A MECHANICAL MODEL OF THE LARGE-DEFORMATION A … · 2007. 9. 4. · Sierra Negra volcano in...

A MECHANICAL MODEL OF THE LARGE-DEFORMATION

2005 SIERRA NEGRA VOLCANIC ERUPTION

DERIVED FROM INSAR MEASUREMENTS

A DISSERTATION

SUBMITTED TO THE DEPARTMENT OF GEOPHYSICS

AND THE COMMITTEE ON GRADUATE STUDIES

OF STANFORD UNIVERSITY

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

Sang-Ho Yun

March 2007

c© Copyright by Sang-Ho Yun 2007

All Rights Reserved

ii

I certify that I have read this dissertation and that, in my opinion, it is fully

adequate in scope and quality as a dissertation for the degree of Doctor of

Philosophy.

(Howard Zebker) Principal Co-Adviser



Philosophy.

(Paul Segall) Principal Co-Adviser



Philosophy.

(David Pollard)

Approved for the University Committee on Graduate Studies.

iii

Abstract

During the past decade, Space-borne Interferometric Synthetic Aperture Radar (InSAR)

has been successfully used to measure millimeter- to meter-level deformation on the sur-

face due to physical process at depth including magma accumulation and migration, pres-

surization, and crystallization. The InSAR data of deformation have been explained using

various source models of simple geometry.

The InSAR data contain the effects of baseline, topography, deformation, and atmo-

sphere. The topography effect is removed using a Digital Elevation Model (DEM). At

Sierra Negra volcano in Isabela Island, Galapagos, both high- and low-resolution DEMs

are available, and their merits and demerits are complementary to each other. We develop

an optimal algorithm to merge the two DEMs to produce a new DEM that is superior to

both of the original DEMs. This DEM is used for InSAR processing throughout this work.

The magma chamber at Sierra Negra is believed to be a sill at a shallow depth. In or-

der to solve for the detailed geometry of the sill, we develop a new modeling technique.

The method uses a uniformly pressurized crack as a forward function in a stochastic inver-

sion scheme of simulated annealing. Binary parameters that represent the locations of the

crack elements are optimized in the simulated annealing. This modeling approach provides

physically more plausible and internally more consistent model than kinematic models.

2005 eruption at Sierra Negra poses a great challenge in forming interferograms inside

the caldera of the volcano. An earthquake of Mw 5.4 occurred 3 hours prior to the onset

of the eruption, and the maximum subsidence during the eruption is about 5.4 m at the

center of the caldera. Due to this large and complex deformation, we were not able to

form an intra-caldera interferogram using standard InSAR processing software. Thus, we

develop a new interferogram formation algorithm that involves more robust SAR image

v

coregistration and range offset image subtraction. The resulting interferogram is used for

modeling the 2005 eruption at Sierra Negra.

The 2005 eruption at Sierra Negra was modeled using InSAR and GPS data. The model

consists of three main parts: trapdoor faulting, dike intrusion, and opening-closing of a sill.

The depth and the detailed geometry of the sill was estimated from an ascending and de-

scending interferogram pair before the eruption. The estimated depth is 1.86 km ± 0.13

km. The GPS data were used to estimate pre- and post-eruptive inflation, and the deforma-

tion due to the inflation was removed from the InSAR data in order to effectively reduce

the temporal baseline of the InSAR data. In the trapdoor faulting model, the estimated

maximum slip ( 1.8 m) is at the bottom of the western end of the fault system, and about

1.5 m toward the surface, which matches the field observation very well. The equivalent

moment magnitude of the total slip was estimated to be Mw 5.7 when the shear modu-

lus is 30 GPa. For shear modulus of 10 GPa, it becomes Mw 5.4, which is the moment

magnitude of the earthquake that occurred 3 hours prior to the onset of the eruption. The

dike model showed average opening of 1.7 m and “reverse faulting” average dip slip of 1.6

m. The large dip slip is due to the interaction with the sill and the free surface. The the

sill model is the sum of two components: interaction with the trapdoor faulting event and

uniformly depressurized closing during the eruption. The interaction component showed a

wedge-like opening distribution close to the fault system, and the uniformly depressurized

sill accounted for the co-eruptive subsidence. The repeating cycle of trapdoor faulting and

eruption can produce an accumulated wedge-like structure at depth as well as on the sur-

face. We believe that the surface expression of this structure is shown as the characteristic

C-shaped sinuous ridge inside the caldera of Sierra Negra. The estimated volume decrease

at the sill was 0.124 km3, and the estimated extruded volume (dense rock equivalent) was

about 0.120 km3. This similarity suggests that there may not have been substantial amount

of volatiles in the magma before the eruption.

vi

Acknowledgments

I deeply thank my advisors, Howard Zebker and Paul Segall, for their generous support,

constructive ideas, and thoughtful encouragement. Working with Falk Amelung nurtured

my research experiences. I am also grateful to David Pollard for his readily organized guid-

ance, and Allan Rubin for his in-depth advice. Great details of fieldwork at Sierra Negra

volcano by Dennis Geist and William Chadwick enriched my perspectives. I thank Jun

Ji for his help on coding. Systematic InSAR work done by Michael Poland helped me to

draw big pictures of Sierra Negra. I appreciate Peter Cervelli’s answers to my questions

on his codes. I would like to thank Gregory Beroza and Goran Ekstrom for comments

and discussion. Special thanks go to Julia Morgan for broadening my research interests. I

acknowledge Ramon Hanssen for his generous consideration, and Yuri Fialko for his sug-

gestions and encouragement. I thank Curtis Chen, Eric Fielding, Paul Rosen, Tim Wright,

and David Schmidt for their helpful comments and tips on InSAR processing. I also thank

Matt Pritchard, Paul Lundgren, Rowena Lohman, Gareth Funning, and Yo Fukushima for

enjoyable discussions on data handling and modeling. I deeply appreciate thoughtful con-

sideration from Chuck Wicks and Zhong Lu.

I owe tons of gratitude to former and current folks at Stanford, Andy Hooper, Fayaz

Onn, Kaj Johnson, Sigurjon Jonsson, Jorn Hoffmann, Leif Harcke, Jessica Murray, Emily

Desmarais, Michael Fleishman, Youngseuk Keehm, Seok Goo Song, David Shelly, Adam

Pidlisecky, Tasha Reddy, James Irving, Bill Curry, Brad Artman, Alejandro Valenciano,

Zhen Liu, Ana Bertran Ortiz, Shadi Oveisgharan, Piyush Shanker Agram, Noa Bechor,

Lauren Wye, Christopher Tsai, Mahar Lagmay, Kyle Anderson, Dorte Mann, Dan Sinnett,

Stuart Schmitt, Eleonora Rivalta, and Takanori Matsuzawa, for being patient to answer my

questions on various topics from homework assignments to political issues, and for sharing

vii

all the sweet academic and non-academic memories for years.

I have been funded in this work by various sources - Korean Government Overseas

Fellowship; National Science Foundation grants EAR-0511035 and EAR-0346240; Orcutt

Fellowship; Dean Fellowship; a few Shell Grants for my travels to conferences. I appreciate

all these generous supports, which made my research possible.

All these things still would not have been possible without the support of my parents,

Chung-Il Yun and Hye-Jin Kwon, and my parents-in-law, Kil-Sang Kim and Moo-Sun Huh.

I received enormous cheer from my siblings Eun-Kyung Yun, Su-Jeong Yun, Jung-A Kim,

and Kwang-Hyun Kim. I wish to express my endless gratitude to my wife, Crispy Jung-

Youn Kim, and my son, Miles Sung-Joon Yun, for their patience, humor, and creativity that

inspired me in many ways.

viii

Contents

Abstract v

Acknowledgments vii

1 Introduction 11.1 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Thesis Roadmap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 InSAR Background 62.1 SAR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 InSAR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Merging Digital Elevation Models 193.1 Image Descriptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.1.1 TOPSAR DEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.1.2 SRTM DEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2 Image Registration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.3 Artifact Elimination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.4 Prediction-Error (PE) Filter . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.4.1 Designing the filter . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.4.2 1-D example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.4.3 The effect of the filter . . . . . . . . . . . . . . . . . . . . . . . . 28

3.5 Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.5.1 PE filter constraint . . . . . . . . . . . . . . . . . . . . . . . . . . 28

ix

3.5.2 SRTM DEM constraint . . . . . . . . . . . . . . . . . . . . . . . . 30

3.5.3 Inversion with two constraints . . . . . . . . . . . . . . . . . . . . 31

3.5.4 Optimal weighting . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.5.5 Simulation of the interpolation . . . . . . . . . . . . . . . . . . . . 34

3.6 Interpolation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.7 Effect on InSAR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4 Constraints on Magma Chamber Geometry at Sierra Negra Volcano, Galapagos 414.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.2 Shape of the Magma Chamber . . . . . . . . . . . . . . . . . . . . . . . . 44

4.3 Estimation of Best-fitting Sill Geometry . . . . . . . . . . . . . . . . . . . 49

4.3.1 Forward Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.3.2 Nonlinear Inversion . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

5 Interferogram Formation in the Presence of Large Deformation 625.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

5.2 Range and Azimuth Offset . . . . . . . . . . . . . . . . . . . . . . . . . . 64

5.3 Unbiased Masking of Noise . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5.4 Smoothing for Resampling . . . . . . . . . . . . . . . . . . . . . . . . . . 68

5.5 Resampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5.6 Range Offset as a Proxy for Interferogram Phase . . . . . . . . . . . . . . 71

5.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

6 2005 Eruption at Sierra Negra Volcano Unveiled by InSAR Observations 786.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

6.2 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

6.3 Magma Chamber Geometry . . . . . . . . . . . . . . . . . . . . . . . . . 83

6.3.1 Depth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

6.3.2 Sill Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

x

6.4 Data for Eruption Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . 88

6.4.1 Data Weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

6.4.2 Removing Pre- and Post-eruptive Deformation . . . . . . . . . . . 93

6.5 Eruption Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

6.5.1 Prior Information . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

6.5.2 Wedge-like Sill . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

6.5.3 Trapdoor Faulting . . . . . . . . . . . . . . . . . . . . . . . . . . 100

6.5.4 Dike Intrusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

6.5.5 Fault-Sill Interaction . . . . . . . . . . . . . . . . . . . . . . . . . 105

6.6 Extruded Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

6.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

7 Thesis Findings and Conclusions 114

xi

List of Tables

3.1 TOPSAR mission vs. SRTM mission . . . . . . . . . . . . . . . . . . . . . 20

4.1 Interferograms Used in This Study . . . . . . . . . . . . . . . . . . . . . . 46

6.1 Interferograms Used in This Study . . . . . . . . . . . . . . . . . . . . . . 83

6.2 RMS errors of two cases in Figure 6.18 . . . . . . . . . . . . . . . . . . . 105

xii

List of Figures

2.1 (a) the physical shape of a SAR antenna and its radiation pattern in the far

field, (b) common imaging geometry of a side-looking imaging radar . . . . 8

2.2 (a) Real aperture antenna, (b) synthetic aperture antenna, and (c) an imagi-

nary real aperture antenna that is equivalent to the synthetic aperture antenna 10

2.3 (a) raw data from Envisat satellite, (b) amplitude image of a SLC, the output

of SAR processing, (c) Sierra Negra volcano shown in the amplitude image 11

2.4 (a) SAR amplitude image of Sierra Negra volcano, (b) SLC pair and the

corresponding interogram. . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.5 InSAR coregistration process. (a) SAR amplitude image pair, (b) sparse

offset vector field, (c) resampling of the second image to register it to the

first image. The range and azimuth offset images are interpolated, so the

resampling can be done for the dense grid of the entire image. . . . . . . . 13

2.6 The effect of coregistration. (a) SLCs for interferograms, (b) an interfer-

ometric phase map from SLC1 and SLC2 that is not registered to SLC1,

(c) an interferometric phase map from SLC1 and SLC2 that is registered to

SLC1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.7 Schematic imaging geometry for space-borne repeat orbit interferometry to

demonstrate the effect of (a) baseline, (b) topography, (c) deformation, and

(d) atmosphere. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.8 (a) interferogram due to deformation, (b) interferometric coherence map.

Once color cycle represents 2.83 cm of range change, which is the surface

displacement projected onto the satellite’s line-of-sight vector. . . . . . . . 18

xiii

3.1 The original TOPSAR DEM of Sierra Negra volcano in Galapagos Islands

(inset for location). The pixel spacing of the image is 10 m. The boxed ar-

eas are used for illustration later in this paper. Note that there are a number

of regions of missing data with various shapes and sizes. Artifacts are not

identifiable due to the variation in topography. . . . . . . . . . . . . . . . . 22

3.2 (a) TOPSAR DEM and (b) SRTM DEM. The tick labels are pixel numbers.

Note the difference in pixel spacing between the two DEMs. (c) Artifacts

obtained by subtracting the SRTM DEM from the TOPSAR DEM. The

flight direction and the radar look direction of the aircraft associated with

the swath with the artifact are indicated with a long and short arrows re-

spectively. Note that the artifacts appear in one entire TOPSAR swath,

while it is not as serious in other swaths. . . . . . . . . . . . . . . . . . . 25

3.3 The flow diagram of the artifact elimination. . . . . . . . . . . . . . . . . . 26

3.4 The effect of a PE filter. (a) original DEM, (b) a 2-D PE filter found from

the DEM, (c) DEM filtered with the PE filter (d), (e), and (f) are the spectra

of (a), (b), and (c) respectively plotted in dB. (a) and (c) are drawn with the

same color scale. Note that in (c) the variation of image (a) was effectively

suppressed by the filter. The standard deviations of (a) and (c) are 27.6 m

and 2.5 m respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.5 Concept of PE filter. The PE filter is estimated by solving an inverse prob-

lem constrained with the remaining part, and the missing part is estimated

by solving another inverse problem constrained with the filter. The ε1 and

ε2 are white noise with small amplitude. . . . . . . . . . . . . . . . . . . . 30

3.6 Example subimages of (a) TOPSAR DEM showing regions of missing data

(black), and (b) SRTM DEM of the same area. These subimages are en-

gaged in one implementation of the interpolation. The grayscale is altitude

in meters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.7 Cross-validation sum of squares. The minimum occurs when λ = 0.16. . . 33

3.8 The results of interpolation applied to DEMs in Fig. 3.6, with various

weights. (a) λ → ∞, (b) λ = 0.16, and (c) λ = 0. Profiles along A-A’ are

shown in the plot (d). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

xiv

3.9 The quality of the CVSS, (a) a sample image that does not have a hole, (b)

a hole was made, (c) interpolated image with an optimal weight, (d) CVSS

as a function of λ . The CVSS has a minimum when λ = 0.062. (e) RMS

error between true image (a) and the interpolated image (c). The minimum

occurs when λ = 0.065. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.10 The original TOPSAR DEM (a) and the reconstructed DEM (b) after in-

terpolation with PE filter and SRTM DEM constraints. The gray scale is

altitude in meters, and the spatial extent is about 12 km across the image. . 38

3.11 Simulated interferograms from (a) the original registered TOPSAR DEM,

(b) the DEM after the artifact was removed, and (c) the DEM interpolated

with PE filter and the SRTM DEM. All the interferograms were simulated

with the C-band wavelength (5.6 cm) and a perpendicular baseline of 452

m. Thus, one color cycle represents 20 m height difference. . . . . . . . . . 40

4.1 Shaded relief topographic map of Galapagos Islands. The study area is in-

dicated with a black box, which includes the caldera of Sierra Negra volcano. 42

4.2 Map view of ascending and descending orbit imaging geometry and ideal-

ized surface displacement of volcano deformation. The open triangle indi-

cates the center of the volcano . . . . . . . . . . . . . . . . . . . . . . . . 47

4.3 Circularly symmetric deformation source was used to estimate the satel-

lite orbit inclination induced error. The vertical and radial component are

reconstructed in the same way as the data was analyzed. The x-axis is

the distance normalized by the depth of the source, and the y-axis is dis-

placement normalized by the maximum vertical displacement. In the case

of circular symmetry, the maximum error due to the orbit inclination not

being 90◦ is about 1%. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

xv

4.4 Interferograms from ascending (a) and descending (b) orbits with temporal

baselines of 1998/10/31 - 1999/02/13 and 1998/11/05 - 1999/02/18 respec-

tively. One color cycle represents 5 cm change of range in LOS direction

of satellite. Interferometric displacements can be separated into vertical

(c) and horizontal (d) components using the imaging geometries of the two

orbits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.5 Profiles of vertical and east component of surface deformation along A-A’

in Fig. 4.4d. Five lines are averaged to produce a smooth plot, and values

are normalized by the maximum vertical component. . . . . . . . . . . . . 49

4.6 Least-squares fitting results. (a) Interferogram for the time period of 1998/10/31

- 1999/02/13. (b) Scaled version of the the interferogram for 1998/09/26 -

1999/03/20. (c) Residual. (d) Profiles through (a) and (b). The blue solid

line is the S-N profile of (a), and the red solid line is the W-E profile of

(a). The black dashed lines are the corresponding profiles of (b). One color

cycle in the interferograms and residual represents 5 cm of LOS displacement. 51

4.7 Simple example of model grid that shows four open sill elements. The

upper right corner sill element will open widest under uniform pressure. . . 52

4.8 Schematic vertical section of the lithosphere. . . . . . . . . . . . . . . . . 55

4.9 Observed interferogram (a) and simulated interferogram (b) from the best-

fit model (d). The residual (c) between the data and the model shows dif-

ferences smaller than 2.5 cm, half the magnitude of one color cycle in (a)

and (b). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.10 (a) Best-fit model of Amelung et al. using a sill model with spatially vary-

ing opening distribution. (b) Best-fit model with uniform pressure bound-

ary condition. Depth was estimated as 1.9 km in both cases. . . . . . . . . 57

xvi

4.11 Asymmetry test using the best-fit model in Fig. 4.10b. The contour lines

show the expected observable eastward deformation. The contour labels

represent its magnitude normalized by the maximum vertical deformation.

The best-fit model was rotated from 0◦ to 150◦ in 30◦ increments. θ is the

angle of counterclockwise rotation. At each angle two interferograms from

ascending and descending orbit were simulated, and they were transformed

into east component as described in section 2. . . . . . . . . . . . . . . . . 58

4.12 (a) A flat-topped diapir with its sides dipping 45◦. The depth to the top

of the diapir is 1.9 km, and the radius of the top of the diapir is 3 km.

(b) Surface deformation due to the diapir and a sill whose geometry is the

same as the top of the diapir. The line of observation points is located on

the surface of the half-space starting from directly above the center of the

diapir. The x-axis is the distance normalized by the depth, and the y-axis is

displacement normalized by the maximum vertical displacement. . . . . . . 59

5.1 Co-eruptive interferogram processed by GAMMA software using Envisat

data (beam IS 5, track 376, 051016 - 051120). . . . . . . . . . . . . . . . 65

5.2 (a) Amplitude, (b) Range offset, (c) Azimuth offset in radar coordinate sys-

tem. Note that the letter N is upside down and the circular map compass is

enlongated to show how features look different compared to georeferenced

frame. The blow-ups of black boxed portion in (a)-(c) are shown in (d)-(f).

The small portion of the bright sinuous ridge of Sierra Negra is shown in

(d), and its effect on offset images are shown in (e) and (f), in which the

color-saturated boxes are 32-pixel wide, the size of the cross-correlation

block . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

5.3 Cumulative historgram of (a) range and (b) azimuth offset values inside the

caldera . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5.4 Mean coherence at caldera and north flank as a function of Gaussian smooth-

ing parameter, σ . Note that the range of coherence variation is very small.

The maximum occurs at (a) σ = 10 pixels and (b) σ = 7 pixels. . . . . . . . 69

xvii

5.5 Coherence histogram for (a),(c) caldera before and after coregistration re-

spectively, and (b),(d) north flank before and after coregistration respec-

tively. (b) and (d) are for σ = 10 pixels. . . . . . . . . . . . . . . . . . . . 70

5.6 (a) Deformation interferogram after the rubber-sheeting SAR coregistra-

tion. (b) Blow-up of the white box in (a). Fringe rate becomes higher close

to critical sampling rate in azimuth direction. In range direction the fringe

rate quickly becomes aliased. After subtracting the range offset from the

interferogram, the fringe rate becomes much lower (c) , and phase unwrap-

ping becomes possible for much larger area. . . . . . . . . . . . . . . . . . 74

5.7 Maximum detectable displacement gradient as a function of coherence

(Baran et al., 2005). Thu rubber-sheeting coregistration increases the co-

herence (A) and the range offst subtraction decreases the phase gradient

(B) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

5.8 Uncertainty of amplitude offset for (a) range and (b) azimuth . . . . . . . . 75

5.9 Final result interferogram from the same SAR images used to produce Fig-

ure 5.1. One color fringe represents 15 cm of range change. . . . . . . . . 76

6.1 Surface temperature image taken at night on November 2, 2005 by ASTER

sensor onboard NASA’s Terra satellite. The image is georeferenced and

draped on the shaded relief image from the SRTM DEM (Farr and Kobrick,

2000). Six continuous GPS stations were deployed inside the caldera at the

time of the eruption. GV04, GV05, and GV06 were used for InSAR data

adjustment. The C-shaped sinuous ridge inside the western side of the

caldera is clearly shown. The outer sides of the ridge are composed of fault

scarps. The southern part of the scarps, which are less clear, extend just

below the GV06 station. . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

6.2 Schematic vertical cross section along the line C-C’ in Figure 6.1. (a) pre-

eruptive inflation, (b) trapdoor faulting, (c) dike intrusion and the fissure

eruption, (d) post-eruptive inflation. . . . . . . . . . . . . . . . . . . . . . 82

xviii

6.3 Interferograms from 1992 to 2006 (a-d). The letter A and D represent

ascending and descending orbit. Note that a-c are before the eruption and

d is after the eruption. For modeling the pressure source geometry a pair

of ascending and descending interferograms (e,f) are used. One color cycle

represents 5 cm of LOS displacement in a, b, e, and f, and 2.83 cm of LOS

displacement in c and d. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

6.4 (a) Reduced data points and (b) Data variance-covariance matrix. The first

block in the matrix is for the ascending interferogram and the second block

is for the descending interferogram. . . . . . . . . . . . . . . . . . . . . . 86

6.5 Joint a posteriori probability density for the penny-shaped crack parame-

ters. The red lines are 95% confidence intervals. Note that depth, radius,

and excess pressure are highly correlated to each other. The depth of 1.86

km is the maximum likelihood solution. The color plots represent the den-

sity clouds of accepted samples weighted by the likelihood . . . . . . . . . 87

6.6 Best-fit sill geometry. Note that the sill is bounded by the C-shaped sinuous

ridge. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

6.7 (a) Ascending interferogram, (b) Descending interferogram, (c) Ascend-

ing residual, (d) Descending residual. (e)-(h) are the profiles of data and

predictions (which are not shown) along the N-S and E-W lines in (a). . . . 89

6.8 (a) Ascending interferogram from beam IS5 (incidense angle = 37.6◦), (b)

Descending interferogram from beam IS2 (incidence angle = 23.0◦), (c)

Azimuth offset from the ascending SAR image pair. One color cycle in

both interferograms represents 20 cm of LOS displacement. The interfero-

grams are composed of several isolated patches, in each of which phase is

internally consistent and continuous. . . . . . . . . . . . . . . . . . . . . . 90

6.9 (a) wrapped interferogram in radar coordinate, (b) azimuth offset in radar

coordinate, (c) residual between ”extreme” kinematic model prediction and

the interferogram, (d) azimuth offset residual for the same model, (e) Un-

certainty plot for the interferogram, (f) Uncertainty plot for the azimuth

offset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

6.10 Schematic plot of uplift and subsidence of the center of Sierra Negra’s caldera. 95

xix

6.11 GPS time series processed by Charles Meertens and Dennis Geist. The tick

marks on the x-axis correspond to the vertical lines in Figure 6.10. . . . . . 97

6.12 Simulated (a) ascending interferogram and (b) descending interferogram

and (c) azimuth offset, which are to be added to the original data (Figure

6.8) in order to account for pre- and post-eruptive inflation. . . . . . . . . . 98

6.13 Prior information for eruption modeling. (a) Map view of the surface con-

straint, where eruption fissure is indicated with a red line and the the fault

trace on the surface, drawn in black, goes along the C-shaped sinuous ridge.

