smoothness however is included
in the Laplacian of the neighboring patches. The minimization in the model is taken
care of by adding a zero to the set of observations.
Assuming that an interface has three square patches in strike and dip direction the
Laplacian design matrix ∇ can be written as:

1 −2 1 0
0
0 0 0
1 0 0 −2 0
0 1 0
1 
0
1
0
1
−4
1 0 1
∇= 2
w 
0 0 1 0
0 −2 0 0
0 0 0 0
0
0 1 −2
66

0
0

0

1
1
(8.11)
8.2. Model constraints
Including the smoothness criteria into the model (equation 8.8), results in
 DS InSAR
 InSAR 
GLOS
dLOS
 DS GP S
S 
 dGP
 GN
 N

 DS GP S
GP S 
E{
}
=
d
 GE
 E


 dGP S 
 GDS GP S
U
U
0
∇
x
0
0
0
0
y
0
0
0
0

1 

 DS
0 
 a 

0
 b 
0 c
0
(8.12)
The degree or amount of smoothness λ, known as the smoothness factor, is taken
into account in the variance.
σ∇ = λ
(8.13)
With Q∇∇ a diagonal matrix, the updated variance-covariance matrix of the model
is given as:


QInSAR
0
0
QGP S
0 
Q= 0
(8.14)
0
0
Q∇∇
Different values of λ result into a different slipping solution. However the question remains, what degree of smoothing should be imposed on the model? In literature different methods are given on how to select the smoothing factor. Examples include visual inspection [Freed, 2007], minimizing the misfit-roughness [Jonsson et al., 2002]
[Wright et al., 2004], a fully Bayesian inversion [Fukuda and Johnson, 2008], and
others.
Better than a visual inspection would be to plot the misfit-roughness curve of the
slip solution and to select a smoothness factor that minimizes both the misfit and
the roughness. However, this is a trade-off, as a minimum is not obtained. Rather
than selecting a single smoothness value, it is better to select the value which maximizes the the joint Probability Density Function (PDF) of the modelled parameters,
maximum likelihood solution, which includes the variation of the smoothness. I applied this method and compared the smoothness of the maximum likelihood with
the position on the roughness-misfit curve.
Misfit-roughness trade-off
The misfit Ω is defined as the weighted residual between the modeled surface deformations and the original observations. In case of equation 8.12 this results in:
−1
T
Ω = εTInSAR Q−1
InSAR εInSAR + εGP S QGP S εGP S
(8.15)
With εSAR and εGP S the residuals of the InSAR and GPS data. The roughness ρ
equals the mean absolute Laplacian of the slip model [Jonsson et al., 2002].
Pk
ρ=
|pi |
k
i
(8.16)
where p=∇DS and k is the number of rows of the Laplacian design matrix ∇. By
computing the misfit and the roughness for various values of λ it is possible to obtain
67
8. Modeling the 2006 SSE
the misfit-roughness curve, as is visualized in figure 8.4a. It is clear that there is not
a single smoothness factor that minimizes both the misfit and the roughness, but
rather a range of acceptable values. Even in this range, the question remains, what
value of λ is best to pick?
Maximum likelihood from Linear Bayesian inversion
Neither the visual inspection nor the misfit-roughness method give a justification
why a specific smoothness factor is best. [Fukuda and Johnson, 2008] presented a
non-linear Bayesian inversion to estimate the maximum likelihood distribution of λ.
From this distribution one is able to select the most likely value for λ. The Bayesian
is given as:
P (d|m)P (m)
−∞ P (d|m)P (m)dx
P (m|d) = R ∞
(8.17)
Here λ is used as hyperparameter in the Bayesian inversion, with P (d|m) the conditional PPDF of the observations d given the model m, P (m) the prior PDF of
the model and P (m|d) the posterior PDF of the model given the observations. The
conditional PDF is computed assuming a Gaussian distribution with mean ∇DS
and variance-covariance matrix Q. Using the model as given in equation 8.12 the
conditional PDF becomes:
P (d|m) = Ce
−εT Q−1 ε
2
P (d|m) = Ce
−1
2
−1
T
(εTInSAR Q−1
InSAR εInSAR +εGP S QGP S εGP S )
(8.18)
With C a normalization constant. Assuming a Gaussian distribution with zero mean
−1
and a variance-covariance of λ2 ∇T ∇
for the prior PDF, gives:
P (m) = C 2πλ2
−n0 /2
e
−λ−2
2
[(DS∇)T DS∇]
(8.19)
Here n0 gives, as before, the number of patches on the interface. Since λ is a
hyperparameter it will be present as scalar factor. Substituting equation 8.18 and
8.19 into 8.17 yields:
−n0 /2 −1 [εT Q−1 ε+λ−2 (DS∇)T DS∇]