(b) 3-D perspective view of the model geometry. . . . . . . . . . . . . . . . 99

6.14 Best-fit sill-only model for the adjusted data. Associated excess pressure

change is -41.7 MPa, with host rock’s shear modulus of 30 GPa. . . . . . . 100

6.15 Residuals between data and model prediction for (a) ascending interfero-

gram, (b) descending interferogram, and (c) azimuth offset due to the best-

fit sill-only model (Figure 6.14). One color fringe represents 20 cm LOS

displacement in (a) and (b). The near-constant-slope phase ramp over large

area inside the caldera suggests a ramp-like feature at depth. . . . . . . . . 101

6.16 Best-fit fault model estimated simultaneously with dike and sill models, (a)

with the same view as in the inset, and (b) when the model is rotated 130◦

counterclock-wise. The inset shows the location of the fault. . . . . . . . . 102

6.17 Schematic vertical cross section as in Figure 6.2. (a) pressurized sill after

the trapdoor faulting, (b) dike intrusion accompanied with “reverse fault-

ing”, (c) co-eruptive subsidence accompanied with “normal faulting”. . . . 104

6.18 Best-fit dike models estimated simultaneously with fault and sill models,

(a) when displacements of dike and sill are not coupled and (b) they are

coupled. The inset shows the location of the dike. . . . . . . . . . . . . . . 106

6.19 (a) Best-fit uniformly depressurized sill model, and (b) best-fit kinematic

sill model (opening), which is the effet of interaction of sill with faulting

and dike intrusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

6.20 (a) pressurized sill (b) magma transport in the sill due to the trapdoor faulting108

6.21 Best-fit composite sill model (uniformly depressurized sill + kinematic

opening due to fault-sill and dike-sill interaction. . . . . . . . . . . . . . . 109

xx

6.22 Residual between the best-fit model prediction and the data for (a) ascend-

ing interferogram, (b) descending interferogram, and (c) azimuth offset.

One color fringe represents 20 cm of LOS displacement in (a) and (b). . . . 109

6.23 Lava coverage simulation using DEM. . . . . . . . . . . . . . . . . . . . . 111

xxi

Chapter 1

Introduction

During the past decade, Space-borne Interferometric Synthetic Aperture Radar (InSAR)

has been successfully used to measure millimeter- to meter-level deformation on the sur-

face due to physical process at depth including magma accumulation and migration, pres-

surization, and crystallization (Massonnet et al., 1995; Lu et al., 1997; Jonsson et al., 1999;

Pritchard and Simons, 2002; Wright et al., 2006; Wicks et al., 2006).

InSAR data has been explained by various source models of simple geometry such as

center of dilation (Mogi, 1958), finite sphere (McTigue, 1987), prolate spheroid (Yang et al.,

1988), rectangular dislocation (Okada, 1992), angular dislocation (Yoffe, 1960; Thomas,

1993), or horizontal circular crack (Fialko et al., 2001a). In some cases, simple geometry

was not sufficient to fit the data, and a combination of the models was used (Fialko and

Simons, 2000; Lundgren et al., 2003). In some other cases, distributed source is used, as

source parameters needed vary within a sill (Amelung et al., 2000) or a dike (Fukushima

et al., 2005; Walter and Amelung, 2006).

In this thesis, we demonstrate a new method to estimate a detailed geometry of uni-

formly pressurized (or depressurized) crack in a mechanically consistent way. The new

method involves binary parameters that represent the locations of crack elements and global

optimization scheme. This method is applied to model the 2005 eruption at Sierra Negra

volcano in Isabela Island, Galapagos.

The 2005 eruption at Sierra Negra volcano impose a challenge in data processing, as

the 8-day co-eruptive subsidence inside the caldera was about 5.4 m across 4 km, too

1

2 CHAPTER 1. INTRODUCTION

large to be imaged using conventional InSAR processing. Moreover, the eruption was

preceded by a Mw 5.4 earthquake, which caused a complex deformation pattern, whose

spatial variation was too large to be properly captured by standard InSAR software. This

thesis describes a new algorithm to form interferograms in the presence of such large and

complex deformation.

In order to produce interferograms of deformation, topography signal has to be re-

moved. This is usually done using a Digital Elevation Model (DEM). At Sierra Negra,

both Topographic Synthetic Aperture Radar (TOPSAR) DEM and Shuttle Radar Topog-

raphy Mission (SRTM) DEM are available and their features are complementary to each

other. We merge the two DEMs using a Prediction-Error Filter (PEF) to create a new DEM

that is superior to either of the original DEMs. The merged DEM is used for the 2005

eruption modeling.

This DEM merging algorithm is described at the beginning of the thesis, for the DEM

effect (i.e. the simulated interferogram of topography) is removed from the InSAR data

and emphasis is placed on deformation through the rest of the thesis. The new modeling

technique is explained in the following chapter to provide the details of the modeling pro-

cess before we apply it to the 2005 eruption at Sierra Negra volcano. Then in the following

chapter we demonstrate the new interferogram formation technique, as the result of the

chapter constitutes a part of the data section of the following chapter, where we model the

eruption.

1.1 Contributions

1. We develop an optimal way to merge high- and low-resolution digital elevation model

(DEM) to produce a DEM that is both high resolution and comprehensive in cover-

age. Using this approach we merge TOPSAR and SRTM DEMs of Sierra Negra

volcano.

2. We develop a new inversion algorithm to estimate the detailed geometry of a pres-

surized crack. This method was then applied to Sierra Negra volcano to constrain the

geometry of the magma chamber of the volcano.

1.2. THESIS ROADMAP 3

3. We develop a new algorithm to form interferograms over regions of very large and

complex deformation patterns. Using this new algorithm, we form the first two inter-

ferograms of the area inside the caldera of Sierra Negra.

4. We estimate the depth and the geometry of a sill at Sierra Negra rigorously using

both the ascending and descending interferograms. We find that the sill geometry

has sustained its shape for over a decade.

5. We estimate the excess magma pressure changes of pre- and post-eruptive uplift at

Sierra Negra and remove the deformation before and after the eruption from the

InSAR data.

6. We derive the slip distribution of the trapdoor faulting event that occurred three hours

prior to the 2005 eruption at Sierra Negra volcano.

7. We estimate the opening and the slip of a dike plane at Sierra Negra prior to and

during the 2005 eruption.

8. We model interaction between the trapdoor faulting and the sill at Sierra Negra. We

find that InSAR data favors wedge-like opening distribution of the sill, where the

maximum opening of the sill is at the bottom of the western part of the fault system.

9. We estimate the volume decrease at the magma chamber and the extruded volume

during the eruption.

1.2 Thesis Roadmap

There are four main chapters in this thesis. Two of them are about data processing (Chapter

3 and Chapter 5) and two of them are about modeling (Chapter 4 and Chapter 6). Chapter

3 through Chapter 5 are supporting Chapter 6, where data and models are derived from

algorithms and results from previous chapters.

Chapter 3 describes a way to merge high- and low-resolution DEMs. When an indepen-

dent lower-resolution version of the DEM is available, the performance of the interpolation

can be improved. As a result, we can merge the two DEMs and produce an output DEM

4 CHAPTER 1. INTRODUCTION

that is better than the two input DEMs. The merged DEM is optimal in least-square’s

with two constraints of 1) valid area of the high-resolution DEM and prediction-error fil-

ter calculated from the high resolution DEM, and 2) the low-resolution DEM. The relative

weight between the two input DEMs can optimally be determined using cross-validation.

The chapter shows application to a digital elevation model (DEM) of Sierra Negra volcano.

The output DEM was used in Chapter 5 and in Chapter 6. The work of this chapter was

published in the IEEE Transactions on Geoscience and Remote Sensing in 2005 (Yun et al.,

2005).

Chapter 4 describes solutions for magma chamber geometry. In particular we develop

a new inversion algorithm to estimate the periphery of any planar magma body under the

assumption of constant hydrostatic pressure. The algorithm uses the simulated annealing

optimization and require a boundary element calculation at each iteration. The new algo-

rithm does not require a smoothing constraint. Rather we use a crack model with a uniform

pressure boundary condition to estimate the geometry of a sill or dike. We apply the new

method to InSAR data acquired over Sierra Negra volcano and compared our solution with

published modeling results. The new inversion method introduced in this chapter was sub-

sequently used to model the 2005 eruption at Sierra Negra volcano (i.e. Chapter 6). The

work of this chapter was published in the Journal of Volcanology and Geothermal Research

in 2006 (Yun et al., 2006).

Chapter 5 introduces a new method that can form interferograms in regions of very

large and complex deformation. As a result, we were able to describe the 2005 eruption

at Sierra Negra using InSAR data. Our solution yields the internally consistent model as

described in Chapter 6, which otherwise would not possible to be done. The work of this

chapter is in review for publication in the Geophysical Research Letters in 2007 (Yun et al.,

in review).

In Chapter 6 we estimate geophysical parameters of i) the trapdoor faulting event that

occurred 3 hours prior to the 2005 eruption at Sierra Negra, ii) dike intrusion that fed the

eruption, iii) sill closing during the eruption, and iv) the interaction between the trapdoor

faulting and the sill. For all of these we used InSAR data derived using the algorithm

described in Chapter 5. In order to make the InSAR data, we used the TOPSAR-SRTM

merged DEM produced using the method described in Chapter 3. Modeling was done

1.2. THESIS ROADMAP 5

using the algorithm described in Chapter 4. The work of this chapter will be submitted to

the Journal of Geophysical Research in 2007.

Chapter 7 provides the findings from this work and summary of this thesis.

Chapter 2

InSAR Background

In this chapter we discuss brief background information on Synthetic Aperture Radar

(SAR) and Interferometric SAR (InSAR). Since InSAR emerged in 1970s (Richman; Zisk,

1972), the technology has been evolved. Zebker and Goldstein (1986) produced topo-

graphic mapping using a radar system mounted on an aircraft. Since European Space

Agency (ESA) launched ERS-1 satellite in 1991, InSAR applications have been expanded.

It was first successfully used to study ground deformation due to earthquake (Massonnet

et al., 1993) and volcanic process (Massonnet et al., 1995). Glacier movement has become

another interesting application of InSAR (Goldstein et al., 1993). Hanssen et al. (1999)

produced high-resolution maps of integrated atmospheric water vapor using spaceborne

radar interferometric delay measurements. InSAR is even capable of capturing dynamic

water level topography in wetland (Wdowinski et al., 2004). Using InSAR Dixon et al.

(2006) constructed a subsidence map of New Orleans and related the subsidence to the

catastrophic flooding by Hurricane Katrina.

2.1 SAR

Antenna theory tells us that the antenna radiation pattern (i.e. a graphical representation of

the intensity of the radiation as a function of the angle from the perpendicular line to the

antenna plane) in the far field can be approximated to a Fourier transform of the physical

shape of the antenna itself. Figure 2.1a shows an example of a rectangular antenna, which

6

2.1. SAR 7

is a common shape of a SAR antenna, and its radiation pattern.

The 3-D shape of the radiation pattern is shown in Figure 2.1b, which illustrates a

typical imaging geometry of a side-looking imaging radar. The rectangular SAR antenna is

loaded either on an aircraft or on a satellite. As the platform moves along with the antenna,

a stream of radar pulses are transmitted from the antenna. The width of the mainlobe

of the radiation pattern defines the beamwidth and the beamwidth defines the footprint

(illuminated area on the ground). The platform’s flight direction is called azimuth direction,

and the transmission direction of radiated pulses in the mainlobe is called range (or slant

range) direction. The length of the footprint perpendicular to the azimuth direction is called

swath, and the angle of range from vertical is called look angle.

Figure 2.2 shows the schematic imaging geometry of a real and a synthetic aperture

radar, viewed in the direction of the large arrow at the lower right corner of Figure 2.1b.

Consider two trees on the ground slightly separated in azimuth direction. The resolution

in range direction is determined by the bandwidth of the transmitted pulse. In case of real

aperture radar, the resolution in azimuth direction is controlled by the size of the footprint

in azimuth direction. Whatever two objects on the ground close enough to be included in a

footprint are not distinguishable from each other.

The size of the footprint in azimuth direction is a function of the length of the antenna

(the dimension of the antenna in azimuth direction). According to the Fourier transform

properties, the beamwidth in azimuth direction is inversely proportional to the antenna

length. Thus, if one wish to acquire narrower beamwidth to increase the resolution, the

length of the antenna should be increased. However, there is a physical limit on the length

of the antenna.

SAR is an alternative solution via signal processing to increase the azimuth resolution

without increasing the length of the antenna. In Figure 2.2b the two trees are well included

in one beamwidth, and both trees are illuminated by many pulses as the platform flies by.

However, the tree on the left always have smaller Doppler shift than the tree on the right.

Because of this property, when we deconvolve the phase history of a point target from the

data, we can acquire fine azimuth resolution and resolve the two trees. The longer phase

history we have, the better the azimuth resolution becomes. The data include a point target’s

phase history as long as the point target is illuminated. Thus, wider beamwidth (i.e. the

8 CHAPTER 2. INSAR BACKGROUND

SAR antenna

-3dB

footprint

azimuth

range

swathlook angle

θx

θy

Antenna

Antenna pattern

U

(a)

(b)

Figure 2.1: (a) the physical shape of a SAR antenna and its radiation pattern in the far field,(b) common imaging geometry of a side-looking imaging radar

2.2. INSAR 9

shorter antenna) will cause finer resolution. One can imagine an imaginary real aperture

antenna (Figure 2.2c) that is equivalent to this synthesized antenna. The word “synthetic

aperture” came from this property.

Once the SAR processing is implemented a SAR image, or a Single-Look Complex

(SLC), is produced. SAR images are 2-D complex number arrays. Thus, they have both

amplitude and phase values for each pixel. In Figure 2.3, (b) is an example of an amplitude

image of an SLC as a result of SAR processing applied to the raw data (a). The image

covers a part of Isabela Island in Galapagos, and the blow-up image of the black box shows

the caldera of Sierra Negra volcano (c). All images are in radar coordinate system in Figure

2.3.

2.2 InSAR

InSAR involves two or more SAR images of the same area acquired either simultaneously

or separately in time. All deformation studies use the latter case, which is called repeat

orbit interferometry for space-borne InSAR. The satellite passes by one place on Earth and

acquires the first scene. After multiple of 35 days (Envisat, ERS-1/2) or 24 days (Radarsat),

the satellite revisits the same area and acquires the second scene. Then SAR processing

produces two SLCs. Figure 2.4a is the amplitude image previously shown, and the two

SLCs of the area indicated with the black box are shown in (b). By multiplying a complex

conjugate of SLC2 to SLC1 pixel by pixel, we get a new complex image, which is called

interferogram. The phase of the interferogram is the phase difference of the two SLCs.

In this example the first SAR scene was acquired before the 2005 eruption of Sierra

Negra, and the second SAR scene was acquired after the eruption. The noisy flow pattern

going from upper left corner to lower right is the lava flow during the eruption. The area

covered with lava completely changed the back scattering signal keeps the corresponding

pixels of the two SLCs from being “coherent”. In this case, the area is called decorrelated.

The coherence of the two SLCs, which is called interferometric coherence, is a crucial

factor that determines the quality of interferograms. The degrade in coherence is cause

not only by the surface disturbing events such as lava flow, but also by pixel misalignment

between the two SLCs. The pixel misalignment is often due to the change of imaging


(a) Real Aperture Radar (b) Synthetic Aperture Radar

…

Synthetic Aperture Radar(c) Equivalent Real Aperture Radar

Figure 2.2: (a) Real aperture antenna, (b) synthetic aperture antenna, and (c) an imaginaryreal aperture antenna that is equivalent to the synthetic aperture antenna

2.2. INSAR 11

SAR processing

Raw data

Sierra Negra 2007/11/20

range

azimuth

(b)

(c)

(a)

Figure 2.3: (a) raw data from Envisat satellite, (b) amplitude image of a SLC, the output ofSAR processing, (c) Sierra Negra volcano shown in the amplitude image

SLC1

Amplitude Phase

SLC2

1

2

Δ Interferogram

(a) (b) Amplitude Phase

Figure 2.4: (a) SAR amplitude image of Sierra Negra volcano, (b) SLC pair and the corre-sponding interogram.


geometry, topography, deformation, and atmospheric distortion. This misalignment has to

be fixed before the two SLCs are “interfered”, and in fact Figure 2.4b was the case.

The process of fixing the misalignment is called SAR image coregistration, and is il-

lustrated in Figure 2.5. Consider a pair of SAR amplitude images. The second one shows

distortion from the first one for some reason (a). Three tie points in both images are indi-

cated with color dots. How much the second image has deformed from the first image is

usually calculated using cross-correlation of two subimages of similar area. For example,

we slide the white box from the first image on the white box from the second image. As

a result we get the a offset vector. Repeated at many locations distributed throughout the

entire image, the set of cross-correlations produces a vector filed. The range and azimuth

components of the vector field are called range offset and azimuth offset image respectively

(b).

Usually the range and azimuth offset estimates are sparser than the two amplitude

images, and interpolating sparse image yields the offset fields at all locations. The full-

resolution offset fields are then used to resample the second SLC, so that each pixel in the

second SLC matches the corresponding pixel in the first SLC (c).

Figure 2.6 shows the effect of the coregistration. The registered and non-registered sec-

ond SLCs may look similar. When they are used to form interferogram, however, the noise

levels of each individual phase difference estimate in both cases are quite different. The

interferogram from non-coregistered pair shows a little hint of fringe pattern, but fringes

are much clearer in the interferogram from coregistered pair.

The phase difference map shown in Figure 2.6c can be decomposed into the following

components.

∆φ = ∆φbase +∆φtopo +∆φdeform +∆φatm +∆φnoise (2.1)

where ∆φbase is the phase difference due to the baseline between two satellite positions

when the data are acquired, ∆φtopo is due to topography, ∆φdeform is due to ground defor-

mation, ∆φatm is due to atmospheric delay, and ∆φnoise is due to other sources of decorrela-

tion, influence of ionosphere, and system noise. In fact, which ones are signal and which

ones are noise depends on applications. For crustal deformation study, only the ∆φde f orm is

2.2. INSAR 13

SLC1 Amplitude SLC2 Amplitude Vector field

range

azimuth

Range offset Azimuth offset

Range component Azimuth component

Resample SLC2

(a) (b)

(c)

Figure 2.5: InSAR coregistration process. (a) SAR amplitude image pair, (b) sparse offsetvector field, (c) resampling of the second image to register it to the first image. The rangeand azimuth offset images are interpolated, so the resampling can be done for the densegrid of the entire image.


SLC1

Amplitude Phase

SLC2

coregistered

SLC2

not

coregistered

(a)b

c

Figure 2.6: The effect of coregistration. (a) SLCs for interferograms, (b) an interferometricphase map from SLC1 and SLC2 that is not registered to SLC1, (c) an interferometric phasemap from SLC1 and SLC2 that is registered to SLC1.

2.2. INSAR 15

considered a signal and the other components are treated as noise.

When a satellite revisits the same area to acquire the second scene, it is not possible for

it to follow exactly the same orbit as the first pass. Thus, there always is a finite baseline

between the two satellite positions. Looking from a slightly different positions not only

causes the two scenes slightly different in shape, but also causes them to have systematic

phase difference. This effect is illustrated in Figure 2.7a, a schematic imaging geometry of

space-borne repeat orbit interferometry. The satellite is flying into the figure and looking

down to the right. The nominal dimensions are given for currently operating C-band satel-

lites. Each solid black arcs represents wavefront at an interval of half-wavelength. If the

two satellite positions are exactly the same, there will be no fringes in the interferogram

due to the imaging geometry. As the baseline (particularly the baseline perpendicular to

the range) increases, more interferometric fringes appear in the interferogram. A schematic

version of the interferogram is shown at the bottom of the figure. As we follow from one

interference line to another, we accumulate 2π of phase difference. The baseline effect,

∆φbase, can be removed using precise orbit information.

If the area has a mountain (Figure 2.7b), the intervals of the fringes change according to

the topography. Topography affects the interferogram in this way. Note that the topography

effect does not appear if the baseline is zero. The sensitivity of fringes to the topography

increases with increasing baseline. The topography effect, ∆φtopo, can be removed using a

DEM.

Suppose that the baseline is zero, or the baseline and topography effects are all removed.

Then consider subsidence of the ground surface causing a constant slope (Figure 2.7c). If

this event occurs between the two moments of data acquisition, the interferogram will

reveal the deformation as a constant phase ramp. When the deformation causes the range

to increase with half the wavelength of the transmitted signal, the wave from the second

acquisition has to travel one wavelength farther, resulting in 2π delay in phase. Therefore,

∆φ = −4πλ

∆r (2.2)

where ∆φ = φ2−φ1 is phase change and ∆r = r2−r1 is range change. Note that when there

is a phase delay the quantity φ2 − φ1 is always negative. For example, the phase delay of


∆φ means that when the recorded signal in the first image is exp[iωt], the recorded signal

in the second image is exp[i(ωt −∆φ)]. Also note that Equation 2.2 holds not only for

deformation but also for the first two terms in Equation 2.1.

The radar wave from satellite has to propagate through the atmosphere of the Earth.

The spatial variation of water vapor content in the atmosphere (predominantly in the tro-

posphere) causes the spatial variation of phase delay (Figure 2.7d). The atmospheric effect

can be suppressed by stacking many interferograms, multiple acquisition InSAR such as

PS-InSAR (Ferretti et al., 2001; Hooper et al., 2004) or small-baseline InSAR (Berardino

et al., 2002; Schmidt and Burgmann, 2003), modeling from independent observations such

as Global Positioning System (GPS) data (Onn and Zebker, 2006), or ignored when the

study area is dry or when the signal-to-noise ratio is high.

An example phase difference map due to deformation is shown in Figure 2.8a, which

was processed using ROI PAC software (Rosen et al., 2004). The fringe pattern shows the

subsidence during the 2005 eruption. One color cycle represents a half-wavelength range

change, and for C-band data it is about 2.8 cm. Inside the caldera is severely decorre-

lated due to poor coregistration and severely aliased due to large deformation. A common

measure of the degree of statistical similarity of two SAR images is the interferometric

coherence (or correlation) defined as

ρ =| < c1c∗2 > |

√

< c1c∗1 >< c2c∗2 >≈

|∑nk=1 c1,kc∗2,k|

√

∑nk=1 c1,kc∗1,k ∑n

k=1 c2,kc∗2,k

(2.3)

where ρ is the interferometric coherence, and c1 and c2 are the two SLCs with * meaning

the complex conjugate, and the subscript k denotes the kth pixel of n neighboring pixels av-

eraged. Note that the ensemble averages are approximated with spatial averages, obtained

over a limited area surrounding the pixel of interest. Figure 2.8b shows the coherence map,

which shows the severe decorrelation inside the caldera. By the way, due to the large defor-

mation the estimate of coherence is biased and underestimated inside the caldera. A better

estimate of the coherence inside the caldera is about 0.3 on average. A method to form

interferograms inside the caldera and to get the coherence estimate is described in Chapter

5 in this thesis.