(8.20)
P (m|d) = C 0 λ2
e2
With C 0 a normalization constant. Figure 8.4b) gives the posterior probability
distribution. The maximum likelihood solution of the smoothness together with the
corresponding misfit-roughness solution is indicated by the red star in figure 8.4.
8.3
Model results and discussion
In this section the maximum likelihood slip distribution obtained after combining
GPS and InSAR is discussed and compared with the GPS solution. Moreover a
comparison is made with the model results of other people [Larson et al., 2007],
[Correa-Mora et al., 2009] and [Radiguet et al., 2010]. As last, the slip location is
evaluated with the location of the Ultra Slow velocity Layer [Song et al., 2009].
68
8.3. Model results and discussion
(a) Misfit-roughness curve
(b) Maximum likelihood estimation
Figure 8.4: Selecting a valid smoothness factor λ. Figure a) shows the misfitroughness curve. Good smoothness values are selected by minimising both the misfit
as well as the roughness. This however does not give a single smoothness value.
Figure b) shows the posterior PDF computed from the Bayesian inversion with λ as
hyperparameter. The maximum likelihood solution, red star, gives the most likely
smoothness value.
GPS + InSAR
Slip is forced to go to zero near the borders. However, increasing the extent of
the modeled interface, both in strike and dip direction, does not change the slip
distribution, indication that the interface is taken large enough. While the depth
of the slip is not well constrained, the spatial location and size of the slip pattern
does not change when changing the interface. For this reason, it is more appropriate to compare the spatial extend of the slip distribution with the rupture zones
of past earthquakes, indicated by the grey shaded polygons, rather than comparing
the depth of slip with a depth range for the seismogenic zone.
Different slip distributions exist that give the same surface displacements. The slip
distribution shown in figure 8.5a) corresponds to the maximum likelihood solution
obtained by combining the InSAR and GPS data. Small amounts of slip are modeled
towards and partly into the seismogenic zone to East and the West. However, the
regions where this occurs are not constrained by the InSAR data, indicated by the
dashed red line. The maximum slip of approximately 0.19 m and the surrounding
slip is constrained by both techniques.
Figure 8.6a) gives the propagated errors (2σ) for the maximum likelihood solution.
Lower standard deviations are observed in the neighborhood of the InSAR and
GPS data. The InSAR and GPS data coincide spatially. For this reason, it is
valuable to process the adjacent track to see the full effect of the InSAR data on the
solutions precision. However, rather than showing the solution of a single value for
the smoothness, errors should be propagated over the full range of smoothness values.
That result, shown in figure 8.6b) for 2σ, gives the largest standard deviations for
69
8. Modeling the 2006 SSE
the regions of maximum slip.
Besides making sure the slip distribution seems reasonable, the modeled surface deformations needs to approximate the original ”observations” or data. Figure 8.7a)
and b) show respectively the original and the modeled surface deformations, when
combining GPS and InSAR. A cross-section of both figures is shown in figure b).
From this profile one can conclude that the modeled deformation (blue dashed line)
correlates well with the observations (red for GPS, gray and black for InSAR). InSAR residuals are present at the smaller wavelengths. This is due to the smoothness
requirement that does not allow low wavelength deformations at the surface. Moreover it appears that the InSAR residuals are uncorrelated and can be regarded as
atmospheric noise.
GPS
In order to see the added value of the InSAR data, it is opted to compare the
slip distribution (GPS+InSAR) with the slip distribution obtained when using GPS
observations only. The InSAR data is down weighted in the model inversion by
increasing the standard deviations by a factor of 1000. This can be observed in the
cross-sectional plot of figure 8.7c), where the model now fits the GPS data. This slip
distribution, shown in figure 8.5b), models slip closer towards the coast and more