In the interferogram shown in Figure 2.8a, the phase difference can only be determined

2.2. INSAR 17

mk

00

8m 005 < mc 38.2

(a) (b)

(c) (d)

Figure 2.7: Schematic imaging geometry for space-borne repeat orbit interferometry todemonstrate the effect of (a) baseline, (b) topography, (c) deformation, and (d) atmosphere.


range

azimuth

(a) (b)

1

0

Figure 2.8: (a) interferogram due to deformation, (b) interferometric coherence map. Oncecolor cycle represents 2.83 cm of range change, which is the surface displacement projectedonto the satellite’s line-of-sight vector.

modulo 2π . In order to obtain a continuous phase difference map, the differential phase

between all neighboring pixels is integrated over the interferogram. This process is called

phase unwrapping. Once phase unwrapping is done, the unwrapped interferogram needs

be transformed from radar coordinate system (i.e. range/azimuth) into georeferenced co-

ordinate system (i.e. either latitude/longitude or UTM). This is the final step of InSAR

processing in most cases.

Chapter 3

Merging Digital Elevation Models

As mentioned in Chapter 1, topography component needs be removed from interferograms

in order to obtain surface deformation maps. Most common way to remove topography is

to use Digital Elevation Model (DEM). Thus, having a reliable DEM is crucial for crustal

deformation study. For our study area, Sierra Negra volcano located at the southern end

of Isabela Island in the Galapagos archipelago, two different DEMs are available. One

is Topographic Synthetic Aperture Radar (TOPSAR) DEM and the other is Shuttle Radar

Topography Mission (SRTM) DEM. Their merits and demerits are complementary to each

other. Hence we provide an optimal method to merge the two DEM to produce a new DEM

that is superior to both DEMs.

In practice, high resolution DEMs are often limited in spatial coverage; they also may

possess other systematic artifacts when compared to comprehensive low-resolution maps.

Here we correct artifacts and interpolate regions of missing data in TOPSAR DEMs using

a low-resolution SRTM DEM. We use PE filters to interpolate and fill missing data so that

the interpolated regions have the same spectral content as the valid regions of the TOPSAR

DEM. In addition, the SRTM DEM is used as a constraint in the interpolation. Using

cross-validation methods we obtain the optimal weighting for the PE filter and SRTM DEM

constraints.

19

20 CHAPTER 3. MERGING DIGITAL ELEVATION MODELS

Table 3.1: TOPSAR mission vs. SRTM mission

Mission TOPSAR SRTM

Platform DC-8 aircraft Space shuttle

Nominal altitude 9 km 233 km

Swath width 10 km 225 km

Baseline 2.583 m 60 m

DEM resolution 10 m 90 m

DEM coord. system none Lat/Lon

3.1 Image Descriptions

InSAR is a powerful tool for generating digital elevation models (DEMs) (Zebker and

Goldstein, 1986). The TOPSAR and SRTM sensors are primary sources for the academic

community for DEMs derived from single-pass interferometric data. Differences in system

parameters such as altitude and swath width (Table 3.1) result in very different proper-

ties for derived DEMs. Specifically, TOPSAR DEMs have better resolution, while SRTM

DEMs have better accuracy over larger areas. TOPSAR coverage is often spatially incom-

plete.

3.1.1 TOPSAR DEM

TOPSAR DEMs are produced from cross-track interferometric data acquired with NASA’s

AIRSAR system mounted on a DC-8 aircraft. Although the TOPSAR DEMs have a higher

resolution than other existing data, they sometimes suffer from artifacts and missing data

due to roll of the aircraft, layover, and flight planning limitations. The DEMs derived from

SRTM have lower resolution, but fewer artifacts and missing data than TOPSAR DEMs.

Thus, the former often provides information in the missing regions of the latter.

We illustrate joint use of these data sets using DEMs acquired over the Galapagos Is-

lands. Fig. 3.1 shows the TOPSAR DEM used in this study. The DEM covers Sierra Negra

3.1. IMAGE DESCRIPTIONS 21

volcano on the island of Isabela. Recent InSAR observations reveal that the volcano has

been deforming relatively rapidly (Amelung et al., 2000; Yun et al., 2006). InSAR analy-

sis often requires use of a DEM to produce a simulated interferogram required to isolate

ground deformation. The effect of artifact elimination and interpolation for deformation

studies will be discussed later in this chapter.

The TOPSAR DEMs have a pixel spacing of about 10 m, sufficient for most geodetic

applications. However, regions of missing data are often encountered (Fig. 3.1), and signif-

icant residual artifacts are found (Fig. 3.2). The regions of missing data are caused by lay-

over of the steep volcanoes and by flight planning limitations. Artifacts are large-scale and

systematic and most likely due to uncompensated roll of the DC-8 aircraft (Zebker et al.,

1992). Attempts to compensate this motion include models of piecewise linear imaging

geometry (Madsen et al., 1993) and estimating imaging parameters that minimize the dif-

ference between the TOPSAR DEM and an independent reference DEM (Kobayashi et al.,

2000). We use a non-parameterized direct approach by subtracting the difference between

the TOPSAR and SRTM DEMs.

3.1.2 SRTM DEM

The recent SRTM mission produced nearly worldwide topographic data at 90-m posting.

SRTM topographic data are in fact produced at 30-m posting (1 arc second), however, high

resolution data sets for areas outside of the United States are not available to the public at

this time. Only DEMs at 90-m posting (3 arc second) are available for download.

For many analyses, finer-scale elevation data are required. For example, a typical pixel

spacing in a spaceborne SAR image is 20 m. If the SRTM DEMs are used for topography

removal in spaceborne interferometry, the pixel spacing of the final interferograms would

be limited by the topography data to at best 90 m. Despite the lower resolution, the SRTM

DEM is useful because it has fewer motion-induced artifacts than the TOPSAR DEM. It

also has fewer data holes.

The merits and demerits of the two DEMs are in many ways complementary to each

other. Thus, a proper data fusion method can overcome the shortcomings of each and

produce a new DEM that combines the strengths of the two data sets: a DEM that has


Figure 3.1: The original TOPSAR DEM of Sierra Negra volcano in Galapagos Islands(inset for location). The pixel spacing of the image is 10 m. The boxed areas are used forillustration later in this paper. Note that there are a number of regions of missing data withvarious shapes and sizes. Artifacts are not identifiable due to the variation in topography.

3.2. IMAGE REGISTRATION 23

a resolution of the TOPSAR DEM and large-scale reliability of the SRTM DEM. In this

paper, we present an interpolation method that uses both TOPSAR and SRTM DEMs as

constraints.

3.2 Image Registration

The original TOPSAR DEM, while in ground-range coordinates, is not georeferenced.

Thus, we register the TOPSAR DEM to the SRTM DEM, which is already registered in

a latitude/longitude coordinate system. The image registration is carried out between the

DEM data sets using an affine transformation. Sscaling and rotation are the two most im-

portant components. We find that the skew component is negligible in these data. Any

higher order transformation between the two DEMs would also be of negligible improve-

ment. The affine transformation we used is as follows,

[

xS

yS

]

=

[

a b

c d

][

xT

yT

]

+

[

e

f

]

(3.1)

where[

xSyS

]

and[

xTyT

]

are tie points in the SRTM and TOPSAR DEM coordinate systems

respectively. Since [a b e] and [c d f] are estimated separately, at least 3 tie points are

required to uniquely determine them. We picked 10 tie points from each DEM based on

topographic features and solved for the six unknowns in a least-squares sense.

Given the six unknowns, we choose new georeferenced sample locations that are uni-

formly spaced; every 9th sample location corresponds to the sample location of SRTM

DEM. Those sample locations form[

xSyS

]

, and[

xTyT

]

is calculated. Then, the nearest TOP-

SAR DEM value is selected and is put into the corresponding new georeferenced sample

location. The intermediate values are filled in from the TOPSAR map to produce the geo-

referenced 10-m data set.

It should be noted that it is not easy to determine the tie points in DEM data sets.

Enhancing the contrast of the DEMs facilitated the process. In general, fine registration

is important for correctly merging different data sets. The two DEMs in this study have

different pixel spacings. It is difficult to pick tie points with higher precision than the pixel


spacing of the coarser image. In our method, however, the SRTM DEM, the coarser image,

is treated as an averaged image of the TOPSAR DEM, the finer image. In our inversion,

only the 9-by-9 averaged values of the TOPSAR DEM are compared with the pixel values

of the SRTM DEM. Thus, the fine registration is less critical in this approach than in the

case where a one-to-one match is required.

3.3 Artifact Elimination

Examination of the georeferenced TOPSAR DEM (Fig. 3.2a) shows motion artifacts when

compared to the SRTM DEM (Fig. 3.2b). The artifacts are not clearly discernible in

Fig. 3.2a because their magnitude is small in comparison to the overall data values. The

artifacts are identified by downsampling the registered TOPSAR DEM and subtracting the

SRTM DEM. Large scale anomalies that periodically fluctuate over an entire swath are

visible in Fig. 3.2c. The periodic pattern is most likely due to uncompensated roll of the

DC-8 aircraft. The spaceborne data are less likely to exhibit similar artifacts, because the

spacecraft is not greatly affected by the atmosphere. Note that the width of the anomalies

correspond to the width of a TOPSAR swath. Because the SRTM swath is much larger

than that of the TOPSAR system (Table 3.1), a larger area is covered under consistent

conditions, reducing the number of parallel tracks required to form an SRTM DEM.

The maximum amplitude of the motion artifacts in our study area is about 20 meters.

This would result in substantial errors in many analyses if not properly corrected. For ex-

ample, if this TOPSAR DEM is used for topography reduction in repeat-pass InSAR using

ERS-2 data with a perpendicular baseline of about 400 meters, the resulting deformation

interferogram would contain one fringe (= 2.8 cm) of spurious signal.

To remove these artifacts from the TOPSAR DEM, we upsample the difference image

with bilinear interpolation by a factor of nine so that its pixel spacing matches the TOP-

SAR DEM. The difference image is subtracted from the TOPSAR DEM. This process is

described with a flow diagram in Fig. 3.3. Note that the lower branch undergoes two low-

pass filter operations when averaging and bilinear interpolation are implemented, while the

upper branch preserves the high frequency contents of the TOPSAR DEM. In this way we

can eliminate the large-scale artifacts while retaining details in the TOPSAR DEM.

3.3. ARTIFACT ELIMINATION 25

Figure 3.2: (a) TOPSAR DEM and (b) SRTM DEM. The tick labels are pixel numbers.Note the difference in pixel spacing between the two DEMs. (c) Artifacts obtained bysubtracting the SRTM DEM from the TOPSAR DEM. The flight direction and the radarlook direction of the aircraft associated with the swath with the artifact are indicated witha long and short arrows respectively. Note that the artifacts appear in one entire TOPSARswath, while it is not as serious in other swaths.


Figure 3.3: The flow diagram of the artifact elimination.

3.4 Prediction-Error (PE) Filter

The next step in the DEM process is to fill in missing data. We use a prediction error

(PE) filter operating on the TOPSAR DEM to fill these gaps. The basic idea of the PE

filter constraint (Claerbout, 1992; Claerbout and Fomel, 2002) is that missing data can be

estimated so that the restored data yield minimum energy when the PE filter is applied.

The PE filter is derived from training data, which is normally valid data surrounding the

missing region. The PE filter is selected so that the missing data and the valid data share

approximately the same spectral content. Hence, we assume that the spectral content of

the missing data in the TOPSAR DEM is similar to that of the regions with valid data

surrounding the missing regions.

3.4.1 Designing the filter

We generate a PE filter such that it rejects data with statistics found in the valid regions of

the TOPSAR DEM. Given this PE filter, we solve for data in the missing regions such that

the interpolated data is also been nullified by the PE filter. This concept is illustrated in Fig.

3.5.

The PE filter, fPE, is found by minimizing the following objective function,

‖fPE ∗ xe‖2 (3.2)

where xe is the existing data from the TOPSAR DEM, and ∗ represents convolution. This

3.4. PREDICTION-ERROR (PE) FILTER 27

expression can be rewritten in a linear algebraic form using the following matrix operation,

‖FPE xe‖2, (3.3)

or equivalently

‖Xe fPE‖2 (3.4)

where FPE and Xe are the matrix representations of fPE and xe for convolution operation.

These matrix and vector expressions are used to indicate their linear relationship.

3.4.2 1-D example

The procedure of acquiring the PE filter can be explained with 1-D example. Suppose that

a data set, x = [x1, . . . ,xn] (where n � 3) is given, and we want to compute a PE filter of

length 3, fPE = [1 f1 f2]. Then we form a system of linear equations as follows.

x3 x2 x1

x4 x3 x2...

......

xn xn−1 xn−2

1

f1

f2

≈ 0 (3.5)

The first element of the PE filter should be equal to one to avoid the trivial solution, fPE = 0.

Note that (3.5) is the convolution of the data and the PE filter. After simple algebra and

with d ≡

x3...

xn

and D ≡

x2 x1...

...

xn−1 xn−2

we get

D

[

f1

f2

]

≈−d (3.6)

and its normal equation becomes

[

f1

f2

]

=(

DT D)−1 DT (−d) (3.7)


Note that (3.7) minimizes (3.2) in a least-squares sense. This procedure can be extended

to 2-D problems, and more details are described in Claerbout (1992) and Claerbout and

Fomel (2002).

3.4.3 The effect of the filter

Fig. 3.4 shows the characteristics of the PE filter in the spatial and Fourier domains. Fig.

3.4a is the sample DEM chosen from Fig. 3.1 (numbered box 1) for demonstration. It

contains various topographic features, and has a wide range of spectral content (Fig. 3.4d).

Fig. 3.4b is the 5-by-5 PE filter derived from 3.4a by solving the inverse problem in (3.3).

Note that the first three elements in the first column of the filter coefficients are 0 0 1.

This is the PE filter’s unique constraint that ensures the filtered output to be white noise

(Claerbout, 1992). In the filtered output (Fig. 3.4c) all the variations in the DEM were

effectively suppressed. The size (order) of the PE filter is based on the complexity of the

spectrum of the DEM. In general, as the spectrum becomes more complex, a larger size

filter is required. After testing various sizes of the filter, we found a 5-by-5 size appropriate

for the DEM used in our study. Fig. 3.4d and Fig. 3.4e show the spectra of the DEM

and the PE filter respectively. These illustrate the inverse relationship of the PE filter to

the corresponding DEM in the Fourier domain, such that their product is minimized (Fig.

3.4f). This PE filter constrains the interpolated data in the DEM to similar spectral content

to the existing data.

All inverse problems in this study were derived using the conjugate gradient method,

where forward and adjoint functional operators are used instead of the explicit inverse

operators (Claerbout, 1992), saving computer memory space.

3.5 Interpolation

3.5.1 PE filter constraint

Once the PE filter is determined, we next estimate the missing parts of the image. As de-

picted in Fig. 3.5, interpolation using the PE filter requires that the norm of the filtered

3.5. INTERPOLATION 29

Figure 3.4: The effect of a PE filter. (a) original DEM, (b) a 2-D PE filter found from theDEM, (c) DEM filtered with the PE filter (d), (e), and (f) are the spectra of (a), (b), and (c)respectively plotted in dB. (a) and (c) are drawn with the same color scale. Note that in (c)the variation of image (a) was effectively suppressed by the filter. The standard deviationsof (a) and (c) are 27.6 m and 2.5 m respectively.


Figure 3.5: Concept of PE filter. The PE filter is estimated by solving an inverse problemconstrained with the remaining part, and the missing part is estimated by solving anotherinverse problem constrained with the filter. The ε1 and ε2 are white noise with small am-plitude.

output be minimized. This procedure can be formulated as an inverse computation mini-

mizing the following objective function:

‖FPE x‖2 (3.8)

where FPE is the matrix representation of the PE filter convolution, and x represents the en-

tire data set including the known and the missing regions. In the inversion process we only

update the missing region, without changing the known region. This guarantees seamless

interpolation across the boundaries between the known and missing regions.

3.5.2 SRTM DEM constraint

As previously stated, 90-m posting SRTM DEMs were generated from 30-m posting data.

This downsampling was done by calculating 3 “looks” in both the easting and northing

directions. In order to use the SRTM DEM as a constraint to interpolate the TOPSAR

DEM, we posit the following relationship between the two DEMs: each pixel value in a

90-m posting SRTM DEM can be considered equivalent to the averaged value of a 9-by-9

pixel window in a 10-m posting TOPSAR DEM centered at the corresponding pixel in the


SRTM DEM.

Solution using the constraint of the SRTM DEM to find the missing data points in the

TOPSAR DEM can be expressed as minimizing the following objective function:

‖y−Axm‖2 (3.9)

where y is an SRTM DEM expressed as a vector that covers the missing regions of the

TOPSAR DEM, and A is an averaging operator generating 9 looks, and xm represents the

missing regions of the TOPSAR DEM.

3.5.3 Inversion with two constraints

By combining two constraints, one derived from the statistics of the PE filter and one from

the SRTM DEM, we can interpolate the missing data optimally with respect to both criteria.

The PE filter guarantees that the interpolated data will have the same spectral properties as

the known data. At the same time the SRTM constraint forces the interpolated data to have

average height near the corresponding SRTM DEM. We formulate the inverse problem as

a minimization of the following objective function:

λ 2 ‖FPE xm‖2 +‖y−Axm‖2 (3.10)

where λ set the relative effect of each criterion. Here xm has the dimensions of the TOP-

SAR DEM, while y has the dimensions of the SRTM DEM. If regions of missing data are

localized in an image, entire image does not have to be used for generating a PE filter. We

implement interpolation in subimages to save time and computer memory space. An exam-

ple of the such subimage is shown in Fig. 3.6. The image is a part of Fig. 3.1 (numbered

box 2). Fig. 3.6a and 3.6b are examples of xe in (3.3) and y respectively.

The multiplier λ determines the relative weight of the two terms in the objective func-

tion. As λ → ∞, the solution satisfies the first constraint only, and if λ = 0, the solution

satisfies the second constraint only.


Figure 3.6: Example subimages of (a) TOPSAR DEM showing regions of missing data(black), and (b) SRTM DEM of the same area. These subimages are engaged in one imple-mentation of the interpolation. The grayscale is altitude in meters.


Figure 3.7: Cross-validation sum of squares. The minimum occurs when λ = 0.16.

3.5.4 Optimal weighting

We used cross-validation sum of squares (CVSS) (Wahba, 1990) to determine the optimal

weights for the two terms in (3.10). Consider a model xm that minimizes the following

quantity

λ 2 ‖FPE xm‖2 +‖y(k)−A(k) xm‖2 (k = 1, ...,N) (3.11)

where y(k) and A(k) are the y and the A in (3.10) with the k-th element and the k-th row

omitted respectively, and N is the number of elements in y that fall into the missing region.

Denote this model x(k)m (λ ). Then we compute the CVSS defined as follows,

CVSS(λ ) =1N

N

∑k=1

(

yk −Ak x(k)m (λ )

)2(3.12)

where yk is the omitted element from the vector y and Ak is the omitted row vector from

the matrix A when the x(k)m (λ ) was estimated. Thus, Ak x(k)

m (λ ) is the prediction based on

the other N − 1 observations. Finally, we minimize CVSS(λ ) with respect to λ to obtain

the optimal weight (Fig. 3.7).

In the case of the example shown in Fig. 3.6, the minimum CVSS was obtained for


λ = 0.16 (Fig. 3.7). The effect of varying λ is shown in Fig. 3.8. It is apparent (see Fig.

3.8) that the optimal weight is a more “plausible” result than either of the end members,

preserving aspects of both constraints.

In Fig. 3.8a the interpolation uses only the PE filter constraint. This interpolation does

not recover the continuity of the ridge running across the DEM in north-south direction,

which is observed in the SRTM DEM (Fig. 3.6b). This follows from a PE filter obtained

such that it eliminates the overall variations in the image. The variations include not only

the ridge but also the accurate topography in the DEM.

The other end member, Fig. 3.8c, shows the result for applying zero weight to the PE

filter constraint. Since the averaging operator A in (3.10) is applied independently for each

9 by 9 pixel group, it is equivalent to simply filling the regions of missing data with 9 by 9

identical values that are the same as the corresponding SRTM DEM (3.8a and 3.8c).

3.5.5 Simulation of the interpolation

The quality of cross-validation in this study is itself validated by simulating the interpo-

lation process with known subimages that do not contain missing data. For example, if a

known subimage is selected from Fig. 3.1 (numbered box 3), we can remove some data

and apply our recovery algorithm. The subimage is similar in topographic features to the

area shown in Fig. 3.6. The process is illustrated in Fig. 3.9. We introduce a hole in

3.9b and calculate the CVSS (Fig. 3.9d) for each λ ranging from 0 to 2. Then we use the

estimated λ , which minimizes the CVSS, for the interpolation process to obtain the image

in 3.9c. For each value of λ we also calculate the RMS error between the known and the

interpolated images. The RMS error is plotted against λ in Fig. 3.9e. The CVSS is mini-

mized for λ = 0.062, while the RMS error has a minimum at λ = 0.065. This agreement

suggests that minimizing the CVSS is a useful method to balance the constraints. Note

that the minimum RMS error in Fig. 3.9e is about 5 meters. This value is smaller than the

relative vertical height accuracy of the SRTM DEM, which is about 10 meters.


Figure 3.8: The results of interpolation applied to DEMs in Fig. 3.6, with various weights.(a) λ → ∞, (b) λ = 0.16, and (c) λ = 0. Profiles along A-A’ are shown in the plot (d).


Figure 3.9: The quality of the CVSS, (a) a sample image that does not have a hole, (b) ahole was made, (c) interpolated image with an optimal weight, (d) CVSS as a function ofλ . The CVSS has a minimum when λ = 0.062. (e) RMS error between true image (a) andthe interpolated image (c). The minimum occurs when λ = 0.065.

3.6. INTERPOLATION RESULTS 37

3.6 Interpolation Results

The method presented in the previous section was applied to the entire image of Fig. 3.1.

The registered TOPSAR DEM contains missing data in regions of various sizes. Small

subimages were extracted from the DEM. Each subimage is interpolated, and the results

are reinserted into the large DEM. The locations and sizes of the subimages are indicated

with white boxes in Fig. 3.10a. Note the largest region of missing data in the middle

of the caldera. This region is not only a simple large gap but also a gap between two

swaths. The interpolation is an iterative process and fills up regions of missing data starting

from the boundary. If valid data along the boundary (boundaries of a swath for example)

contain edge effects, error tends to propagate through the interpolation process. In this

case, expanding the region of missing data by a few pixels before interpolation produces

better results. If there is a large region of missing data, the spectral content information

of valid data can fade out as the interpolation proceeds toward the center of the gap. In

this case, sequentially applying the interpolation to parts of the gap is one solution. Due to

edge effects along the boundary of the large gap, the interpolation result does not produce

topography that matches the surrounding terrain well. Hence, we expand the gap by 3

pixels to eliminate edge effects. We divided the gap into multiple subimages, and each

subimage was interpolated individually.

3.7 Effect on InSAR

Finally, we can investigate the effect of the artifact elimination and the interpolation on

simulated interferograms. It is often easier to see differences in elevation in simulated

interferograms than in conventional contour plots. In addition, simulated interferograms

provide a measure of how sensitive the interferogram is to the topography. Fig. 3.11 shows

georeferenced simulated interferograms from three DEMs; the registered TOPSAR DEM,

the TOPSAR DEM after the artifact elimination, and the TOPSAR DEM after the inter-

polation. In all interferograms, a C-band wavelength is used, and we assume a 452 m

perpendicular baseline between two satellite positions. This perpendicular baseline is real-

istic (Amelung et al., 2000). The fringe lines in the interferograms are approximately height


Figure 3.10: The original TOPSAR DEM (a) and the reconstructed DEM (b) after interpo-lation with PE filter and SRTM DEM constraints. The gray scale is altitude in meters, andthe spatial extent is about 12 km across the image.

3.8. CONCLUSION 39

contour lines. The interval of the fringe lines is inversely proportional to the perpendicular

baseline (Zebker et al., 1997), and in this case one color cycle of the fringes represents

about 20 meters. Note in Fig. 3.11a that the fringe lines are discontinuous across the long

region of missing data inside the caldera. This is due to artifacts in the original TOPSAR

DEM. After eliminating these artifacts the discontinuity disappears (Fig. 3.11b). Finally

the missing data regions are interpolated in a seamless manner (Fig. 3.11c).

3.8 Conclusion

The aircraft roll artifacts in the TOPSAR DEM were eliminated by subtracting the differ-

ence between the TOPSAR and SRTM DEMs. A 2-D PE filter derived from the existing

data and the SRTM DEM for the same region are then used as interpolation constraints.

Solving the inverse problem constrained with both the PE filter and the SRTM DEM pro-

duces a high-quality interpolated map of elevation. Cross-validation works well to select

optimal constraint weighting in the inversion. This objective criterion results in less biased

interpolation and guarantees the best fit to the SRTM DEM. The quality of many other

TOPSAR DEMs can be improved similarly. The new DEM created in this chapter is used

in Chapter 5 and Chapter 6 of this dissertation. In Chapter 4 we provide a new inversion

method to model the detailed geometry of uniformly pressurized crack constrained by In-

SAR observations. The SRTM DEM is used in the chapter for simplicity, as the chapter

concerns more about modeling algorithm.


Figure 3.11: Simulated interferograms from (a) the original registered TOPSAR DEM, (b)the DEM after the artifact was removed, and (c) the DEM interpolated with PE filter andthe SRTM DEM. All the interferograms were simulated with the C-band wavelength (5.6cm) and a perpendicular baseline of 452 m. Thus, one color cycle represents 20 m heightdifference.

Chapter 4

Constraints on Magma ChamberGeometry at Sierra Negra Volcano,Galapagos

The previous chapter provided an algorithm to improve the quality of data (i.e. InSAR

data). This chapter describes a new algoritm to explain the data in a mechanically con-

sistent way. In this chapter we investigate the problem of estimating magma chamber

geometry using InSAR observations of Sierra Negra volcano, Galapagos. Ascending and

descending interferograms are combined to determine vertical and one horizontal compo-

nent of displacement. The ratio of maximum horizontal to vertical displacement suggests a

sill-like source. Spherical or stock-like bodies are inconsistent with the data. We estimate

the geometry of the sill assuming a horizontal, uniformly pressurized crack with unknown

periphery and depth. The sill is discretized into small elements that either open, and are

subjected to the pressure boundary condition, or remain closed. We find the best-fitting sill

to be located beneath Sierra Negra’s inner caldera at a depth of about 2 km. Using bound-

ary element calculations we show that any magma chamber with a flat top coincident with

the sill model fits the data equally well. The data are insensitive to the sides and bottom of

the magma chamber.

41

42 CHAPTER 4. MAGMA CHAMBER GEOMETRY AT SIERRA NEGRA

Figure 4.1: Shaded relief topographic map of Galapagos Islands. The study area is indi-cated with a black box, which includes the caldera of Sierra Negra volcano.

4.1 Introduction

Six volcanoes in the western Galapagos islands of Fernandina and Isabela (Fig. 4.1) have

been actively deforming since 1992 (Amelung et al., 2000). Among these Sierra Negra

is by far the most voluminous and has been one of the most active. It has experienced

11 historical eruptions, including most recently the 0.9 km3 eruption on the north flank

of the volcano in 1979 (Reynolds et al., 1995). Sierra Negra’s shallow caldera (110 m in

maximum depth) is the largest by area in the western Galapagos (59.8 km2) (Munro and

Rowland, 1996). The caldera is characterized by a C-shaped sinuous ridge, composed of a

complex set of normally faulted blocks with steep (60◦ - 90◦) outward dipping fault scarps

(Reynolds et al., 1995).

The center of Sierra Negra’s caldera uplifted from 1992 to 1997, causing a near-vertical

line-of-sight (LOS) displacement of about 1.6 meters in InSAR observations (Amelung

4.1. INTRODUCTION 43

et al., 2000). The rapid inflation was followed by trapdoor faulting sometime in 1997-

1998, which occurred along the pre-existing fault system inside the south moat of the

caldera. This event was discovered using InSAR observations (Amelung et al., 2000). The

faulting was also confirmed by field study and range offset measurements with maximum

observed slip of about 1.5 meters (Jonsson et al., 2005). After the trapdoor faulting event,

Sierra Negra resumed uplift from September 1998 to March 1999 with a maximum LOS

displacement of 30 cm. This uplift event was successfully modeled as a sill with spatially

varying opening distribution by Amelung et al. (2000). A GPS network on Sierra Negra

showed that subsidence initiated sometime between late 2000 and early 2001, at rates of up

to 9 cm/yr (Geist et al., in review 2004).

In this chapter, we analyze InSAR data acquired over Sierra Negra, incorporating a

rigorous investigation of the geometry of the putative magma body. While we refer to the

deformation source as a magma body, it may include a hydrothermal component. Although

the sill model of Amelung et al. (2000) fits the observations well, it is non-unique primarily

because of the near vertical one-dimensional sensitivity of the data (Dieterich and Decker,

1975; Fialko et al., 2001c). Given vertical and horizontal displacement data, however, it

is possible to distinguish sills from spherical chambers or stocks (Fialko et al., 2001a).

Here we first constrain the shape of the magma chamber by reconstructing vertical and

horizontal components of the surface displacement using two interferograms, one from an

ascending orbit and the other from a descending orbit. This approach is similar to that of

Fialko et al. (2001b) but does not require calculation of azimuth offsets because we need

only two components to constrain the shape.

Amelung et al. (2000) allow the sill opening to vary spatially to fit the data, but they

did not impose any additional physical constraints on the shape. We explore below a new

inversion approach, in which the deformation source is restricted to be a uniformly pres-

surized sill whose plan outline is unknown, appropriate for a magmatic intrusion. In this

case the geometry of the sill and the magma pressure are the unknowns. Compared to the

previous kinematic model (Amelung et al., 2000), in which sill opening is constrained only

by a Laplacian smoothing constraint, our new approach dramatically reduces the number

of degrees of freedom in the inversion.

An approach with a pressure boundary condition is advantageous because the magma


pressure is directly relevant to fluid processes in the magma chamber. In particular, the

total magma pressure pm is the sum of the excess magma pressure ∆p and the lithostatic

pressure pL,

pm = ∆p+ pL. (4.1)

We estimate ∆p from the InSAR data. However, it should be noted that the estimated ∆p

represents only the change in pressure during the time period of the InSAR observations.

Assuming a lithostatic stress state, pL = ρ gd, where ρ is the density of overlying rock, d is

the depth of the magma body, measured from the caldera floor. Since the magma chamber is

known to be located inside the flat caldera of Sierra Negra at shallow depth (Amelung et al.,

2000), we simplify the problem and assume an elastic half-space, ignoring topographically

induced stresses (Pinel and Jaupart, 2003). Given the excess magma pressure, we can

calculate the stress in the neighborhood of the sill using the pressure as boundary condition.

Knowledge of the stress field allows us to consider crack initiation and propagation criteria,

since cracks may propagate at roughly constant pressure. Moreover, by calculating stress

fields around the magma chamber caused by magma pressurization, we should be able to

predict the direction of crack propagation, and thus the evolution of the system. Overall,

the new method relates to physical properties of the magma itself, and will better describe

the mechanical interaction of the magma intrusion with the surrounding rock.

Later in this paper we find that even with both vertical and horizontal displacement

data at the Earth’s surface we cannot uniquely determine the shape of the lower part of the

magma chamber. We use the boundary element technique to explore the range of models

consistent with the InSAR data, providing clearer constraints on magma chamber geometry.

4.2 Shape of the Magma Chamber

Non-uniqueness of the shape of subsurface magma chambers (e.g. Dieterich and Decker,

1975; Fialko et al., 2001c) is greatly reduced if both vertical and horizontal deformation

data are available (Fialko et al., 2001a). Sills produce little horizontal displacement rel-

ative to the peak uplift, as compared to equi-dimensional magma bodies which generate

4.2. SHAPE OF THE MAGMA CHAMBER 45

more horizontal deformation. Since imaging radar satellites are side-looking, interfero-

grams record horizontal deformation as well as vertical deformation. Using data from both

ascending and descending orbits, it is possible to determine the vertical displacement and

one component of horizontal displacement.

Satellites can acquire data when they pass from south to north (ascending) and from

north to south (descending). Data from ascending and descending orbits have different

imaging geometry (look direction), providing two linearly independent LOS measure-

ments. In principle, we can calculate the 3-D displacement field, using data from both

ascending and descending orbits (Fialko et al., 2001b), by solving for three orthogonal

components of the surface displacement field from the two LOS displacements and also

the interferogram azimuth offsets, which are acquired by cross-correlating two ampli-

tude images. However, in the case of Sierra Negra for the time period when the ascend-

ing/descending pair is available, the maximum horizontal surface displacement is less than

10 cm. The accuracy of the azimuth offsets is about 12.5 cm, if we assume that we can

measure pixel offsets locally between two images to 1/32 of a pixel (Jonsson et al., 2002).

Hence, the signal-to-noise ratio is not sufficient for the azimuth offsets to be meaningful,

and we cannot fully reconstruct the 3-D displacement field. Consequently, the vertical

component and only one horizontal component can be determined. In the case of volcano

deformation, assuming near-circular symmetry allows the one component of horizontal

displacement to be considered as a radial component. This is adequate since we only need

vertical and radial components to reduce the non-uniqueness of the shape of magma cham-

bers.

The ERS-2 data we used are shown in Table 4.1. Fig. 4.2 shows a map view of the

ascending and descending orbits and the idealized surface displacement vectors due to

volcano deformation. Using the same notation as Fialko et al. (2001b), the LOS vector can

be expressed in terms of displacements, Ui, as:

dlos = [Un sin φ − Ue cos φ ] sin λ + Uu cos λ , (4.2)

where φ is the azimuth of the satellite heading vector (positive clockwise from the North),

and λ is the radar incidence angle. Since the orbit inclination of the ERS-2 satellite is 98.5◦,


Table 4.1: Interferograms Used in This Study

Orbit direction Scene 1 Scene 2 T⊥ B⊥

Ascending 1998/10/31 1999/02/13 3.5 months 152 m

Descending 1998/11/05 1999/02/18 3.5 months 448 m

Ascending 1998/09/26 1999/03/20 6 months 50 m

which is the angle between the heading vector of the ascending satellite and the easting

vector at the equator, the LOS displacement fields contain horizontal components that are

off by 8.5◦ from easting or westing vectors. Thus, the φ ’s for ascending and descending

orbits are -8.5◦ and 188.5◦ respectively. We approximate these values to 0◦ and 180◦. With

this approximation, Eq. 4.2 reduces to,

dlos = ∓Ue sin λ + Uu cos λ , (4.3)

where the minus sign is for data from ascending orbit and the plus sign is for data from

descending orbit. Since there are two unknowns and two equations we can solve for Uu and

Ue. We verify the accuracy of this approximation by reconstructing the vertical and radial

components from a well known circularly symmetric deformation source. Fig. 4.3 shows

the true vertical and radial components due to the Mogi point source (Mogi, 1958) and

the reconstructed components, derived from simulated interferograms from ascending and

descending orbits using the ERS-2 satellite’s actual orbit inclination. The interferograms

were transformed into vertical and easting components. The approximation-induced error

is about 1% in this case.

The actual ascending and descending interferograms and the reconstructed vertical and

horizontal components are shown in Fig. 4.4. The interferograms were processed using the

Jet Propulsion Laboratory/Caltech repeat orbit interferometry package ROI PAC. A 90-

meter posting Digital Elevation Model (DEM) from the Shuttle Radar Topography Mission

(SRTM) was used to subtract the topographic signal.

Fig. 4.5 shows the profiles of vertical and east components along A-A’ in Fig. 4.4d. The

line A-A’ was chosen so that it is parallel to E-W axis and includes the maximum vertical

4.2. SHAPE OF THE MAGMA CHAMBER 47

ground deformation

Figure 4.2: Map view of ascending and descending orbit imaging geometry and idealizedsurface displacement of volcano deformation. The open triangle indicates the center of thevolcano

Figure 4.3: Circularly symmetric deformation source was used to estimate the satellite orbitinclination induced error. The vertical and radial component are reconstructed in the sameway as the data was analyzed. The x-axis is the distance normalized by the depth of thesource, and the y-axis is displacement normalized by the maximum vertical displacement.In the case of circular symmetry, the maximum error due to the orbit inclination not being90◦ is about 1%.


Figure 4.4: Interferograms from ascending (a) and descending (b) orbits with temporalbaselines of 1998/10/31 - 1999/02/13 and 1998/11/05 - 1999/02/18 respectively. One colorcycle represents 5 cm change of range in LOS direction of satellite. Interferometric dis-placements can be separated into vertical (c) and horizontal (d) components using the imag-ing geometries of the two orbits.

4.3. ESTIMATION OF BEST-FITTING SILL GEOMETRY 49

Figure 4.5: Profiles of vertical and east component of surface deformation along A-A’ inFig. 4.4d. Five lines are averaged to produce a smooth plot, and values are normalized bythe maximum vertical component.

displacement. The ratio of the maximum horizontal to maximum vertical displacement is

about 0.3, consistent with a sill, but inconsistent with a spherical (Mogi) source for which

the ratio is about 0.4 (Fialko et al., 2001a). Thus, this observation supports the idea that the

magma reservoir beneath Sierra Negra is a sill, or a sill-like body.

4.3 Estimation of Best-fitting Sill Geometry

We determined the sill geometry using a 3-D boundary element method plus a non-linear

inversion scheme. In this approach, the geometry of the sill is described by its periphery or

fracture tip-line and its depth. We used the same SAR scenes (1998/09/26 and 1999/03/20)

that Amelung et al. (2000) used to form an interferogram, rather than the ascending and

descending pair used in the previous section (Table 4.1), to facilitate comparison of our

results with the kinematic model of Amelung et al. (2000). The deformation pattern in the


previous section (Fig. 4.4a) is similar in shape to the interferogram used for modeling (Fig.

4.4b) but smaller in amplitude. We compared the two interferograms by scaling the one

for inversion (s1) to fit the one in the previous section (s2). To do this we estimate a scale

factor a and phase offset b minimizing,

mina,b

||y − (ax + b)||2, (4.4)

Fig. 4.6 shows that the surface deformation is nearly identical and implies that the defor-

mation sources in the two time periods share the same geometry. Thus, the only difference

between the two time periods should be the magnitude of the excess magma pressure. For

this reason, we can now model the deformation source using the third interferogram (Table

4.1), starting from the result of the previous section; that is, the deformation source is a sill.

4.3.1 Forward Modeling

We assume that uniform pressure acts everywhere on the surface of the sill, with no shear

traction. A uniform pressure boundary condition is physically more reasonable than a

kinematic displacement boundary condition, because Sierra Negra commonly erupts low

viscosity basaltic magma which should be close to hydrostatic pressure equilibrium.

We divide the crack into elements that either opened or remained closed in order to

determine the sill geometry. The uniform pressure boundary condition is enforced on all

open elements. Various combinations of open and closed elements are selected to form a

sill. Once a candidate sill periphery, depth, and uniform pressure are selected, the opening

distribution of the entire sill is uniquely determined by a boundary element calculation.

This reduces the number of degrees of freedom of the boundary value problem relative to

the kinematic inversion.

In order to use the boundary value problem as the forward function in an inversion, we

introduce a set of binary parameters that describe whether each element in a grid is open

or closed. Fig. 4.7 shows a simple example of a model grid. A value of zero at a grid cell

means that the sill element is closed (i.e. not a part of the sill) and a “one” means that the

sill element is open. Only the open elements are subject to the uniform internal pressure

condition. Determining the geometry of the sill is equivalent to determining the appropriate


Figure 4.6: Least-squares fitting results. (a) Interferogram for the time period of 1998/10/31- 1999/02/13. (b) Scaled version of the the interferogram for 1998/09/26 - 1999/03/20. (c)Residual. (d) Profiles through (a) and (b). The blue solid line is the S-N profile of (a), andthe red solid line is the W-E profile of (a). The black dashed lines are the correspondingprofiles of (b). One color cycle in the interferograms and residual represents 5 cm of LOSdisplacement.


Figure 4.7: Simple example of model grid that shows four open sill elements. The upperright corner sill element will open widest under uniform pressure.

combination of zeros and ones.

We choose depth, excess magma pressure, and one combination of the binary parame-

ters. These suffice to determine the opening distribution in the entire model grid uniquely,

using a 3-D boundary element calculation. Surface deformation is then derived from the

sill opening using Green’s functions integrated over a rectangular element in an isotropic

homogeneous linear elastic half-space (e.g. Okada, 1992). The shear modulus of the elastic

half-space was assumed to be 30 GPa, and 0.25 was assumed for Poisson’s ratio. Since the

InSAR data are sensitive only to the ratio of the excess magma pressure to the shear mod-

ulus, decreasing the shear modulus by a factor of 3, would decrease the estimated excess

magma pressure by the same factor.

4.3.2 Nonlinear Inversion

Once the forward methodology is defined, we can proceed to select the best-fit model

through inversion. The goal here is to estimate the best-fitting sill geometry, excess magma

pressure, and depth. The binary nature of the crack opening makes the problem highly

nonlinear. In this case, the misfit or objective function is not quadratic, thus convergence to

the global minimum is not guaranteed. Moreover, binary parameters are discrete and thus

not differentiable. Therefore a gradient-based inversion scheme is not possible. As our

model grid size is 16 by 18, the number of possible combination of the binary parameters

is 2288. Because it is not feasible to test all possibilities, we employed simulated annealing,


a stochastic nonlinear inversion scheme (Metropolis et al., 1953).

Simulated annealing was proven to converge with an infinite number of iterations at

a constant (metaphorical) temperature parameter (Rothman, 1986). The cooling schedule

is important as it determines whether the global minimum can be achieved or not, and it

also affects the speed of convergence. We use a simulated annealing code (Cervelli et al.,

2001) that applies Basu and Frazer type rapid determination of critical temperature (Basu

and Frazer, 1990). For sampling at each temperature we use the heat bath algorithm,

where one parameter is perturbed at one iteration while the others are fixed. The code was

modified to accommodate the binary parameters.

The data vector in our inversion is the observed LOS displacements dlos. The size

of the original InSAR data set (16384 pixels) was reduced to 674 points using quadtree

partitioning (Jonsson et al., 2002) to make the problem manageable. The model vector mcontains the set of binary parameters and the excess magma pressure ∆p. Thus,

dlos = g(m) + ε, (4.5)

where ε is data error plus errors in the forward model, and the function g includes the

dislocation model and the LOS projection of the surface displacement. We adopt an L2

objective function,

Φ = ||d − g(m) ||2. (4.6)

The other dislocation model parameters were fixed in the sill plane. The depth of the sill

was estimated by examining the residual pattern as well as the L2-norm of the residual at

depths ranging from 1 km to 3 km in 100 m depth increments.

Solutions with isolated individual open elements require physically unrealistic magma

pressure (hundreds of MPa). In order to preclude this from our models, we put an upper

bound on the excess magma pressure. If a certain choice of geometry yields a magma

pressure that is greater than the upper bound, then the candidate geometry is rejected.

We used physical reasoning to derive an initial value for the upper bound on melt pres-

sure. Fig. 4.8 shows a simplified vertical cross-section containing the magma chamber.

Assuming the magma pressure is at equilibrium at depth D, the pressure in the magma


chamber pm propagated through the melt column can be written as,

pm = ρs gD − ρm g(h + D − d) (4.7)

where ρs is the density of solid rock, and ρm is the melt density. The excess magma pressure

can then be calculated by subtracting the lithostatic pressure due to the overloading rock:

∆p = pm − ρs gd. (4.8)

For example, if we take ρs = 2.9gcm−3 (Hill and Zucca, 1987), ρm = 2.6gcm−3 (Savage,

1984), h = 2.4 km, D = 37 km, and d = 2 km, we compute an excess magma pressure of

7.6 MPa. Here the h was derived from the difference between the altitude of Sierra Negra’s

caldera and the average altitude of a circular region about 100 km in radius, centered at

Sierra Negra. The thickness of the lithosphere D was calculated using the thermal diffusiv-

ity κ = 1mm2 s−1 (Turcotte and Schubert, 2002) and the age of the lithosphere t = 8 Myr

(Sallares and Charvis, 2003) in the following equation (Turcotte and Schubert, 2002),

D = 2.32√

κ t. (4.9)

Starting from the initial value of excess magma pressure, several choices of the upper bound

on excess magma pressure were tested for each depth, and we selected the one that pro-

duced reasonable connectivity and the best fit.

Fig. 4.9 shows the interferogram used for modeling in this part of the study and the

best-fit model derived from simulated annealing. An upper bound of 5 MPa was placed on

the excess magma pressure.

Although disconnected and isolated open elements were effectively suppressed by the

upper bound, a few isolated segments remained. Eliminating these did not change the fit

to the data significantly because the isolated segments open very little given the estimated

excess magma pressure. The best-fitting model after removing the isolated segments is

shown in Fig. 4.10b, in comparison to Amelung et al.’s best-fitting kinematic model (Fig.

4.10a). Note that the estimated sill is restricted to the inner caldera and is bounded by the

sinuous ridge. The depth and excess magma pressure are estimated as 1.9 km and 4.5 MPa


Figure 4.8: Schematic vertical section of the lithosphere.

respectively. The colors of the sill elements represent the inferred opening distribution,

with the maximum being 0.5 m. The total volume change is calculated as 6.7 million cubic

meters. If we attribute this volume change to magma influx, the average magma filling rate

is about 1.1 million cubic meters per month for the time period 1998/09/26 to 1999/03/20.

The overall distribution of estimated sill opening is in good agreement with Amelung et

al.’s kinematic model as expected.

Using the best-fit model the effect of asymmetric deformation source on the method

described in section 2 is simulated. Note in Fig. 4.4 that neither the vertical nor the east-

ing component are perfectly symmetric, implying a lack of symmetry in the deformation

source. In order to check the effect of this asymmetry, we implemented a simulation which

reconstructed the east component of displacement for different orientations of the deforma-

tion source (Fig. 4.11). We rotated the best-fit model, and reconstructed the east component

from the rotated models. This figure shows how far the model is from circular symmetry.

Specifically, each subfigure demonstrates the symmetry about N-S axis through the center

of the image. Note that a 60◦ model rotation shows the best symmetry about N-S axis, im-

plying that the deformation source has an axis of symmetry whose strike is about 60◦ from

the north. This axis of symmetry can also be seen in the vertical component (Fig. 4.4c).