to the Northwest compared to the slip solution in figure b). A significant part of
the slip enters into the seismogenic zone and the Guerrero gap. However, energy
appears not to be released over the full locked area. The maximum slip is a bit
higher for the combination GPS and InSAR. By visual inspection of figure 8.5a) and
b) it is clear that the regions in absence of InSAR data fit the GPS. Moreover, it can
be concluded that the InSAR constrains slip to be more downdip of the seismogenic
zone away from the Guerrero gap and is able to constrain the Eastern extend of
the slip solution. Constraining the Western extend of the slip, requires the adjacent
InSAR track to be processed.
Comparing to other models
Until now, modeling has only be done using the GPS technique. [Larson et al., 2007]
modeled the 2006 SSE by modeling the subduction interface of [Iglesias et al., 2004]
in four patches and by applying a Network Inversion Filter on the time series data
of 10 GPS stations. Their model results indicate most of the dip slip to be occurring
on the horizontal stage of the interface. They suggest that part of the slip is released
into the seismogenic zone. However, due to the large size of their modeled patches
it is not clear till what depth slip extends.
[Correa-Mora et al., 2009] confirms the results of [Larson et al., 2007], by applying
inverse modeling of continuous GPS data using fewer observations, and also indicated
slip to reach shallow depths of 25 km. However, they also noted that in case slip is
forced to depths of 31 km and lower, the misfit increases to only 10%. Figure 8.8a)
shows the slip distribution, having the smallest misfit, obtained from Finite Element
70
8.3. Model results and discussion
(a) GPS+InSAR
(b) GPS
Figure 8.5: Both figures give the modeled maximum likelihood slip distribution of
the 2006 SSE, when combining InSAR and GPS, figure a), and in case of only GPS
data, figure b). The later is obtained by down weighting the InSAR observations by
a factor of 1000 in the estimation. Colors indicate the amount of dip slip, defined
positive towards the trench. Depth contours indicate the subducting interface. The
thick red solid line indicates the transition of the modeled interface’s dipping angle
from 15 to 0◦ , while the dashed red line shows the contours of the InSAR data. Grey
shaded polygons, modified from [Franco et al., 2005], give the rupture zones of past
large earthquakes. The locations in absence of InSAR data tend to model slip closer
towards the seismogenic zone. Both slip distributions have approximately the same
amount of maximum slip.
71
8. Modeling the 2006 SSE
(a) GPS+InSAR
(b) GPS
Figure 8.6: Both figures give 2σ of the modeled slip distribution displayed in figure
8.5. Figure a) represents the 2σ bound of the maximum likelihood solution (InSAR
and GPS) only. Regions with observations show lower standard deviations. In order
to see the full effect of the InSAR data on the solutions precision it is valuable to
see the effect of an adjacent InSAR track. Figure b) shows the case when errors are
propagated for the full range of smoothness values. Clearly the regions that slip the
most are most affected by a change in smoothness. Colors indicate the magnitude
for each patch on the interface. Depth contours indicate the subducting interface.
The thick red solid line gives the transition of the modeled interface’s dipping angle
from 15 to 0◦ , while the dashed red line shows the contours of the InSAR data. Grey
shaded polygons, modified from [Franco et al., 2005], give the rupture zones of past
large earthquakes.
72
8.3. Model results and discussion
(a) ”Observations”
(b) Modeled (GPS+InSAR)
73
8. Modeling the 2006 SSE
(c) Modeled (GPS)
Figure 8.7: Comparing the ”observations”, estimated slow slip deformations from
InSAR and GPS, shown in figure a), with the modeled slow slip deformations obtained by inverting the combination GPS and InSAR (figure b)), and GPS alone
(figure c)). The later one is obtained by down weighting the InSAR data by a factor
of 1000. The black arrows indicate horizontal GPS displacements. Vertical GPS
displacements are drawn vertically, upward is positive, in black solid lines. The
thick red solid line indicates the transition of the modeled interface’s dipping angle
from 15 to 0◦ . The cross-sectional plot indicates the InSAR observations (grey and
black) and GPS observations (red) for 2σ confidence. The dashed blue line gives the
estimated modeled surface deformations.
Modeling. Figure b) shows the maximum likelihood slip distribution of figure 8.5a)
with a saturated colorbar approximating the colorbar of [Correa-Mora et al., 2009].
The assumed modeled interface, indicated by the depth contours, is different for
both models. The experience, however, learned me that the spatial location of the
slip solution is not significantly affected by a different interface. Both inversions
were done with a different set of GPS stations. Slip to the East comes from a small
slip event just prior to the 2006 SSE.
[Radiguet et al., 2010] models the 2006 SSE giving zero displacements to the Eastern
GPS stations (Oaxaca, Pino and Huat) that slipped just prior the 2006 SSE (Oaxaca
state). This however, will only force the slip solution to go to zero near the East, but
does not change the overal slip distribution due to the 2006 SSE. The slip solution
shown in figure 8.9 assums approximately the same subducting interface as mine.
Slip tends to be modeled closer towards the coast and into the seismogenic zone
compared to the maximum likelihood solution, shown in figure 8.5a).
74
8.3. Model results and discussion
(a) [Correa-Mora et al., 2009]
(b) Maximum likelihood slip distribution
Figure 8.8: Figure a) shows the 2006 slip distribution by and modified from
[Correa-Mora et al., 2009]. Figure b) shows the maximum likelihood slip distribution (GPS+InSAR) with a saturated colorbar that approximates the colorbar of
[Correa-Mora et al., 2009]. The set of continuous GPS stations, different for each
model, are indicated by black triangles. Both models assume a different subduction
interface, indicated by the depth contours. Grey shaded polygons, modified from
[Franco et al., 2005], give the rupture zones of past large earthquakes. The red line
shows the contours of the InSAR data used for constraining the slip model in figure
8.5.
75
8. Modeling the 2006 SSE
Figure 8.9: The 2006 slip distribution by and modified from [Radiguet et al., 2010].
Figure 8.5a) shows the maximum likelihood slip distribution (GPS+InSAR). The
set of continuous GPS stations, almost identical for both inversion models, are
indicated by black triangles. Both modeled interfaces compare well, having the
transition of the 40 km depth approximately at the same location. The red
solid line indicates the latter. For shallow depths [Radiguet et al., 2010] uses
[Perez-Campos et al., 2008], while in my modeling I use the interface defined by
[Kim et al., 2010]. [Radiguet et al., 2010] used zero slip for the three Eastern
GPS station (Pino, Oaxaca and Huat). Grey shaded polygons, modified from
[Franco et al., 2005], give the rupture zones of past large earthquakes. The red
line shows the contours of the InSAR data used in my maximum likelihood model.
76
8.3. Model results and discussion
Ultra Slow velocity Layer
[Song et al., 2009] showed, by studying SP arrivals and teleseismic reflections, the
existence of an Ultra Slow velocity Layer (USL) or a High Pore-Fluid Pressure
(HPFP) layer near the coast of Guerrero. They concluded the USL to be well correlated with the spatial extent of earlier documented SSEs. Figure 8.10a) shows
the location of the USL together with the modeled maximum likelihood slip distribution of the 2006 SSE. Regions with slip coincide well with the USL, confirming
the conclusion of [Song et al., 2009] and strengthening the hypothesis of high pore
pressures allowing slow slip to occur. Figure b) and c) give the histogram of slip
coinciding with the ”no USL” (yellow dots) and ”medium USL to USL” (green and
light blue dots). Only the slip of the patches coinciding with the middle of the dots
are considered for the histograms. Remark that not all regions with medium USL to
USL need to have slipped. However, the regions that have slipped should correlate
with the spatial location of the ”medium USL to USL”, as is the case.
77
8. Modeling the 2006 SSE
(a) Ultra Slow velocity Layer (USL)
(b) Slip coinciding with no USL
(c) Slip coinciding with medium USL to USL