Figure 4.9: Observed interferogram (a) and simulated interferogram (b) from the best-fitmodel (d). The residual (c) between the data and the model shows differences smaller than2.5 cm, half the magnitude of one color cycle in (a) and (b).

4.4. DISCUSSION 57

Figure 4.10: (a) Best-fit model of Amelung et al. using a sill model with spatially varyingopening distribution. (b) Best-fit model with uniform pressure boundary condition. Depthwas estimated as 1.9 km in both cases.

4.4 Discussion

In this section we further explore the non-uniqueness of the shape of the magma chamber.

Although an equi-dimensional magma chamber or stock was rejected previously, our anal-

ysis does not prove that the source of deformation is a thin sill. If the radius of the sill is

large compared to the depth, the surface deformation is dominated by displacement of the

sill’s upper surface. We thus suspect that surface deformation would be insensitive to the

sides and bottom of the chamber.

We test this using a boundary element code, Poly3D (Thomas, 1993). The Poly3D

superposes the solution for an angular dislocation (Yoffe, 1960; Comninou and Dundurs,

1975) to calculate the displacements, strains, and stresses induced in an elastic whole-

or half-space by planar polygonal elements of displacement discontinuity and boundary

element method. Fig. 4.12 shows the geometry of a flat-topped diapir used in this test.

The depth to the top is 1.9 km, and the radius of the top is 3 km, resembling the estimated

geometry of the sill at Sierra Negra. The sides of the diapir are dipping inward at an angle

of 45◦. The lower part of the diapir has a hole, whose radius is 600 m. This is done to avoid

numerical instability due to rotation of the inner region with respect to the outer region.


Figure 4.11: Asymmetry test using the best-fit model in Fig. 4.10b. The contour lines showthe expected observable eastward deformation. The contour labels represent its magnitudenormalized by the maximum vertical deformation. The best-fit model was rotated from0◦ to 150◦ in 30◦ increments. θ is the angle of counterclockwise rotation. At each angletwo interferograms from ascending and descending orbit were simulated, and they weretransformed into east component as described in section 2.

4.4. DISCUSSION 59

Figure 4.12: (a) A flat-topped diapir with its sides dipping 45◦. The depth to the top of thediapir is 1.9 km, and the radius of the top of the diapir is 3 km. (b) Surface deformationdue to the diapir and a sill whose geometry is the same as the top of the diapir. The line ofobservation points is located on the surface of the half-space starting from directly abovethe center of the diapir. The x-axis is the distance normalized by the depth, and the y-axisis displacement normalized by the maximum vertical displacement.

The surface deformation is compared to the surface deformation due to a circular disk or a

sill. The geometry of the sill is simply the top part of the diapir. From Fig. 4.12 it is clear

that flat-topped diapirs produce almost identical surface deformation to sills, as long as the

depth is small compared to the radius. Thus, while we have greatly narrowed the class

of viable magma chamber shapes, the deformation data alone cannot uniquely resolve this

question.

To differentiate between the two models, we consider thermal interaction of the intruded

magma body with the host rock. If we consider a single intrusion event, the time required

for complete solidification of the sill can be easily calculated (Turcotte and Schubert, 2002).

For example, a newly intruded sill that has a uniform thickness of 0.5 m and an initial

temperature difference of 1000 K between the magma and the host rock would completely

solidify in about 8 hours. The same sill that intrudes into already heated host rock, having

an initial temperature difference of 500 K, would require about 16 hours to completely

solidify.

The uplift observed at Sierra Negra by InSAR data since 1992 has shown a consistent

spatial pattern for several years except when there was trapdoor faulting. In particular, the


pattern prior to and after the trapdoor faulting was very similar (Amelung et al., 2000).

Therefore, it is unlikely that numerous thin sills intruded and solidified during this time pe-

riod. If separate sills intruded and solidified, there would be no particular reason that they

would each produce the same surface deformation pattern. Instead, it is more likely that

a thicker continuously liquid magma chamber experienced pressure increases, or equiva-

lently volume increases, which in turn produced the surface deformation. If this was the

case, we can conversely calculate the minimum thickness of the magma chamber for it to

have remained liquid over the time period of observation. Taking the observation period

equal to 7 years, the thickness of the sill at the beginning of the observation period must

have been at least about 40 m.

Another way to distinguish between different magma chamber geometries, is to con-

sider the perturbation in the stress field due to magmatic intrusions. Although the surface

displacement fields due to a sill and a diapir are not distinguishable, the stress fields gener-

ated by the two different magma bodies are different. The stress state is difficult to measure

directly. However, the orientation of fissure eruptions on the flank of Sierra Negra is an im-

portant clue to the stress field, since dikes and sills are known to propagate perpendicular

to the least compressional principal stress. Interestingly Chadwick and Dieterich (1995),

who compared stress directions with the orientations of dikes on Sierra Negra, favor a flat-

topped diapir, equivalent to one of the models we find to be consistent with the InSAR

data.

4.5 Conclusions

We have narrowed the class of candidate models consistent with the InSAR observations

of Sierra Negra. We have shown the data require a flat-topped magma body at a depth

of about 2 km restricted to the inner caldera. Deformation data alone, however, cannot

uniquely determine the geometry of the sides or feeder conduit – both a diapir and a sill

with uniform internal pressure provide reasonable fits to the data. The InSAR data do

constrain the lateral geometry of the magma chamber to a high resolution.

We solved for the shape of the sill, or equivalently the flat top of the diapir. A physically

reasonable uniform pressure boundary condition was used in the inversion. The estimated

4.5. CONCLUSIONS 61

sill opening was similar to that estimated by Amelung et al. (2000). Fitting the data equally

well with reduced number of degrees of freedom suggests that the physical constraint used

in this study was reasonable. It gives physical insight to conditions in and around the

magma chamber. The estimated excess magma pressure can be used to infer the stress field

in the surrounding rock. This will be useful for estimating the crack propagation criteria

and predicting the direction of crack propagation, which may lead to an eruption.

The technique that uses data from ascending and descending orbits to resolve the shape

of magma chambers can be applied to any type of deformation source particularly those

with radial symmetry. The binary-parameter inversion scheme used to resolve the detailed

geometry of the sill can be applied to any type of uniform pressure planar deformation

source.

This new algorithm explained in this chapter is applied in Chapter 6 to model the 2005

eruption at Sierra Negra volcano. The eruption caused huge subsidence that is predom-

inantly due to a uniformly depressurized magma chamber. Our new modeling method

successfully account for the detailed geometry that involved the eruption and the excess

pressure change at the magma chamber. By the way, the huge subsidence during the erup-

tion imposed a great challenge in forming an interferogram of the event. We were not able

to form an interferogram that spans the eruption using standard InSAR software. Hence

we deveolped a new interferogram forming algorithm in the presence of large deformation,

which will be discussed in the next chapter.

Chapter 5

Interferogram Formation in thePresence of Large Deformation

In this chapter we provide a new interferogram forming algorithm when the ground de-

formation is large and complex. The results of this chapter become the starting point of

Chapter 6. Sierra Negra volcano erupted from October 22 to October 30 in 2005. During

the 9 days of eruption, the center of Sierra Negra’s caldera subsided about 5.4 meters. Three

hours prior to the onset of the eruption, an earthquake (Mw 5.4) occurred, near the caldera.

Because of the large and complex phase gradient due to the huge subsidence and the earth-

quake, it is difficult to form an interferogram inside the caldera that spans the eruption. The

deformation is so large and spatially variable that the approximations used in existing In-

SAR software (ROI, ROI PAC, DORIS, GAMMA) cannot properly coregister SAR image

pairs spanning the eruption. We have developed here a two-step algorithm that can form

intra-caldera interferograms from these data. The first step involves a “rubber-sheeting”

SAR image coregistration. In the second step we use range offset estimates to mitigate the

steep phase gradient. Using this new algorithm, we retrieve an interferogram with the best

coverage to date inside the caldera of Sierra Negra.

62


5.1 Introduction

The phase difference between two SAR images is directly proportional to range change,

and it depends on imaging geometry, topography, deformation, and atmospheric delay. A

critical step in InSAR processing is SAR image coregistration. In order to form a high-

quality interferogram, SAR image coregistration is required to sub-pixel accuracy. This is

usually implemented by cross-correlating small blocks of one SAR image (here denoted the

“slave” image) with the other SAR image (“master” image). Repeated at many locations

distributed throughout the entire image, the set of cross-correlations produces a range offset

and an azimuth offset image. Interpolating sparse image using a polynomial surface model

(Equation 5.1) yields the registration parameters at all locations.

p(x,y) = ∑i+ j≤n

ckxiy j (5.1)

Here x and y are the locations of each pixel, i and j are non-negative integer expo-

nents, and ck is a constant for each xi and y j combination. For example, ROI PAC uses

a quadratic polynomial (i.e. n = 2), while DORIS and GAMMA can accomodate up to a

4th order polynomial. Once the best-fit polynomial surface is determined, the slave image

is coregistered by resampling it to the master image coordinates. However, for the Sierra

Negra data the 4th order polynomial is not sufficiently spatially variable to properly register

the images, and thus the offset fields of SAR images that span the eruption could not fit by

a simple polynomial.

This is due to complex deformation. The eruption was preceded by an earthquake

somewhere near its caldera. The magnitude of the earthquake was Mw 5.4, large enough

to produce significant amount of surface deformation assuming that the depth of the earth-

quake is similar to that of the previous earthquakes (Amelung et al., 2000; Chadwick et al.,

2006), where about 0.8 to 1.2 m of line-of-sight displacement was measured by examining

the range offset. Moreover, the eruption occurred through a fissure along the northern rim

of caldera. Thus, there was also deformation associated with the dike intrusion. The total

deformation is the sum of all these events.

Without exception, all the interferograms that span the eruption failed to show fringes

64 CHAPTER 5. INTERFEROGRAM FOR LARGE DEFORMATION

inside the caldera, when processed with polynomial registration algorithms. A typical in-

terferogram before unwrapping is shown in Figure 5.1, produced using GAMMA software.

Hence we developed a new approach to achieve InSAR coregistration that is more robust

than the conventional methods for large and complex deformation. This new algorithm

consists of two distinct steps: 1) rubber-sheeting coregistration, and 2) range offset fringe

subtraction. The first step is to coregister a slave single-look complex (SLC) image to a

master SLC, and the second step is for mitigating the steep phase gradient due to the large

deformation.

5.2 Range and Azimuth Offset

Rubber-sheeting coregistration is a non-parameterized method as opposed to the polyno-

mial fit of standard InSAR software, where handful number of polynomial coefficients

explain entire offset field. Consider a slave image printed on a rubber sheet, and overlay it

with a master image. Then distort the slave image to match the features in the master im-

age. This is the concept of rubber sheeting. In our application, we use range and azimuth

offsets to get the distortion information.

We use square subimage blocks to cross-correlate two single-look SAR amplitude im-

ages. The size of the blocks affects the accuracy and the resolution of the cross-correlation

results. Increasing the block size increases the accuracy while reducing the resolution. We

find that 32-by-32 pixel blocks produce a good result. For this study we form the densest

possible offset vector field by implementing the cross-correlation at every single pixel in

the image, as we want to see the effect of smoothing. For practical purpose one can produce

a sparser offset field by evaluating every 8 pixels or every 16 pixels.

Figure 5.2 shows a single look range and azimuth offset image in the radar coordinate

system. We work with single look images in radar coordinates through the phase unwrap-

ping step, in order to minimize the risk of spatial aliasing due to the large deformation. The

elongated circular map compass in Figure 5.2a gives a sense of what transformation would

map the images onto the georeferenced frame.

The range and azimuth offset images are noisy, and they show a characteristic cross-hair

artifact pattern (Figure 5.2). Bright scatterers tend to dominate the offset results for every

5.2. RANGE AND AZIMUTH OFFSET 65

2 km

Figure 5.1: Co-eruptive interferogram processed by GAMMA software using Envisat data(beam IS 5, track 376, 051016 - 051120).


N

Azimuth

Range

Figure 5.2: (a) Amplitude, (b) Range offset, (c) Azimuth offset in radar coordinate system.Note that the letter N is upside down and the circular map compass is enlongated to showhow features look different compared to georeferenced frame. The blow-ups of black boxedportion in (a)-(c) are shown in (d)-(f). The small portion of the bright sinuous ridge of SierraNegra is shown in (d), and its effect on offset images are shown in (e) and (f), in which thecolor-saturated boxes are 32-pixel wide, the size of the cross-correlation block

5.3. UNBIASED MASKING OF NOISE 67

(a) (b)

Figure 5.3: Cumulative historgram of (a) range and (b) azimuth offset values inside thecaldera

block in which they are included. In Figure 5.2e and f, the size of the cross-hair patterns

is about the size of the cross-correlation block. We apply a Gaussian smoothing filter to

suppress the high-frequency noise, which will be described later in this paper.

5.3 Unbiased Masking of Noise

Before smoothing the offset images, we mask out noise-dominant areas, where offset values

are unrealistically small or large. Determining the boundary of the mask is not a trivial

task, because coherence information is not readily available due to the large and complex

deformation. We find that the cumulative distribution function (cdf) of offset values provide

a clear guide for masking. Figure 5.3 shows the cumulative histograms of (a) range and (b)

azimuth offset inside the caldera. The slope of the cdf in the noisy areas is much smaller

than that for valid areas. We use piecewise linear fitting (red lines) to the cdf; the thresholds

for masking are given by the intersections of the fitting lines.


5.4 Smoothing for Resampling

Once the noisy areas are masked, we smooth the offset fields by filtering. We tested a

moving average filter, a median filter, and a Gaussian smoothing filter. We found that

a Gaussian smoothing filter produced interferograms with the highest correlation. Our

Gaussian smoothing filter is circular and with width σ . The radius of the filter was always

set as 2.634σ , where the elevation of the kernel at its edge becomes smaller than 1/32, the

nominal theoretical accuracy of cross-correlation with 32 x 32 pixel block.

In order to determine the optimal size of the filter (or interpolation kernel), we calcu-

late the mean of the interferometric coherence after resampling. Estimating the accurate

coherence is challenging when steep and complex phase gradients are present.

When calculating coherence, we correct the deformation phase (because we consider

this as signal, not noise) with a low-pass version of our final unwrapped interferogram as

follows.

ρ =|∑n

k=1 c1,kc∗2,ke−iφ |√

∑nk=1 c1,kc∗1,k ∑n

k=1 c2,kc∗2,k

(5.2)

where ρ is the interferometric coherence, and c1 and c2 are the two SLCs with * meaning

the complex conjugate, and the subscript k denotes the kth pixel of n neighboring pixels

averaged, and φ is the smoothed unwrapped interferogram phase. We smooth the inter-

ferogram by applying a circular Gaussian filter, whose σ and maximum radius are both 4

pixels to roughly match the size of the subimage (8 x 8 pixels) over which we calculate the

coherence. This method of calculating the coherence is similar to Zebker and Chen (2005)

in that it corrects the phase by subtracting the low frequency component. Note, however,

that this method does not overestimate the true coherence.

Figure 5.4 shows the mean coherence plots as a function of the smoothing parameter σ .

We calculate the mean coherence for a poorly coherent area (caldera) and a highly coherent

area (north flank). We find that the coregistration based on rubber-sheeting interpolation

is robust for a wide range of σ . As σ varies from 0 to 10 pixels, the mean coherence

increases rapidly from 0.329 to 0.454 for caldera and from 0.555 to 0.809 for north flank.

After hitting the maxima (σ = 10 pixels for caldera and σ = 7 pixels for north flank), the

5.5. RESAMPLING 69

σ σ

(a) (b)

Figure 5.4: Mean coherence at caldera and north flank as a function of Gaussian smoothingparameter, σ . Note that the range of coherence variation is very small. The maximumoccurs at (a) σ = 10 pixels and (b) σ = 7 pixels.

mean coherence does not vary significantly. Note that the Gaussian smoothing kernel with

σ = 10pixels and radius = 26.34 pixels is an efficient filter size that reduces the cross-hair

artifacts of 32 x 32 pixels. Using a circular median filter, the maximum mean coherence

occurs when the radius is 16 pixels, which is again of comparable size to a 32 x 32 block.

5.5 Resampling

Most SAR data are near-critically sampled in both the range and azimuth directions. In

other words, the sampling frequencies are slightly greater than twice the signal bandwidth

in both directions. Thus, sinc-type interpolation is useful for SAR image registration. We

use a raised cosine (RC) interpolation kernel suggested by Cho et al. (2005), which is a

sinc-type interpolation but has smaller phase error due to resampling than the plain sinc

interpolation. They combined a sinc with a raised cosine function used in digital commu-

nications, the 2-D version impulse response i(x,y) of which is written as

i(x,y) = sinc(x,y)cos(απx)cos(βπy)

(1−4α2x2)(1−4β 2y2)rect(

xL,

yL) (5.3)


µ

µ

µ

µ

(a) (b)

(c) (d)

Figure 5.5: Coherence histogram for (a),(c) caldera before and after coregistration respec-tively, and (b),(d) north flank before and after coregistration respectively. (b) and (d) arefor σ = 10 pixels.

5.6. RANGE OFFSET AS A PROXY FOR INTERFEROGRAM PHASE 71

where sinc(x,y) means sinc(x)sinc(y), L is the kernel size, and α and β are roll-off factors

with a value between 0 and 1. When α and β = 0, Equation (5.3) becomes a 2-D sinc

interpolator. As α and β grow larger, the interferometric phase error becomes larger, but

the sidelobes of i(x,y) are suppressed, relaxing the effect of the finite kernel size. The

optimum values of α and β that satisfy the Nyquist criterion can be calculated as follows

(Cho et al., 2005).

α = 1− Br

fsr= 1− 1

χr(5.4)

β = 1− Ba

fsa= 1− 1

χa(5.5)

where χr and χa are oversampling factors (i.e. sampling frequency divided by bandwidth)

in range and azimuth respectively. In this study χr = 1.2005 and χa = 1.1588 were used.

Prior to the resampling, we estimate and subtract the carrier phase both in range and az-

imuth direction. This is to ensure that dominant energy is not lost during resampling, which

is a low-pass filter. After resampling, we add the estimated carrier phase back to the data.

This step finishes the rubber-sheeting coregistration. The result of the coregistration is

shown in Figure 5.4. Coherence is improved within the coregistered image both in poorly

cohrerent area (caldera) and highly coherent areas (north flank).

5.6 Range Offset as a Proxy for Interferogram Phase

All phase unwrapping algorithms are based on the assumption that the input interferogram

phase is not aliased most of the time. This assumption does not hold in regions of large

deformation. Aliasing can be reduced by subtracting an estimate of the interferogram phase

before unwrapping. When there is no independent information on the rough shape of the

deformation, the range offset image can be used to construct the estimated interferogram.

A range offset field includes the same information as an interferogram (Amelung et al.,

2000; Jonsson et al., 2005), except that it has a different noise character and magnitude.

Bamler (2000) derived the standard deviation of the estimated amplitude offset as follows


for homogeneous (i.e. featureless) image patches.

σampoffset =

√

32N

√

1−ρ2

πρχ3/2 (5.6)

where ρ is the coherence of the interferometric data pair, and N is the number of samples

in the cross-correlation block, and χ is the oversampling factor of the data. Using the

oversampling factors used for resampling and the pixel spacings (range = 7.8 m, azimuth =

3.23 m) in our study, we plot the range and azimuth offset error in Figure 5.6.

Note that the plot shows the lower bound, as in reality features such as topography in

images will cause more noise (Figure 5.2e,f). Nevertheless, this analysis shows that the

range offset is useful for mitigating the steep phase gradient. Within the coherence range

of our study (around 0.4) the level of noise from the range offset is small enough to be used

as a proxy for interferogram phase.

We smooth the range offset image with a Gaussian smoothing kernel of σ = 50 pixels

and r =132 pixels. The smoothed range offset is subtracted from the interferogram before

phase unwrapping and later added back, after unwrapping. When smoothing the range

offset, one should keep two things in mind. First, all the natural features including de-

formation in a single-look interferogram are elongated in the azimuth direction. Thus, the

shape of the interpolation kernel should also be elongated by the same factor. Alternatively,

one can take looks in azimuth to make square-pixel image, and apply an equidimensional

smoothing kernel, and then stretch the result back to the single-look coordinate. Usually,

the latter approach will be more efficient.

Second, deformation signals in interferograms are projections of 3-D surface displace-

ment vector field onto a slant line of sight. Therefore, the fringe patterns are in general

slightly skewed to one side, depending on the look angle of the radar. Convolving the

skewed signal with a large symmetric smoothing kernel can modify the overall skewness,

and this can result in a poor fit of the range offset to the interferogram. The distortion in

the skewness can also be caused by errors in the range offset that were not properly masked

out in the first step. In this study we were able to identify a small faint concentric peak

signal at the center of caldera in the interferogram before subtracting the range offset, and

we simply shift (about 40 pixels toward an increasing range direction) the smoothed range


offset to match the peak of the range offset with the observed peak.

Figure 5.6 shows the effect of range offset subtraction. The single-look interferogram

of deformation (Figure 5.6a) is the result of rubber-sheeting coregistration. Despite the

improved coherence, the fringes are still not visible due to the large phase gradient. In the

blow-up of the white box (Figure 5.6b) the maximum subsidence is shown in the upper right

corner. The fringe rate becomes higher and higher, getting close to the critical sampling

rate in azimuth direction. This problem is more severe in range direction and the fringes

become quickly aliased toward the left. When the range offset was subtracted (Figure 5.6c)

the observed fringe rate was much lower.

The overall improvement during the process of the new algorithm is illustrated in Fig-

ure 5.6. The theoretical limit on the maximum detectable displacement gradient can be

expressed as following (Massonnet and Feigl, 1998).

dx =λ

2D(5.7)

where dx is the maximum detectable displacement gradient, and λ is the wavelength of

radar pulse, and D is the pixel spacing. For Envisat ASAR (C-band) strip mode image

with nominal pixel spacing of 20 m, dx is about 1.4× 10−3. For real data that suffer from

decorrelation (Zebker and Villasenor, 1992) dx becomes smaller as a funciton of interfero-

metric coherence. Based on simulated and real data, Baran et al. (2005) derived a bound

on whether or not an interferogram can be formed (see Figure 5.6). The 2005 eruption

at Sierra Negra falls into the filled dot indicated in the figure. The rubber-sheeting coreg-

istration increases the coherence (A) and the range offst subtraction decreases the phase

gradient (B), shifting the 2005 eruption at Sierra Negra into the hollow dot, where we can

apply phase unwrapping and successfully form an interferogram.

SNAPHU (Chen and Zebker, 2001) was used for phase unwrapping. Figure 5.6 shows

the final interferogram with fairly good coverage inside the caldera. The eastern part of

the caldera was completely decorrelated due to the lava flow during the eruption, and the

western part of the caldera was decorrelated possibly due to the earthquake occurred 3

hours prior to the eruption.


a

b c

Azimuth

Range

Figure 5.6: (a) Deformation interferogram after the rubber-sheeting SAR coregistration.(b) Blow-up of the white box in (a). Fringe rate becomes higher close to critical samplingrate in azimuth direction. In range direction the fringe rate quickly becomes aliased. Aftersubtracting the range offset from the interferogram, the fringe rate becomes much lower (c), and phase unwrapping becomes possible for much larger area.