Figure 8.10: Comparing the 2006 slip distribution with the Ultra Slow velocity Layer (USL). Figure a) show the maximum likelihood slip distribution
(GPS+InSAR), overlaid with the USL from [Song et al., 2009]. By studying SP
arrivals and teleseismic reflections, they were able to identify three types of USL:
no USL (yellow dots), medium USL (green dots) and USL (light blue dots). Regions with slip coincide well with the USL strengthening the hypothesis of high pore
pressures allowing slow slip to occur. Remark that not all regions with USL need
to have slipped. Depth contours indicate the subducting interface. The thick red
solid line indicates the transition of the modeled interface’s dipping angle from 15
to 0◦ , while the dashed red line shows the contours of the InSAR data. Grey shaded
polygons, modified from [Franco et al., 2005], give the rupture zones of past large
earthquakes. Figure b) and c) give respectively an histogram of the dip slip for the
patches coinciding with no USL and an other one for the combination medium USL
and USL. Large slip values are observed for the combination medium USL and USL.
78
Chapter 9
Conclusions and recommendations
For the InSAR technique, noise due to decorrelation and atmosphere is a limitation.
The Guerrero InSAR dataset is large (100 by 350 km) and consists out of a broad
range of ground scatter characteristics, ranging from coastal and mountainous regions to cities and forests. This all makes Guerrero a challenging region for applying
InSAR to.
By applying the Persistent Scatterer InSAR time series technique, decorrelation
noise is reduced. Net atmospheric delays are estimated for the long wavelength
tropospheric signal as well as the master atmosphere. The slave atmosphere, on the
other hand, is included as a stochastic process in the model.
From a total of 19 interferograms, from which 5 interferograms spanning the 2006
SSE, I was able to estimate the deformation signal. These deformation estimates
correlate well with the GPS estimates, obtained from the analysis of continuous GPS
data.
Until now, slip models, using only GPS data as observations, show slip to be extending into the seismogenic zone. However due to poor spatial resolution of the
continuous GPS stations, the slip is not well constrained. My modeling results, assuming slip towards subduction only and smoothness in a Bayesian framework, show
that slip does not extend into the seismogenic zone, for the regions where InSAR
is combined with GPS. Locations with slip coincide well with the location of the
Ultra Slow velocity Layer, strengthening the hypothesis of slip caused by high pore
pressures.
Recommendations
• The slip distribution can be better constrained when adjacent satellite tracks
are included as well. Moreover, different platforms can be combined in order
to better constrain the extent of the slip.
• Since Guerrero is subjected to high noise due to decorrelation, mainly caused
by vegetation growth and increasing perpendicular baseline, it can be valuable
to assess the region using L-band SAR platforms, like for example ALOS,
reducing the decorrelation due to vegetation. However, when using L-band,
the signal is more sensitive to ionospheric distortion.
79
9. Conclusions and recommendations
• The time series analysis can be better constrained when the number of interferograms that span the SSE is increased. However, large baselines form a
limitation when applying a PS-InSAR. By applying the Small Baseline technique, temporal and perpendicular baseline decorrelation is reduced considerably. This makes the unwrapping in the coastal areas more reliable.
• Use of more sophisticated models that include strike slip, surface topography
and even variations in the material properties of the crust.
• Investigate the effect that the slow slip has on the stress field, in the seismogenic
zone and the seismic gap, and consequences for potentially future earthquakes.
• Investigate the temporal and spatial relation between Non-Volcanic Tremor
and SSEs. Is there a relation like is the case for Cascadia region?
• Investigate the temporal and spatial behaviour of SSEs. Are the slip events
periodic and do they occur at the same location?
• Increased spatial density of the continuous GPS stations would provide a more