Sierra Negra

2005 Eruption

A

B

Figure 5.7: Maximum detectable displacement gradient as a function of coherence (Baranet al., 2005). Thu rubber-sheeting coregistration increases the coherence (A) and the rangeoffst subtraction decreases the phase gradient (B)

(a) (b)

Figure 5.8: Uncertainty of amplitude offset for (a) range and (b) azimuth


2 km

Figure 5.9: Final result interferogram from the same SAR images used to produce Figure5.1. One color fringe represents 15 cm of range change.

5.7. CONCLUSION 77

5.7 Conclusion

Using a rubber-sheeting coregistration scheme, we were able to improve the interferomet-

ric coherence inside the caldera for InSAR data that span the 2005 eruption at Sierra Negra

Volcano, Galapagos. Based on the smoothed range and azimuth offset images, resam-

pling was done with a raised cosine interpolation kernel. We subtracted even smoother and

slightly shifted version of the range offset from the interferogram of the coregistered pair.

These steps enabled us to form a useful interferogram inside the caldera of the volcano (see

Figure 5.6. The fringe on the south flank is from separate regular process. The interfer-

ogram forming algorithm developed in this chapter is used in the nex chapter to produce

more interferograms and better constrain the eruption event.

Chapter 6

2005 Eruption at Sierra Negra VolcanoUnveiled by InSAR Observations

All the previous chapters support this final chapter. Chapter 3 produces a DEM that is used

to remove topography from the interferograms spanning the 2005 eruption. We produce the

interferograms using a new algorithm introduced in Chapter 5. Then we model the InSAR

data using our new algorithm described in Chapter 4. Thus, this chapter applies all the

novel techniques to explain the eruption event at Sierra Negra.

Sierra Negra volcano erupted from October 22 to October 30 in 2005. During the 9

days of the eruption, the center of Sierra Negra’s caldera subsided about 5.4 meters. Three

hours prior to the onset of the eruption, an earthquake (Mw 5.4) occurred, somewhere near

the caldera. We analyze here a pair of interferometric synthetic aperture radar (InSAR)

images from ascending and descending orbits of the Envisat satelite that temporally span

the eruption. The interferograms plus the azimuth offset image from the ascending pair are

used to model the euption. The data display several different events overlapped in time and

space; pre-eruptive deformation, trapdoor faulting, dike intrusion and eruption, co-eruptive

subsidence, and post-eruptive uplift. The pre- and post-eruptive uplift is modeled using

the InSAR data plus GPS observations. The faulting, dike intrusion, and subsidence are

modeled using a combination of displacement and uniform pressure boundary conditions.

We estimate that the pre- and post-eruptive pressurization rate of a sill-like magma body

are 29 MPa/year (23 days average) and 79 MPa/year (21 days average) respectively.

78


In the trapdoor faulting model, the estimated maximum slip ( 1.8 m) is at the bottom

of the western end of the fault system, and about 1.5 m toward the surface, which matches

the field observation very well. The equivalent moment magnitude of the total slip was

estimated to be Mw 5.7 when the shear modulus is 30 GPa. For shear modulus of 10 GPa,

it becomes Mw 5.4, which is the moment magnitude of the earthquake that occurred 3 hours

prior to the onset of the eruption. The dike model showed average opening of 1.7 m and

“reverse faulting” average dip slip of 1.6 m. The large dip slip is due to the interaction with

the sill and the free surface. The the sill model is the sum of two components: interaction

with the trapdoor faulting event and uniformly depressurized closing during the eruption.

The interaction component showed a wedge-like opening distribution close to the fault

system, and the uniformly depressurized sill accounted for the co-eruptive subsidence. We

find significant interaction between the trapdoor faulting and the sill; When the faulting

occurred, the sill underwent a sudden stress perturbation, which caused an uneven opening

and closing of the sill accompanied by horizontal magma transport. This effect combined

with the co-eruptive uniform depressurization provides an opening-closing map with about

-8.8 m of maximum sill closing.

The repeating cycle of trapdoor faulting and eruption can produce an accumulated

wedge-like structure at depth as well as on the surface. We believe that the surface ex-

pression of this structure is shown as the characteristic C-shaped sinuous ridge inside the

caldera of Sierra Negra. The estimated volume decrease of the sill was 0.124 km3, and the

estimated extruded volume (dense rock equivalent) was about 0.120 km3. This similarity

suggests that there may not have been substantial amount of volatiles in the magma before

the eruption.

6.1 Introduction

Sierra Negra is an active basaltic shield volcano located at the southern end of Isabela

Island in the Galapagos archipelago (Figure 4.1). The volcano has experienced 12 historical

eruptions; the most recent one occurred in 2005. The penultimate eruption prior to 2005

occurred in 1979. That eruption produced lava through fissures on the northern flank close

to the caldera, with an erupted lava volume of almost 0.9 km3 (Reynolds et al., 1995).

80 CHAPTER 6. 2005 ERUPTION AT SIERRA NEGRA

Since InSAR observations began in 1992, the center of the volcano had uplifted nearly 5

m preceding the 2005 eruption. The source of uplift has been successfully modeled as a

sill located at a depth of about 2 km (Amelung et al., 2000; Yun et al., 2006). The uplift

was accompanied by trapdoor faulting events, two of which were observed in InSAR data

and modeled by Amelung et al. (2000) and Chadwick et al. (2006). The trapdoor faulting

is believed to be near vertical dip-slip induced by pressurization of the underlying magma

body. The excess pressure at the magma body causes the overlying crust to hinge upward

like a trapdoor.

Figure 6.1 shows the lava coverage (displayed through surface temperature) of the erup-

tion as measured by the ASTER (Advanced Spaceborne Thermal Emission and Reflection

Radiometer) sensor flying onboard NASA’s Terra satellite on November 2, 2005, three days

after the eruption ended. The image, draped on an SRTM DEM, clearly shows the extent

of the erupted lava flow, which filled the lowest, eastern part of the caldera. The extent of

initial fissure is indicated in red subparallel to the northern rim of caldera (Geist et al., in

press).

Sierra Negra’s wide (7×10.5km) and shallow (100 m in depth) caldera contains a char-

acteristic C-shaped sinuous ridge, composed of a complex set of normally faulted blocks

with steep (60◦ - 90◦) outward dipping fault scarps (Reynolds et al., 1995). This sinuous

ridge is believed to have formed by repeated trapdoor faulting events. Recent trapdoor

faulting occurred along the southern part of the scarps sometime in 1997-1998 (Amelung

et al., 2000) and on 16 April 2005 (Chadwick et al., 2006). Chadwick and Geist found a

fresh fault scarp (delineated with red curves if Figure 6.1) along the southwestern part of

the sinuous ridge during their field trip in June 2006, and the measured slip at Spot A was

from 1.4 to 1.5 m. Geist and others had another field trip in January 2007 and measured a

slip of 1.5 m at Spot B (Geist, pers. comm.).

The InSAR data used in this study span the 2005 eruption, including deformation due

to pre-eruptive uplift, trapdoor faulting, dike intrusion that fed the eruption, co-eruptive

subsidence, and post-eruptive uplift. These series of events are illustrated in Figure 6.2,

where schematic vertical sections are shown in chronological order. Later in this paper, the

deformation due to pre-eruptive uplift (a) and the post-eruptive uplift (d) is eliminated from

InSAR data using GPS data.


A

B

C

C’

GV03

GV01

GV05

GV04

GV02

GV06

2 km

GV03

GV01

GV05

GV04

GV02

GV06

2 km

Figure 6.1: Surface temperature image taken at night on November 2, 2005 by ASTER sen-sor onboard NASA’s Terra satellite. The image is georeferenced and draped on the shadedrelief image from the SRTM DEM (Farr and Kobrick, 2000). Six continuous GPS stationswere deployed inside the caldera at the time of the eruption. GV04, GV05, and GV06 wereused for InSAR data adjustment. The C-shaped sinuous ridge inside the western side of thecaldera is clearly shown. The outer sides of the ridge are composed of fault scarps. Thesouthern part of the scarps, which are less clear, extend just below the GV06 station.


t

e )

n

(a) (b) (c) (d)

Figure 6.2: Schematic vertical cross section along the line C-C’ in Figure 6.1. (a) pre-eruptive inflation, (b) trapdoor faulting, (c) dike intrusion and the fissure eruption, (d) post-eruptive inflation.

6.2 Data

For this study we use both InSAR and GPS data for modeling. The InSAR data are derived

from the European satellite Envisat; details on the scenes are summarized in Table 6.1,

where ∆T is temporal baseline or time span, and B⊥ is perpendicular baseline. The first

two interferograms precede the eruption and yield a model of the detailed geometry of the

pressurized magma chamber. The last two interferograms span the eruption and are used

to model the faulting and co-eruptive events.

Due to the complex and large deformation of the faulting and the eruption, it was

difficult to form the co-eruptive interferograms. Using existing standard software (ROI,

ROI PAC, DORIS, and GAMMA) we were not able to obtain interferometric fringes in

the caldera, where displacement gradients were most severe. Instead, we developed a new

algorithm, as described in Chapter 5, to form the interferograms. The new algorithm ap-

plies rubber-sheeting image coregistration to improve interferometric coherence and range

offset subtraction to mitigate the steep phase gradient.

Six continuous GPS stations operated at the time of the earthquake and eruption (see

Figure 6.1). Unfortunately, however, several hours prior to the earthquake the GPS network

failed due to the loss of the main computer collecting the GPS data and was not restored

until 24 October 2005 (Geist et al., in press). Thus, it is not possible to retrieve the temporal

evolution of the deformation and to separate the effects of faulting, dike intrusion, and co-

eruptive subsidence. Although the GPS data are not available during the faulting and the

initial stage of the eruption, they do yield the pre- and post-eruptive uplift, which we can

eliminate from the interferogram observations.

6.3. MAGMA CHAMBER GEOMETRY 83

Table 6.1: Interferograms Used in This Study

Beam Orbit direction Track No. Scene 1 Scene 2 ∆T B⊥

IS2 Ascending 61 2005/05/07 2005/07/16 70 days 118 m

IS2 Descending 140 2005/05/12 2005/07/21 70 days -76 m

IS5 Ascending 376 2005/10/16 2005/11/20 35 days 100 m

IS2 Descending 140 2005/09/29 2005/11/03 35 days 433 m

6.3 Magma Chamber Geometry

One remarkable feature of the uplift at Sierra Negra is that the extent and pattern of de-

formation has been surprisingly consistent for a long period of time. Figure 6.3 demon-

strates this consistency as shown in several interferograms acquired between 1992 and

2006. Given that the uplift at Sierra Negra has been successfully explained with a sill

model, it is likely that the sill at Sierra Negra is reasonably thick and thermally stable. The

pattern of uplift remained mainly unchanged even after the 2005 eruption.

Based on this observation, we assume that the magma chamber geometry did not change

prior to or during the eruption. We further assume that the host rock can be modeled as a

homogeneous linear elastic half space, and walls of the magma chamber subject to uni-

form magma pressure. We use a pair of ascending and descending interferograms acquired

before the eruption (Figure 6.3e,f) to solve for the magma chamber geometry. These two

interferograms were formed using ROI PAC software (Rosen et al., 2004). Topographic

phase was removed using a digital elevation model (DEM) created by merging Topographic

Synthetic Aperture Radar (TOPSAR, Zebker and Goldstein (1986)) DEM and SRTM DEM

(Yun et al., 2005), as described in Chapter 3.

6.3.1 Depth

In order to describe the geometry of the magma chamber one must specify location, depth,

shape, size, and excess pressure. Since there are strong trade-offs between depth, size, and

the excess pressure, we first estimate the a posteriori distributions of the highly correlated


Courtesy of Poland

A 2004/05 - 2004/09

(c)

Courtesy of Poland

A 2005/12 - 2006/01

(d)

Jonsson, 2002

D 1992/07 - 1997/10

(a)

,

A 1998/09 - 1999/03

(b)

Jonsson, 2002

4 km

A 2005/05/07 - 2005/07/16 D 2005/05/12 - 2005/07/21

Look

direction

Lookdirection5 cm / fringe

(e) (f )

,

Figure 6.3: Interferograms from 1992 to 2006 (a-d). The letter A and D represent ascendingand descending orbit. Note that a-c are before the eruption and d is after the eruption. Formodeling the pressure source geometry a pair of ascending and descending interferograms(e,f) are used. One color cycle represents 5 cm of LOS displacement in a, b, e, and f, and2.83 cm of LOS displacement in c and d.


parameters with a simple model. For this purpose, we use a pressurized horizontal circu-

lar crack (Fialko et al., 2001a) to produce simulated surface displacement fields. In the

inversion we form the following quadratic form:

Q = (d−dLOS)TΣ

−1(d−dLOS) (6.1)

where d is the computed and vectorized LOS displacement, and dLOS is the vectorized

observed LOS displacement, and Σ is the data variance-covariance matrix, and the scalar

Q is the L-2 norm of the misfit weighted by the inverse of the variance-covariance matrix.

The variance-covariance matrix is defined using an exponential 1-D covariance function

with a variance of 40 mm2 and falls to 1/e of the variance at a distance of 2 km (Jonsson,

2002; Wright et al., 2004).

Rather than minimizing Q to get the best-fit model, we adopt Markov Chain Monte

Carlo sampling, where current sampling (and associated norm Qi) depends only on the

previous sample (and associated norm Qi−1), with the Metropolis rule (Metropolis et al.,

1953; Tarantola, 2004; Hooper, 2006). Figure 6.4 shows the reduced data set and its

variance-covariance matrix. Various InSAR data reduction schemes have been proposed

for geophysical inversion (Jonsson, 2002; Simons et al., 2002; Lohman and Simons, 2005),

all of which are adaptive to data. Considering the geometry of the horizontal circular crack,

the reduced data shown in Figure 6.4 mimic the data resolution-based approach of Lohman

and Simons (2005), which is an optimal scheme for linear least squares problems. When

multiple interferograms are used for inversion, fixing the data point locations for all inter-

ferograms makes the inversion run faster.

The resulting posterior distributions for the circular crack parameters are shown in Fig-

ure 6.5. Although depth, radius, excess pressure are highly correlated to one another, the

depth is relatively well defined at 1.86 km ±0.13 km, which is the maximum likelihood

solution. This depth will be fixed in the following step, where we estimate the shape of the

sill.


Variance-covariance Matrix

100 200 300 400 500 600

100

200

300

400

500

600

40

0

mm2

Data Points

Figure 6.4: (a) Reduced data points and (b) Data variance-covariance matrix. The firstblock in the matrix is for the ascending interferogram and the second block is for the de-scending interferogram.

6.3.2 Sill Geometry

From the fringe pattern of the interferograms at Sierra Negra, we know that the sill is not

circularly symmetric. For this reason we solve for the detailed geometry of the sill using

the binary parameter inversion introduced by Yun et al. (2006). The binary parameter in-

version assumes a uniform pressure inside the crack and estimates the best-fit combination

of the binary parameters that represent “on/off” crack elements. “on” or “1” means that the

element is open (i.e. part of the crack) and is subject to the uniform pressure, and “off” or

“0” means that the element remains closed and not a part of the crack. The best-fit com-

bination of the binary parameters is searched with simulated annealing methods (Cervelli

et al., 2001) that applies Basu and Frazer type rapid determination of the critical temper-

ature (Basu and Frazer, 1990). The elements that are open define the shape of the sill.

Figure 6.6 shows the best-fit model. Note that the boundary of the sill is surrounded by

the C-shaped sinuous ridge on the west and south. Fit to the InSAR data for this model is

shown in Figure 6.7. Figure 6.7a,b are the same as Figure 6.3e,f but in unwrapped form.

The best-fit sill model explains about 99% of data variance. We fix this sill geometry for

the subsequent modeling procedure.


North Depth Radius Excess pressureEast

Ra

diu

s (k

m)

De

pth

(k

m)

No

rth

(k

m)

Ea

st (

km

)

km

−0.15 −0.1 −0.05 0

−0.3 −0.25 −0.2

1.7 1.8 1.9 2

3.1 3.2 3.3 3.4

6.3 6.4 6.5

log10

(Pa)

1.7

1.8

1.9

2

3.1

3.2

3.3

3.4

−0.3

−0.25

−0.2

−0.15

−0.1

−0.05

0

km

km

km

Figure 6.5: Joint a posteriori probability density for the penny-shaped crack parameters.The red lines are 95% confidence intervals. Note that depth, radius, and excess pressureare highly correlated to each other. The depth of 1.86 km is the maximum likelihoodsolution. The color plots represent the density clouds of accepted samples weighted by thelikelihood


4 km0.15 0.6

Opening (m)

Figure 6.6: Best-fit sill geometry. Note that the sill is bounded by the C-shaped sinuousridge.

6.4 Data for Eruption Modeling

As mentioned previously, forming an interferogram that spans the eruption was challeng-

ing because of the complex and large deformation caused by the faulting and the eruption.

Using our new algorithm explained in Chapter 5, we form two interferograms from ascend-

ing and descending orbits. In Chapter 5, phase unwrapping was done using an algorithm

called SNAPHU, a maximum a posteriori probability estimation approach (Chen and Ze-

bker, 2001).

SNAPHU can easily be customized with a number of controlling parameters. Its mask-

ing function automatically creates a number of separate but internally consistent coherent

patches. This is useful for noisy data, where coherent patches are found as isolated islands

in a ”sea of decorrelation”. However, one should be careful with the masking function,

as the shape and distribution of the resulting mask can look quite different depending on

one’s choice of masking parameters. Because of this feature, SNAPHU’s result can be

subjective to user’s preference, especially when the data is noisy and the shape, extent, and

connectivity of coherent patches are critical.

This is the case in the 2005 eruption at Sierra Negra: the data is noisy inside the caldera,

6.4. DATA FOR ERUPTION MODELING 89

�0.2 0.2

LOS (m)

�10 0 100

0.1

0.2

0.3

0.4

LO

S d

isp

lac

em

en

t (m

)

Ascending Profiles

�10 0 100

0.1

0.2

0.3

0.4

Distance (km)

LO

S d

isp

lac

em

en

t (m

)

�10 0 100

0.1

0.2

0.3

0.4

Descending Profiles

�10 0 100

0.1

0.2

0.3

0.4

Distance (km)

g h

S - N

W - E

S - N

W - E

4 km

2005/05/07 - 2005/07/16 2005/05/12 - 2005/07/21

Look

direction

Lookdirection

a b

c d

e f

�0.2 0.2

LOS (m)

N

S

W E

Figure 6.7: (a) Ascending interferogram, (b) Descending interferogram, (c) Ascendingresidual, (d) Descending residual. (e)-(h) are the profiles of data and predictions (which arenot shown) along the N-S and E-W lines in (a).


4 km−4.9 3.9m

a b c

Figure 6.8: (a) Ascending interferogram from beam IS5 (incidense angle = 37.6◦), (b)Descending interferogram from beam IS2 (incidence angle = 23.0◦), (c) Azimuth offsetfrom the ascending SAR image pair. One color cycle in both interferograms represents 20cm of LOS displacement. The interferograms are composed of several isolated patches, ineach of which phase is internally consistent and continuous.

and we want to be careful with any discontinuity in the region, as we do not know the ex-

tent of surface-breaking faulting before we process the data. For this reason, we use a

residue-cut phase-unwrapping algorithm (Goldstein et al., 1988). The residue-cut algo-

rithm requires multiple runs for noisy data, as we have to manually provide seed location

for each coherent patch. Figure 6.8 shows the resulting interferograms. The ascending

interferogram (a) is composed of 22 separate patches and the descending interferogram (b)

has 7 separate patches.

The fringes on the southern flank in the interferograms are from independent data pro-

cessing. Sierra Negra’s southern flank is heavily vegetated and usually heavily decorrelated

as well. However, the huge subsidence caused by the 2005 eruption increased the signal-

to-noise ratio on the southern flank and provided a hint of fringes in the interfergram before

unwrapping (see Figure 5.1). We use ROI PAC software, applying an adaptive power spec-

trum filter (Goldstein and Werner, 1998) twice to enhance the narrow-band signal (the hint

of fringes) against the broad-band noise (vegetation-induced decorrelation). The azimuth

offset (Figure 6.8) is from an ascending SAR image pair that was used to produce Figure

6.8a.


6.4.1 Data Weights

We calculate uncertainties of the interferograms and azimuth offset in order to determine

the weights on each data type for inversion. Assuming that we can use spatial averages as

a proxy for ensemble averages, we take a square sub-image of each data and calculate the

standard deviation of the pixel values in the sub-images. Figure 6.9 shows the procedure

to calculate the uncertainties. There (a) and (d) are the wrapped interferogram and azimuth

offset image in radar coordinates. The data uncertainty should be calculated in radar coor-

dinates because geocoding involves interpolation which is a low-pass filter that suppresses

the data noise. For the same reason, the uncertainty should be calculated before applying

any filter such as power-spectrum filter (Goldstein et al., 1988), a commonly used filter be-

fore phase unwrapping in InSAR data processing, which enhances the dominant frequency

content, thereby biasing the uncertainty.

For noisy data, for example inside the caldera of Sierra Negra, phase unwrapping is

difficult without applying such a filter. Thus, we calculate the uncertainty from wrapped

interferograms. The challenge now is to avoid crossing 2π jumps, as the phase of wrapped

interferograms is the 2π modulo of the continuous phase. If the jump is included in the

sub-image over which we calculate the standard deviation, the standard deviation will be

overestimated. Unfortunately, inside the caldera the wrapped interferogram is saturated

with fringes. One can subtract a smoothed version of range offset to mitigate the phase

gradient in the wrapped interferogram as in Chapter 5. However, the problem is that the

smoothed range offset subtraction produces an unwrappable interferogram, which still con-

tains many fringes.

To better flatten the wrapped interferogram, we use a two-step kinematic inversion to

produce a model that fits the data extremely well, although the model does not have any

physical meaning. Using the interferogram produced with a method described in Chapter 5,

we first estimate the best-fit distributed sill model at a depth of 2 km and a uniform grid size

of 500 m. The associated model prediction is then transformed onto radar coordinate and

subtracted from the interferogram to form residual. Then we fit the residual with another

sill model at a depth of 300 m and with a finer grid size of 300 m. The residuals from the

second step are shown in (c) and (d) of Figure 6.9. Note that one color cycle represents 2.83


cm in all the interferograms in Figure 6.9. With this flattened interferogram (Figure 6.9c),

it is now possible to avoid the 2π jumps and estimate the uncertainty of the interferogram.

We applied the same flattening to the azimuth offset image in order to estimate the relative

weight correctly.

We take sub-images of 1) inside the caldera, 2) less noisy northern flank, and 3) com-

pletely noisy area from Figure 6.9c. The locations of the sub-images are carefully chosen

so they do not contain the 2π jump fringe. The size of the sub-image is increased from 2

to 100 pixels, and for each step we remove any remaining phase gradient and calculate the

standard deviation of the pixel values of the sub-image. The results are shown in Figure

6.9e, where standard deviations of Box 1 (inside the caldera) and Box 2 (northern flank)

converge to 0.65 cm and 0.15 cm respectively. The increase in Box 1 plot around 70 to 80

pixels is due to the inclusion of a 2π jump that is shown in Figure 6.9c. The black boxes

shown in the image are 100×100 pixels big, and the boxes was enlarged starting from their

upper left corner.

Box 3 is chosen in a completely decorrelated (or random) area, where we know what

value the standard deviation should converge to. For a completely decorrelated area, the

phase behaves like a random variable with a uniform distribution. The variance of the

random variable X that is uniformly distributed in the interval [a,b] can be calculated as

follows.

Since the mean of X is (a+b)/2,

VAR[X ] =1

b−a

∫ b

a

(

x− a+b2

)2

dx

Let y = x− (a+b)/2,

VAR[X ] =1

b−a

∫ (b−a)/2

−(b−a)/2y2dy =

(b−a)2

12

The phase in Box 3 is uniformly distributed in the interval [-φ , φ ]. Thus, its standard

deviation of the phase σφ becomes

σφ =π − (−π)√

12(6.2)


Converting this to range change, we get

σr =2π√12

λ4π

=λ

4√

3(6.3)

where σr is the uncertainty in range change (or line-of-sight displacement) λ is the wave-

length of the radar signal. Substituting Envisat’s wavelength, σr becomes about 0.8 cm.

Therefore, Figure 6.9e suggests that sub-image of at least 20×20 to 30×30 pixels should

be used to estimate the uncertainty of interferograms.

The same locations are chosen in azimuth offset and their standard deviations are cal-

culated (Figure 6.9). By comparing Figure 6.9e and Figure 6.9f, we get a constant relative

weight of 35 (35 for interferograms and 1 for azimuth offset), which is used for the follow-

ing eruption modeling. We find that both inside the caldera and the northern flank give the

similar ratio.

6.4.2 Removing Pre- and Post-eruptive Deformation

The earthquake and the eruption occurred within an 8 day time window, whereas the short-

est temporal baseline of the interferogram from Envisat ASAR strip mode is 35 days. Since

the time span of the data is much larger than the event period, we need to remove the defor-

mation before and after the eruption from the InSAR data, so they contain only deformation

due to the events during these 8 days. To do this, we utilize continuous GPS data that are

available until several hours before the earthquake and from a few days after the onset of

the eruption.

Figure 6.10 shows a schematic plot of the uplift and subsidence of the center of the

caldera. Except for the eight days of the eruption, the caldera floor uplifted monotonically

during this period. Note that both ascending and descending interferograms span not only

the eruption but also include deformation occurring before and after the eruption.

In a linear elastic problem, the magnitude of surface displacement due to a pressurized

crack is proportional to the excess pressure in the crack. Using this property, we construct a

linear set of equations to solve for the excess pressure increase before and after the eruption.

First, we define a composite baseline vector d in Equation 6.4, which consists of east,

north, and up components of the baseline GV04-GV05, GV04-GV06, and GV05-GV06


200 400 600 800 1000

100

200

300

400

500

600

700

800

900

1000

1100

1

2

200 400 600 800 1000

100

200

300

400

500

600

700

800

900

1000

1100

200 400 600 800 1000

100

200

300

400

500

600

700

800

900

1000

1100

1

2

3

200 400 600 800 1000

100

200

300

400

500

600

700

800

900

1000

1100

0 20 40 60 80 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Square block size (pixels)

Sta

nd

ard

de

via

tio

n (

cm

)

Box 1

Box 2

Box 3

0 20 40 60 80 1000

5

10

15

20

25

Square block size (pixels)

Sta

nd

ard

de

via

tio

n (

cm

)

Box 1

Box 2

-3

3

(m)

-3

3

(m)

(a) (b)

(c) (d)

(e) (f )

Azimuth

Range

Figure 6.9: (a) wrapped interferogram in radar coordinate, (b) azimuth offset in radar coor-dinate, (c) residual between ”extreme” kinematic model prediction and the interferogram,(d) azimuth offset residual for the same model, (e) Uncertainty plot for the interferogram,(f) Uncertainty plot for the azimuth offset


17 days 6 days 8 days 4 17 days

272

(9/29)

289

(10/16)

295

(10/22)

303

(10/30)

307

(11/3)

324

(11/20)

Pre-eruptive Uplift Eruption Post-eruptive Uplift

Ascending Interferogram

Descending Interferogram

Figure 6.10: Schematic plot of uplift and subsidence of the center of Sierra Negra’s caldera.

(see Figure 6.1). Then we construct the G matrix that maps the excess pressure to the

displacements at GV04, GV05, and GV06 as in the following equation,

d =

x4 −x5

x4 −x6

x5 −x6

,

u4

u5

u6

= G∆p (6.4)

where xi and ui are the position and the displacement vectors respectively at station i. The

G matrix is constructed by applying a unit excess pressure and by solving for the opening

distribution in the best-fit crack model (i.e. Figure 6.6) using boundary element method,

and by calculating the surface displacement with Green’s function for elastic half-space due

to rectangular dislocation (Okada, 1992) of all the elements shown in Figure 6.6. Using the

following differencing operator, D, we relate the composite baseline vector to the surface

displacement. Finally, if we know the composite baseline vector at time t1 and t2, we can

estimate the excess pressure change during the time t2 − t1 (Equation 6.7).


D =

I −I OI O −IO I −I

(6.5)

dt2 −dt1 = DG(∆pt2−t1) (6.6)

where I and O are 3-by-3 identity matrix and zero matrix respectively, and ∆pt2−t1 is the

excess pressure change during the time t2 − t1. In this study, we want to estimate and

subtract the effect of pre- and post-eruptive excess pressure change from both ascending

and descending interferograms. In particular, we use the following formulae,

d295 −d289 +d324 −d303 = DG(∆p295−289 +∆p324−303) (6.7)

d295 −d272 +d307 −d303 = DG(∆p295−272 +∆p307−303) (6.8)

where the first expression is for ascending interferogram and the second for descending

interferogram, and subscripts represent days of the year indicated in the GPS time series

(Figure 6.11). The left-hand sides are readily available from GPS data. Once we esti-

mate ∆p in the right-hand side, we impose the excess pressure change on the sill geometry

estimated in Section 6.3.2 and calculate the predicted surface deformation in an elastic

half-space. Then, the displacement field is projected onto the LOS directions of ascending

and descending beams and along track direction of the ascending orbit. Finally, these pro-

jections are subtracted from ascending and descending interferograms and from azimuth

offset. After this adjustment the interferograms and azimuth data represent the narrower

time span of from 10/22 to 10/30.

The excess pressure increase that accounts for the pre- and post-eruptive uplift during

the time span of the ascending interferogram is estimated as 4.31 MPa, and for descending

interferogram as 3.06 MPa. The effect of these excess pressure increase is shown in Figure

6.12, where sign is flipped for easier perception of the amount of displacement that should

be added. These simulated interferograms and azimuth offset are added to the original data


295 289 295 303 307 324�1.5

�1

�0.5

0

0.5

1

GV04 � GV05

Date

Dis

pla

ce

me

nt

(m)

East

North

Up

�3.5

�3

�2.5

�2

�1.5

�1

�0.5

0

0.5

1

1.5

GV04 � GV06

Date

Dis

pla

ce

me

nt

(m)

East

North

Up

�4

�3

�2

�1

0

1

2

GV05 � GV06

Date

Dis

pla

ce

me

nt

(m)

East

North

Up

295 289 295 303 307 324

295 289 295 303 307 324

Figure 6.11: GPS time series processed by Charles Meertens and Dennis Geist. The tickmarks on the x-axis correspond to the vertical lines in Figure 6.10.


4 km -38 0cm

a b c

-46 0cm -15 15cm

Figure 6.12: Simulated (a) ascending interferogram and (b) descending interferogram and(c) azimuth offset, which are to be added to the original data (Figure 6.8) in order to accountfor pre- and post-eruptive inflation.

(Figure 6.8).

6.5 Eruption Modeling

The adjusted interferograms and azimuth offset contains three major events: trapdoor fault-

ing, dike intrusion, and co-eruptive subsidence due to closing of the sill. The trapdoor

faulting may have triggered the dike intrusion (Geist et al., in press), which then fed the

eruption. All these events are confined within the caldera and within the first several hours

of the time span of the adjusted data. We do not have the temporal resolution of the events.

However, with their separation in space and simple boundary conditions, we construct a

physically plausible model of the events that fits the data.

6.5.1 Prior Information

Two surface constraints are used for this study. The eruption started as a curtain of fire

through a 2 km-long fissure inside the northern rim of the caldera (Geist et al., in press).

This observation, indicated with a red line in Figure 6.13, constrains the location of the

dike on the free surface of an elastic half-space in our model. We assume that the faulting

occurred somewhere along the C-shaped sinuous ridge. The sinuous ridge is composed

6.5. ERUPTION MODELING 99

�4

�3

�2

�1

0

1

2

3 �3

�2

�1

0

1

2

3

4

�3

�2

�1

0

northing (km)easting (km)

z (

km

)

4 km

(a) (b)

Figure 6.13: Prior information for eruption modeling. (a) Map view of the surface con-straint, where eruption fissure is indicated with a red line and the the fault trace on thesurface, drawn in black, goes along the C-shaped sinuous ridge. (b) 3-D perspective viewof the model geometry.

of a complex set of normally faulted blocks with steep (60◦ - 90◦) outward dipping fault

scarps (Reynolds et al., 1995). This characteristic topography suggests that the ridge has

formed by repeated faulting events, two of which were discovered by InSAR observations

(Amelung et al., 2000; Jonsson et al., 2005; Chadwick et al., 2006).

6.5.2 Wedge-like Sill

Surface deformation due to the faulting and the dike intrusion is swamped by the large

co-eruptive subsidence. Thus, it is worth looking at the residual after subtracting the effect

of the best-fit subsidence-only model. Given that we fix the geometry of the sill, this is a

linear inverse problem with only one parameter, the excess pressure, to be estimated. The

best-fit subsidence-only model (Figure 6.14) requires an excess pressure change of -41.7

MPa. As we include faulting and dike intrusion in the following sections, the estimated

excess pressure change at the sill will change. At this stage, we inspect in the pattern of

the residual, which is shown in Figure 6.15. The residuals for the interferograms show

near-constant-slope phase ramp over large area inside the caldera. This suggests that there


4 km-7.8 -2

Closing (m)

Figure 6.14: Best-fit sill-only model for the adjusted data. Associated excess pressurechange is -41.7 MPa, with host rock’s shear modulus of 30 GPa.

may be a wedge-like structure at depth perhaps where the sill is located. Thus, we intro-

duce an additional component of deformation not modeled by uniform pressure drop. The

interpretation of this additional component will be discussed later in this chapter.

6.5.3 Trapdoor Faulting

Trapdoor faulting events at Sierra Negra have been previously modeled once with an out-

ward dipping high-angle normal fault (Amelung et al., 2000) and once with an inward

dipping high-angle reverse fault (Chadwick et al., 2006). The near-vertical fault geometry

is also supported by the estimated sill periphery that is collocated with the sinuous ridge in

a map view (Figure 6.6). In this study, for simplicity, we assume vertical fault planes and

allow only dip-slip on each element. At the northern and southern edge of the fault system,

the slip is constrained to be zero. Along the top edge of the fault planes, where they break

the surface, zero displacement gradient boundary condition is applied to accommodate a

stress-free interface. Inner fault elements are subject to Laplacian smoothing. At the bot-

tom of the fault planes, the dip-slips are matched to the opening at the tip of the sill (see

Figure 6.2b). This condition is further explained later in this chapter.


a b c

4 km−4.9 3.9m

Figure 6.15: Residuals between data and model prediction for (a) ascending interferogram,(b) descending interferogram, and (c) azimuth offset due to the best-fit sill-only model(Figure 6.14). One color fringe represents 20 cm LOS displacement in (a) and (b). Thenear-constant-slope phase ramp over large area inside the caldera suggests a ramp-likefeature at depth.

Our best-fit fault slip model is shown in Figure 6.16. The model is estimated simultane-

ously with dike intrusion and sill closing. Previous trapdoor faulting has been concentrated

along the southern part of the fault system, whereas this event is more concentrated along

the western part of the fault system. The maximum slip of about 1.8 m is located at the

bottom of the western-most fault planes and the amount of slip becomes gradually smaller

along the bottom of the fault planes towards the north. The estimated slip toward the sur-

face at the western end of the fault system is about 1.5 m. This matches very well with the

field measurement at Spot B (1.5 m) by Geist and others (Geist, pers. comm.). At Spot A

the model slightly underestimates the field measurement (1.4 - 1.5 m).

The seismic moment of this fault model is about 4.32×1024 dyne-cm, which is equiv-

alent to moment magnitude Mw 5.7. According to the Harvard CMT catalog, the moment

magnitude of the earthquake several hours prior to the onset of the eruption was Mw 5.4

(equivalent to seismic moment of 1.41× 1024 dyne-cm). This may suggest that substan-

tial amount of the slip occurred aseismically. Note, however, that we assume that the shear

modulus of the host rock is 30 GPa. If the shear modulus is 10 GPa, the moment magnitude

of the model becomes Mw 5.4 and matches the observation.


0

1

2

55

500.511.5

5

5

0

easting (km)

northing (km)

z (

km

)

0

1

2

0

1

5

5

0

northing (km)

easting (km)

z (

km

)

Dip slip (m)

1.80

(a) (b)

0

1

2

3

0

1

2

3

4

0


z (

km

)

Figure 6.16: Best-fit fault model estimated simultaneously with dike and sill models, (a)with the same view as in the inset, and (b) when the model is rotated 130◦ counterclock-wise. The inset shows the location of the fault.


6.5.4 Dike Intrusion

We assume that the dike propagation initiated close to the northern edge of the sill and

daylighted where the fissure was observed. As in the case of faulting we take the simplest

planar path between the starting and the ending positions. Unlike faulting, however, we

estimate all three components of displacement discontinuity in each element. This is to

accommodate slips that release any pre-existing ambient shear stresses, pressurized sill-

induced shear stresses, shear stresses due to an interaction between the dike and the free

surface, and finally shear stresses due to the slip on the neighboring fault system.

Figure 6.18 shows the best-fit dike model estimated simultaneously with faulting and

sill closing. The color indicates the amount of opening and the arrows show the magnitude

and direction of the slip along the dike plane on each element. Eastern and western edges

of the dike are clamped and along the top edge displacement gradients are encouraged to

be zero. Inner dike elements are subject to Laplacian smoothing. At the bottom edge of the

dike we try two different boundary conditions: (a) zero displacement gradient, and (b) an

adaptive boundary condition that matches the vertical components of the slip vectors with

the openings at the northern edge of the sill.

A possible scenario that may explain the two best-fit models is illustrated in Figure

6.17. The scenario starts with a pressurized sill that induces a dike propagation (Johnson

and Pollard, 1973; Pollard and Johnson, 1973). Once the dike opens up, the interaction

between the sill and the dike may favor a “reverse faulting” along the dike plane, given

the high-angle dike geometry (b). As the dike gets closer to the surface it starts feeling the

existence of the free surface, creating shear stress. This shear stress favors more “reverse

faulting” on the dike plane. Once the dike leads to an eruption, the sill gets depressurized

and starts closing, resulting in “normal faulting”. Accordingly, the amount of slip becomes

smaller at the bottom of the dike, as the sill closes and pulls down the bottom edge of

the dike (c). In other words, the “reverse faulting” is partially negated by the “normal

faulting”. Note that the faulting at the dike is reversible process as long as the magma in

the dike remains liquid.

Geodetic data are sensitive only to the final state of the dike at the moment it solidi-

fies. Thus, depending on how quickly the dike solidifies, one should carefully choose the


(a) (b) (c)

Figure 6.17: Schematic vertical cross section as in Figure 6.2. (a) pressurized sill after thetrapdoor faulting, (b) dike intrusion accompanied with “reverse faulting”, (c) co-eruptivesubsidence accompanied with “normal faulting”.

boundary conditions in the model. Up until the onset of the eruption the dike experiences

the “reverse faulting”. Once the eruption initiates and the sill starts to close, a “normal

faulting” begins on the dike plane, reducing the amount of slip due to the “reverse fault-

ing”.

Once the dike starts solidifying, the wall of the frozen part of the dike does not slip

any more. If the dike freezes quickly and coalesces into a few conduits early during the

eruption, one can consider the dike and the sill decoupled when modeling (Delaney and

Pollard, 1981). If the dike solidifies slowly and the sill pulls the hanging wall down until

the end of the eruption, the dike and the sill are fully coupled and should be treated so in a

model. Figure 6.18a and Figure 6.18b represent these decoupled and coupled end member

models respectively. Geist et al. (in press) observed that the curtain-of-fire coalesced into

a single vent by day 4, and on day 6 of the eruption the easternmost vent reopened. When

the fissure became a single vent, at least part of the dike solidified.

The RMS errors for each data in the two cases are summarized in Table 6.2. The decou-

pled model has a slightly better fit to the descending interferogram, whereas the coupled

model fits the azimuth offset slightly better. Given that the uncertainties in the interfero-

grams and azimuth offset are about 0.65 cm and 22 cm respectively, we cannot say one

model is favored against the other based on the data fitting. The aforementioned field ob-

servations by Geist et al. (in press) suggest that the model should be somewhere between

the two end members. Especially at the bottom of the dike, however, it is likely that the

opening was sustained for a long time, perhaps close to the end the eruption. Thus, model

(b) seems to be physically more plausible than model (a). In model (b), the maximum dike

opening is about 3.6 m and the average opening is about 1.7 m.


Table 6.2: RMS errors of two cases in Figure 6.18

Data Dike-sill decoupled Dike-sill coupled

Ascending interferogram 5.15 cm 5.17 cm

Descending interferogram 4.58 cm 4.68 cm

Azimuth offset 81.69 cm 81.40 cm

Thus, if the dike solidified quickly after the onset of the eruption, it must have frozen

with substantial dip-slip at the bottom of the dike (Figure 6.18a), due to the shear stress

release. In this case the sill does not mechanically interact directly with the dike any more,

and basically they are decoupled. However, if the lava in the dike sustained its liquid phase

for a longer period of time, it may have been subject to the sill closing perhaps large enough

to negate the initial slip and even reverse the sense of motion. When the shallow part of

the dike solidifies first, the effect of the sill at the initial stage (i.e. the reverse faulting) is

recorded and frozen along with the shallow part of the dike. By the time the deeper part

of the dike solidifies the sill has closed substantially, pulling the hanging wall of the dike

back toward the sill and closing the dike at the bottom (Figure 6.18b). The slight closing

at the bottom of the dike in Figure 6.18b may be due to the poor resolution of the model at

the boundary of the dike and the sill. For example, each element of the sill has a dimension

of 500 m by 500 m, and its displacement discontinuity is calculated at the center of the

element, where the amount of closing is already a few meters. This value is matched to the

displacement discontinuity at the dike.

6.5.5 Fault-Sill Interaction

As previously mentioned, the data hint at a ramp or wedge-like structure at depth. Thus,

we separate the sill model into two components: a uniformly depressurized sill (i.e. crack

solution) and a sill that can freely open or close with a smoothing constraint (i.e. kinematic

solution). These two sill models share the same depth and the same detailed geometry (i.e.

Figure 6.14). Opening distributions of the two models are simultaneously estimated with

faulting and dike intrusion models, and the sum of the openings (or closings) from the two


0

0.5

1

1.5 2.5

3

3.5

�1.8

�1.6

�1.4

�1.2

�1

�0.8

�0.6

�0.4

�0.2

0

northing (km)

easting (km)

z (

km

)

30.5

Opening (m)Slip

4 m3.5-0.5

Opening (m)Slip

3.4 m

�4

�3

�2

�1

0

1

2

3 �3

�2

�1

0

1

2

3

4

�3

�2

�1

0


z (

km

)

0

0.5

1

1.5 2.5

3

3.5

�1.8

�1.6

�1.4

�1.2

�1

�0.8

�0.6

�0.4

�0.2

0

northing (km)

easting (km)

z (

km

)

(a) (b)

Figure 6.18: Best-fit dike models estimated simultaneously with fault and sill models, (a)when displacements of dike and sill are not coupled and (b) they are coupled. The insetshows the location of the dike.


models constitute the final opening distribution of the sill. Here we discuss the second

component, the kinematic solution.

The Laplacian smoothing is applied to inner elements of the sill and the zero displace-

ment discontinuity gradient condition is applied at the edge elements. In addition, we apply

a physical boundary condition along the edge of the sill that is close to the bottom edge of

the fault. We assume that the bottom of the fault system is just touching the edge of the sill,

which is plausible for a pressure-induced trapdoor faulting. When the faulting happens,

the shear stress drop on the fault plane induces normal stress drop on the wall of the sill,

redistributing the openings at the sill. In order to avoid large singularities at the edge of

the sill, or equivalently to avoid material gap at depth, the slip distribution at the bottom of

the fault system is matched with the openings along the corresponding edge of the sill as

illustrated in Figure 6.2b.

The inversion results are shown in Figure 6.19. There, (a) is the uniformly depressur-

ized sill model, and (b) is the kinematic model of the sill coupled with the fault and the

dike. The kinematic model suggests the interaction between the fault and the sill. The east-

ern part of the sill shows a strong contraction or closing, whereas the western part shows an

opening along the edge. We believe that this anomaly shows the effect of magma transport

toward the fault plane when a sudden opening was forced on the western end of the sill due

to the faulting event. The trapdoor faulting occurs nearly instantaneously compared to the

magma influx. Hence, the existing magma in the chamber has to move in response to the

pressure gradient caused by the faulting event (Figure 6.20). By superposing the uniformly

depressurized model and the kinematic model of the sill, we can create a composite open-

ing/closing sill model (Figure 6.21). The maximum closing of the composite sill model is

about 8.9 m, and the total volume decrease at the sill is about 0.124 km3.

The residuals between the best-fit model prediction and the data are shown in Figure

6.22. The model is composed of the trapdoor faulting, dike intrusion, and the composite

sill, all of which are coupled and simultaneously estimated. A few fringes shown in (a)

may be due to the simplified fault and dike geometry or dipping of the fault planes. About

94% of data variance is explained by the best-fit model.


(a) (b)

3

2

1

0

1

2

3 2

1

0

1

2

3

2.5

2

1.5

1

0.5

0

z−(km)

northing−(km)

easting−(km)

3

2

1

0

1

2

3 2

1

0

1

2

3

2.5

2

1.5

1

0.5

0

z−(km)

northing−(km)

easting−(km)

Opening−(m)

-6.4 -1.5

Opening−(m)

-2.2 2.2

4

3

2

1

0

1

2

3 3

2

1

0

1

2

3

4

3

2

1

0

northing−(km)easting−(km)

z−(km)

Figure 6.19: (a) Best-fit uniformly depressurized sill model, and (b) best-fit kinematic sillmodel (opening), which is the effet of interaction of sill with faulting and dike intrusion.

(a) (b)

Figure 6.20: (a) pressurized sill (b) magma transport in the sill due to the trapdoor faulting


4 km-8.9 0.4

Opening (m)

Figure 6.21: Best-fit composite sill model (uniformly depressurized sill + kinematic open-ing due to fault-sill and dike-sill interaction.

a b c

4 km−4.9 3.9m

Figure 6.22: Residual between the best-fit model prediction and the data for (a) ascend-ing interferogram, (b) descending interferogram, and (c) azimuth offset. One color fringerepresents 20 cm of LOS displacement in (a) and (b).


6.6 Extruded Volume

The estimated volume decrease at the sill during the eruption was about 0.124 km3. We

compare this with extruded volume estimated by simulating the lava coverage using a

TOPSAR-SRTM merged DEM. We pour lava inside the caldera and let it flow to reach

the lowest elevation. Practically this was done simply by subtracting a preset elevation

from the DEM and define the negative area as ”flooded”. Then we calculate the volume of

the flooded area.