reliable correction of the InSAR data for the long wavelength errors, like for
example ionospheric and orbital errors.
• Apply modeling methods to other slow slip regions.
80
Appendix A
GPS time series Guerrero
Figure A.1: Guerrero GPS time series of North component [Vergnolle et al., 2010].
Filled circles are measured daily coordinates. The grey shaded areas represent the
monitored slow slip events. One large SSE has been observed approximately once
every 4 year. Small SSEs are numbered. GPS station data is property of UNAM.
81
A. GPS time series Guerrero
Figure A.2: Guerrero GPS time series of East component [Vergnolle et al., 2010].
Filled circles are measured daily coordinates. The grey shaded areas represent the
monitored slow slip events. One large SSE has been observed approximately once
every 4 year. Small SSEs are numbered. GPS station data is property of UNAM.
82
Figure A.3: Guerrero GPS time series of Up component [Vergnolle et al., 2010].
Filled circles are measured daily coordinates. The grey shaded areas represent the
large observed SSEs. One large SSE has been observed approximately once every 4
year. Small SSEs are numbered. GPS station data is property of UNAM.
83
Appendix B
Multiscale tropospheric delay estimation
26 Nov 2004
31 Dec 2004
84
4 Feb 2004
11 Mar 2005
15 Apr 2005
85
B. Multiscale tropospheric delay estimation
20 May 2005
24 Jun 2005
29 Jul 2005
86
2 Sep 2005
7 Oct 2005
11 Nov 2005
87
B. Multiscale tropospheric delay estimation
20 Jan 2006
24 Feb 2006
5 Jan 2007
88
9 Feb 2007
16 Mar 2007
9 May 2008
89
B. Multiscale tropospheric delay estimation
11 Sep 2009
Figure B.1: Multiscale estimation of the stratification coefficient K. The correlation between phase and topography is shown in each figure for different wavelength
bands. Prior to filtering phase is corrected for DEM errors and master atmosphere
(u − dm). The top row in each figure gives from left to right, a low pass filter (> 128
km) and two band pass filters of (64-128 km) and (32-64 km). The bottom row gives
two band pass filters of (16-32 km) and (8-16 km) together with a high pass filter
(< 8 km). Slow slip and secular deformations have long wavelength deformations
and are not expected to have deformations with wavelengths smaller than 100 km.
The bands of 32-64, 16-32 and 8-16 km have estimates for K which are consistent
compared to the longer wavelengths (64-128 km), the high pass and low pass filter.
90
Appendix C
2006 SSE Small Baseline model
The SB approach differs from the Single Master approach in a few ways. First
of all there is not a single master image, implying that there will be no master
atmosphere. Moreover the amount of interferograms in case of SB is a lot higher,
resulting in redundancy. Figure C.1 gives a visualization of the time series for a
random point, together with the three cases that can occur for SB interferograms:
1. Master and slave are both acquired before the SSE
2. Master is acquired before and slave after the SSE
3. Master and slave are both acquired after the SSE
The modeling equation given by equation 7.1 remain valid. By considering the three
cases as given above the change in phase can be written as:
(
vsec ∆t
tm , ts ≷ tSSE
∆φ =
(C.1)
vsecm ∆t + φSSE , tm < tSSE < ts
The contribution of the DEM errors can be taken into account by including an error
that scales with perpendicular baseline. This all can be written in matrix format
as:
∆φ = Ax + e




∆t1
0
Bperp1
∆φ1




..
..
..
..




.
.
.
.




 ∆φn 
 ∆tn
0
Bperpn 





 ∆tn+1
 ∆φn+1 
1 Bperpn+1 



 vsec
(C.2)




..
..
..
..
E{
} = 
 φSSE 
.
.
.
.




 ∆tn+m 1 Bperpn+m  ctopo
 ∆φn+m 




∆tn+m+1 0 Bperp
∆φn+m+1 

n+m+1 







..
..
..
..




.
.
.
.
∆φn+m+k
∆tn+m+k 0 Bperpn+m+k
Here the three cases are indicated by respectively n, m and k. The values of these
variables depend on the amount of SB interferograms satisfying the criteria as set
before. The remaining estimation process is identical to the steps as described for
the PS-InSAR modeling, and differs only by the fact that the master atmosphere is
excluded in equation 7.8.
91
C. 2006 SSE Small Baseline model
Figure C.1: Small Baseline time series model of 2006 SSE.
92
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