We choose 15 elevations from 900 m to 950 m with a uniform interval. For each

elevation of the flat lava surface, the flooded area and the associated volume is plotted

(Figure 6.23). This flood chart was compared with the real lava coverage observed by

ASTER sensor mounted on NASA’s Terra satellite (Figure 6.1), which shows the surface

temperature. Note that the most similar pattern of the flooded area can be found around

0.141 km3. Assuming an average porosity of lava pile of 15%, we estimate the dense rock

equivalent volume of extruded lava as 0.120 km3. This roughly matches the estimated

volume decrease of the sill (0.124 km3), implying that there may not have been substantial

amount of volatiles in the magma chamber before the eruption, but enough to make 15%

porosity. The estimated volume increase of the dike intrusion is about 7.8 million cubic

meters, negligible compared to the volume loss of the sill and the extruded volume. In case

of the flood model with 0.141 km3 of extruded volume, the average lava depth is about 11.4

m and the maximum depth is about 44.9 m.

This simplified flood model does not consider the strength of lava that resists flowing.

For example, the extent of real lava flow along the southern moat (the valley between the

southern part of the fault scarp and the southern caldera rim) is about 1 km shorter than

the extent of the simulated lava coverage. Another source of error is the subsidence of

the caldera floor, which is not included in the DEM. Nevertheless, the overall similarity

between the observed and simulated flood patterns indicates that this simple flood model

may provide a first-order estimate on the extruded volume, especially when the lava flow is

restricted by a confining topography such as the caldera rim.

6.6. EXTRUDED VOLUME 111

0.009 km3

0.020 km3

0.039 km3

0.065 km3

0.100 km3

0.141 km3

0.189 km3

0.243 km3

0.302 km3

0.364 km3

0.430 km3

0.499 km3

0.572 km3

0.648 km3

0.728 km3

Figure 6.23: Lava coverage simulation using DEM.


6.7 Conclusions

2005 eruption as Sierra Negra volcano underwent complex and large deformation due to

a trapdoor faulting event that preceded the eruption by 3 hours, dike intrusion that fed the

eruption, and the huge co-eruptive subsidence. By using a newly developed algorithm, we

were able to form one ascending interferogram and one descending interferogram that span

the eruption. Azimuth offset image was formed from ascending SAR image pair. These

data were adjusted using GPS data, so the data span only the eruption period but not the

pre- and post-eruptive uplifting period. Then the adjusted data were used for modeling the

faulting event, dike intrusion, and the sill closing simultaneously.

The trapdoor faulting event and the dike intrusion were modeled with kinematic dis-

tributed dislocation models. The estimated seismic moment of the faulting event (Mw 5.7)

suggests that a substantial amount of slip occurred aseismically. Most of the slips were

concentrated along the western part of the fault system. The average dike opening was

estimated as about 1.7 m. The dike must have undergone large dip-slip due to the pres-

surized sill and the interaction with the free surface. However, while the dike contained

liquid magma, the initial slip seemed to be negated mostly at depth by the closing sill. This

suggests that lower part of the dike must have remained liquid at least during the 8 days of

the eruption.

The periphery of the sill was estimated independently using a pre-eruptive uplift in-

teferograms. The depth of the sill was estimated as 1.86 km ± 0.13 km, and the best-fit

sill periphery model explained about 99% of the data variance of interferograms before the

eruption. The fixed geometry of the sill enabled us to parameterize the sill with only one

parameter of excess magma pressure. We estimate geophysical parameters of three ma-

jor events: faulting, dike intrusion, and sill closing. A kinematic sill model is used along

with a uniformly depressurized sill. The kinematic sill model was mechanically coupled at

its edge with the fault system and the dike plane. We believe that the estimated opening-

closing distribution of the kinematic sill model show the change in opening distribution of

the sill due to the trapdoor faulting.

By superposing the kinematic sill model and the uniformly depressurized sill, we pro-

duced a composite sill model, whose volume decrease was estimated as about 0.124 km3.

6.7. CONCLUSIONS 113

This volume loss roughly matches our estimate of the extruded volume, which was cal-

culted by DEM analysis. The best-fit model of trapdoor faulting, dike intrusion, and sill

opening-closing explain about 94% of the geodetic data.

Chapter 7

Thesis Findings and Conclusions

In this work we used Space-borne Interferometric Synthetic Aperture Radar (InSAR) to

estimate geophysical parameters of the 2005 eruption at Sierra Negra volcano in Isabela

Island, Galapagos. In order to achieve this goal, we developed three main novel algorithms

to implement data processing and modeling.

First, we produced the best quality DEM to date of Sierra Negra volcano by merging

a high-resolution TOPSAR DEM and a low-resolution SRTM DEM. The pixel spacing

of TOPSAR DEM is 10 m, but it has some artifacts and many “data holes”. The pixel

spacing of the SRTM DEM is 90 m, but it has fewer holes and has more comprehensive

coverage and reliable data values. We merged the two DEMs by solving an inverse prob-

lem constrained with a Prediction-Error filter and the SRTM DEM. The pixel spacing of

the merged DEM is 10 m, it does not have a hole, and its data values have reliability of

SRTM DEM (Yun et al., 2005). We used this DEM to subtract topography effect from

interferograms that span the 2005 eruption at Sierra NEgra.

Second, we developed a new inversion algorithm to solve for the detailed geometry of a

uniformly pressurized crack. Using this inversion we can produce a mechanically plausible

and internally consistent crack model that fits InSAR observations. We also constrained

the geometry of the magma chamber at Sierra Negra. Using a ascending and descending

interferogram pair, we showed that the magma chamber at Sierra Negra had a flat top.

However, the vertical extent of the magma chamber cannot be well constrained by surface

displacement. Thermal analysis suggested that the thickness of the magma chamber is at

114

115

least 40 m (Yun et al., 2006).

Third, we developed a new interferogram forming algorithm, which is useful when the

deformation is large and complex. Using the algorithm we were able to produce intra-

caldera interferograms that span the 2005 eruption at Sierra Negra, which was not possible

with existing standard InSAR software. The 2005 eruption at Sierra Negra caused about 5.4

meter subsidece and was preceded by a Mw 5.4 earthquake. These two events caused com-

plex and steep displacement gradients, which made SAR image coregistration and phase

unwrapping very difficult. The new algorithm involves a rubber-sheeting coregistration

and range offset image subtraction to solve this problem (Yun et al., in review). The output

interferograms were used to model Sierra Negra’s 2055 eruption.

The 2005 eruption at Sierra Negra was modeled using InSAR and GPS data. The

model consists of three main parts: trapdoor faulting, dike intrusion, and opening-closing

of a sill. The depth and the detailed geometry of the sill was estimated from an ascending

and descending interferogram pair before the eruption. The estimated depth was 1.86 km

± 0.13 km. The GPS data were used to estimate pre- and post-eruptive inflation, and the

deformation due to the inflation was subtracted from InSAR data in order to effectively

reduce the temporal baseline of the InSAR data.

In the trapdoor faulting model, the maximum slip ( 1.8 m) occurred along the western

and bottom end of the fault system. The slip becomes about 1.5 m toward the surface, which

matches the field observation very well. The equivalent moment magnitude of the total slip

was estimated to be Mw 5.7 when the shear modulus is 30 GPa. For shear modulus of 10

GPa, it becomes Mw 5.4, which is the moment magnitude of the earthquake that occurred

3 hours prior to the onset of the eruption. The dike model showed average opening of 1.7 m

and “reverse faulting” average dip slip of 1.6 m. The large dip slip is due to the interaction

with the sill and the free surface.

The the sill model is the sum of two components: a uniformly depressurized sill and a

kinematic solution of openings at the sill. The uniformly depressurized sill accounted for

the co-eruptive subsidence. The kinematic model showed a wedge-like opening distribu-

tion, and this suggested the interaction of the trapdoor faulting and the sill. The repeating

cycle of trapdoor faulting and eruption can produce an accumulated wedge-like structure

at depth as well as on the surface. We believe that the surface expression of this structure is

116 CHAPTER 7. THESIS FINDINGS AND CONCLUSIONS

shown as the characteristic C-shaped sinuous ridge inside the caldera of Sierra Negra. The

estimated volume decrease at the sill was 0.124 km3, and the estimated extruded volume

(dense rock equivalent) was about 0.120 km3. This similarity suggests that there may not

have been substantial amount of volatiles in the magma before the eruption.

Bibliography

Amelung, F., S. Jonsson, H. Zebker, and P. Segall (2000), Widespread uplift and ‘trapdoor’

faulting on galapagos volcanoes observed with radar interferometry., Nature, 407(6807),

993 – 6.

Bamler, R. (2000), Interferometric stereo radargrammetry: Absolute height determination

from ers-envisat interferograms, International Geoscience and Remote Sensing Sympo-

sium (IGARSS), 2, 742 – 745.

Baran, I., M. Stewart, and S. Claessens (2005), A new functional model for determining

minimum and maximum detectable deformation gradient resolved by satellite radar in-

terferometry., IEEE Transactions on Geoscience and Remote Sensing, 43(4), 675 – 82.

Basu, A., and L. N. Frazer (1990), Rapid determination of the critical temperature in sim-

ulated annealing inversion., Science, 249(4975), 1409 – 12.

Berardino, P., G. Fornaro, R. Lanari, and E. Sansosti (2002), A new algorithm for surface

deformation monitoring based on small baseline differential sar interferograms., IEEE

Transactions on Geoscience and Remote Sensing, 40(11), 2375 – 83.

Cervelli, P., M. H. Murray, P. Segall, Y. Aoki, and T. Kato (2001), Estimating source param-

eters from deformation data, with an application to the march 1997 earthquake swarm

off the izu peninsula, japan., Journal of Geophysical Research, 106(B6), 11,217 – 37.

Chadwick, W., and J. Dieterich (1995), Mechanical modeling of circumferential and radial

dike intrusion on galapagos volcanoes, J. Geophys. Res., 66(1-4), 37 – 52.

117

118 BIBLIOGRAPHY

Chadwick, W., D. Geist, S. Jonsson, M. Poland, D. Johnson, and C. Meertens (2006), A

volcano bursting at the seams: Inflation, faulting, and eruption at sierra negra volcano,

galapagos, Geology, 34(12), 1025–1028.

Chen, C. W., and H. A. Zebker (2001), Two-dimensional phase unwrapping with use of

statistical models for cost functions in nonlinear optimization., Journal of the Optical

Society of America A (Optics, Image Science and Vision), 18(2), 338 – 51.

Cho, B. L., Y. K. Kong, and Y. S. Kim (2005), Interpolation using optimum nyquist filter

for sar interferometry., Journal of Electromagnetic Waves and Applications, 19(1), 129

– 35.

Claerbout, J. F. (1992), Earth sounding analysis: Processing versus inversion, Blackwell

Scientific Publications.

Claerbout, J. F., and S. Fomel (2002), Image estimation by example: Geophysical sound-

ings image construction (class notes).

Comninou, M., and J. Dundurs (1975), The angular dislocations in a half space., Journal

of Elasticity, 5(3/4), 203 – 16.

Delaney, P. T., and D. D. Pollard (1981), Deformation of host rocks and flow of magma dur-

ing growth of minette dikes and breccia-bearing intrusions near ship rock, new mexico,

Geological Survey Professional Paper, 1202.

Dieterich, J., and R. Decker (1975), Finite element modeling of the surface deformation

associated with volcanism, J. Geophys. Res., 80.

Dixon, T. H., F. Amelung, A. Ferretti, F. Novali, F. Rocca, R. Dokka, G. Sella, S. W.

Kim, S. Wdowinski, and D. Whitman (2006), Subsidence and flooding in new orleans.,

Nature, 441(7093), 587 – 8, doi:10.1038/441587a.

Farr, T., and M. Kobrick (2000), Shuttle radar topography mission produces a wealth of

data, Eos Trans. AGU, 81(48), 583,585.

BIBLIOGRAPHY 119

Ferretti, A., C. Prati, and F. Rocca (2001), Permanent scatterers in sar interferometry., IEEE

Transactions on Geoscience and Remote Sensing, 39(1), 8 – 20.

Fialko, Y., and M. Simons (2000), Deformation and seismicity in the coso geothermal area,

inyo county, california; observations and modeling using satellite radar interferometry,

J. Geophys. Res., 105(B9), 21,781–21,793.

Fialko, Y., Y. Khazan, and M. Simons (2001a), Deformation due to a pressurized horizontal

circular crack in an elastic half-space, with applications to volcano geodesy., Geophysi-

cal Journal International, 146(1), 181 – 90.

Fialko, Y., M. Simons, and D. Agnew (2001b), The complete (3-d) surface displacement

field in the epicentral area of the 1999 msub w7.1 hector mine earthquake, california,

from space geodetic observations., Geophysical Research Letters, 28(16), 3063 – 6.

Fialko, Y., M. Simons, and Y. Khazan (2001c), Finite source modelling of magmatic unrest

in socorro, new mexico, and long valley, california., Geophysical Journal International,

146(1), 191 – 200.

Fukushima, Y., V. Cayol, and P. Durand (2005), Finding realistic dike models from in-

terferometric synthetic aperture radar data: the february 2000 eruption at piton de la

fournaise., Journal of Geophysical Research, 110(B3), 15, doi:10.1029/2004JB003268.

Geist, D., W. Chadwick, and D. Johnson (in review 2004), Results from new gps and

gravity monitoring networks at fernandina and sierra negra volcanoes, galapagos, 2000-

2002, Journal of Volcanology and Geothermal Research.

Geist, D. J., K. S. Harpp, T. R. Naumann, M. Poland, and W. W. Chadwick (in press), The

2005 eruption of sierra negra volcano, galapagos, ecuador, Bulletin of Volcanology.

Goldstein, R. M., and C. L. Werner (1998), Radar interferogram filtering for geophysical

applications., Geophysical Research Letters, 25(21), 4035 – 8.

Goldstein, R. M., H. A. Zebker, and C. L. Werner (1988), Satellite radar interferometry:

two-dimensional phase unwrapping., Radio Science, 23(4), 713 – 20.

120 BIBLIOGRAPHY

Goldstein, R. M., H. ENGELHARDT, B. KAMB, and R. M. FROLICH (1993), Satellite

radar interferometry for monitoring ice sheet motion: application to an antarctic ice

stream., Science, 262(5139), 1525 – 30.

Hanssen, R. F., T. M. Weckwerth, H. A. Zebker, and R. Klees (1999), High-resolution water

vapor mapping from interferometric radar measurements., Science, 283(5406), 1297 – 9.

Hill, D. P., and J. J. Zucca (1987), Geophysical constraints on the structure of kilauea and

mauna loa volcanoes and some implications for seismomagnetic processes, Volcanism in

Hawaii, 2, 903 – 917.

Hooper, A. (2006), Persistent scatterer radar interferometry for crustal deformation studies

and modeling of volcanic deformation, Ph.D. thesis, Stanford University, Department of

Geophysics, Stanford, CA.

Hooper, A., H. Zebker, P. Segall, and B. Kampes (2004), A new method for measuring

deformation on volcanoes and other natural terrains using insar persistent scatterers.,

Geophysical Research Letters, 31(23), 5, doi:10.1029/2004GL021737.

Johnson, A. M., and D. D. Pollard (1973), Mechanics of growth of some laccolithic in-

trusions in the henry mountains, utah. i. field observations, gilbert’s model, physical

properties and flow of the magma., Tectonophysics, 18(3/4), 261 – 309.

Jonsson, S. (2002), Modeling volcano and earthquake deformation from satellite radar in-

terferometric observations, Ph.D. thesis, Stanford University, Department of Geophysics,

Stanford, CA.

Jonsson, S., H. Zebker, P. Cervelli, P. Segall, H. Garbeil, P. Mouginis-Mark, and S. Row-

land (1999), A shallow-dipping dike fed the 1995 flank eruption at fernandina volcano,

galapagos, observed by satellite radar interferometry., Geophysical Research Letters,

26(8), 1077 – 80.

Jonsson, S., H. Zebker, P. Segall, and F. Amelung (2002), Fault slip distribution of the 1999

msub w 7.1 hector mine, california, earthquake, estimated from satellite radar and gps

measurements., Bulletin of the Seismological Society of America, 92(4), 1377 – 89.

BIBLIOGRAPHY 121

Jonsson, S., H. Zebker, and F. Amelung (2005), On trapdoor faulting at sierra negra vol-

cano, galapagos, Journal of Volcanology and Geothermal Research, 144, 59 – 71.

Kobayashi, Y., K. Sarabandi, L. Pierce, and M. C. Dobson (2000), An evaluation of the

jpl topsar for extracting tree heights., IEEE Transactions on Geoscience and Remote

Sensing, 38(6), 2446 – 54.

Lohman, R. B., and M. Simons (2005), Some thoughts on the use of insar data to constrain

models of surface deformation: Noise structure and data downsampling, GEOCHEM-

ISTRY GEOPHYSICS GEOSYSTEMS, 6, Q01,007 –.

Lu, Z., R. Fatland, M. Wyss, S. Li, J. Eichelberer, K. Dean, and J. Freymueller (1997), De-

formation of new trident volcano measured by ers-1 sar interferometry, katmai national

park, alaska., Geophysical Research Letters, 24(6), 695 – 8.

Lundgren, P., P. Berardino, M. Coltelli, G. Fornaro, R. Lanari, G. Puglisi, E. Sansosti, and

M. Tesauro (2003), Coupled magma chamber inflation and sector collapse slip observed

with synthetic aperture radar interferometry on mt. etna volcano., Journal of Geophysical

Research, 108(B5), ECV4 – 1.

Madsen, S. N., H. A. Zebker, and J. Martin (1993), Topographic mapping using radar

interferometry: processing techniques., IEEE Transactions on Geoscience and Remote

Sensing, 31(1), 246 – 56.

Massonnet, D., and K. L. Feigl (1998), Radar interferometry and its application to changes

in the earth’s surface., Reviews of Geophysics, 36(4), 441 – 500.

Massonnet, D., M. Rossi, C. Carmona, F. Adragna, G. Peltzer, K. Feigl, and T. Rabaute

(1993), The displacement field of the landers earthquake mapped by radar interferome-

try., Nature, 364(6433), 138 – 42.

Massonnet, D., P. Briole, and A. Arnaud (1995), Deflation of mount etna monitored by

spaceborne radar interferometry., Nature, 375(6532), 567 – 70.

McTigue, D. F. (1987), Elastic stress and deformation near a finite spherical magma body

- resolution of the point-source paradox, J. Geophys. Res., 92(B12), 12,931 – 12,940.

122 BIBLIOGRAPHY

Metropolis, N., A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller (1953),

Equation of state calculations by fast computing machines, Journal of Chemical Physics,

21(6), 1087 – 1092.

Mogi, K. (1958), Relations between the eruptions of various volcanoes and the deforma-

tions of the ground surfaces around them, Bull. Earthquake Res. Inst. Univ. Tokyo, 36,

111–123.

Munro, D. C., and S. K. Rowland (1996), Caldera morphology in the western galapagos

and implications for volcano eruptive behavior and mechanisms of caldera formation,

Journal of Volcanology and Geothermal Research, 72(1-2), 85 – 100.

Okada, Y. (1992), Internal deformation due to shear and tensile faults in a half-space.,

Bulletin of the Seismological Society of America, 82(2), 1018 – 40.

Onn, F., and H. A. Zebker (2006), Correction for interferometric synthetic aperture radar

atmospheric phase artifacts using time series of zenith wet delay observations from a

gps network, JOURNAL OF GEOPHYSICAL RESEARCH-SOLID EARTH, 111(B9),

B09,102 –.

Pinel, V., and C. Jaupart (2003), Magma chamber behavior beneath a volcanic edifice - art.

no. 2072, Journal of Gephysical Research-Solid Earth, 108(B2), 2072 – 2072.

Pollard, D. D., and A. M. Johnson (1973), Mechanics of growth of some laccolithic in-

trusions in henry mountains, utah .2. bending and failure of overburden layers and sill

formation, TECTONOPHYSICS, 18(3-4), 311 – 354.

Pritchard, M. E., and M. Simons (2002), A satellite geodetic survey of large-scale defor-

mation of volcanic centres in the central andes., Nature, 417(6894), 167 – 71.

Reynolds, R. W., D. Geist, and M. D. Kurz (1995), Physical volcanology and structural

development of sierra negra volcano, isabela island, galapagos archipelago, Geological

Society of America Bulletin, 107(12), 1398 – 1410.

Richman, D. (), Three dimensional azimuth-correcting mapping radar., Tech. rep., USA:

United Technologies Corporation.

BIBLIOGRAPHY 123

Rosen, P. A., S. Hensley, G. Peltzer, and M. Simons (2004), Updated repeat orbit interfer-

ometry package released, Eos Trans. AGU, 85(5), 47.

Rothman, D. H. (1986), Automatic estimation of large residual statics corrections., Geo-

physics, 51(2), 332 – 46.

Sallares, V., and P. Charvis (2003), Crustal thickness constraints on the geodynamic evolu-

tion of the galapagos volcanic province, Earth and Planetary Science Letters, 214(3-4),

545 – 559.

Savage, J. C. (1984), Local gravity-anomalies produced by dislocation sources, Journal of

Geophysical Research, 89(NB3), 1945 – 1952.

Schmidt, D. A., and R. Burgmann (2003), Time-dependent land uplift and subsidence in

the santa clara valley, california, from a large interferometric synthetic aperture radar

data set, JOURNAL OF GEOPHYSICAL RESEARCH-SOLID EARTH, 108(B9), 2416 –.

Simons, M., Y. Fialko, and L. Rivera (2002), Coseismic deformation from the 1999 msub

w 7.1 hector mine, california, earthquake as inferred from insar and gps observations.,

Bulletin of the Seismological Society of America, 92(4), 1390 – 402.

Tarantola, A. (2004), Inverse Problem Theory and Methods for Model Parameter Estima-

tion, SIAM, Philadelphia, PA.

Thomas, A. L. (1993), Poly3d: A three-dimensional, polygonal element, displacement dis-

continuity boundary element computer program with applications to fractures, faults, and

cavities in the earth’s crust, M.S. Thesis, Stanford University, Department of Geological

and Environmental Sciences, Stanford, CA.

Turcotte, D. L., and G. Schubert (2002), Geodynamics, second ed., 166 – 168 pp., Cam-

bridge University Press, Cambridge, UK.

Wahba, G. (1990), Spline models for observational data, no. 59 in Applied Mathematics,

SIAM, Philadelphia, PA.

124 BIBLIOGRAPHY

Walter, T. R., and F. Amelung (2006), Volcano-earthquake interaction at mauna loa vol-

cano, hawaii., Journal of Geophysical Research-Part B-Solid Earth, 111(B5), 17, doi:

10.1029/2005JB003861.

Wdowinski, S., F. Amelung, F. Miralles-Wilhelm, T. H. Dixon, and R. Carande (2004),

Space-based measurements of sheet-flow characteristics in the everglades wetland,

florida, Geophysical Research Letters, 31(15), L15,503 – 5.

Wicks, C. W., W. Thatcher, D. Dzurisin, and J. Svarc (2006), Uplift, thermal un-

rest and magma intrusion at yellowstone caldera., Nature, 440(7080), 72 – 5, doi:

10.1038/nature04507.

Wright, T. J., Z. Lu, and C. Wicks (2004), Constraining the slip distribution and fault ge-

ometry of the msub w 7.9, 3 november 2002, denali fault earthquake with interferometric

synthetic aperture radar and global positioning system data, Bulletin of the Seismological

Society of America, 94(6 SUPPL. B), S175 – S189.

Wright, T. J., C. Ebinger, J. Biggs, A. Ayele, G. Yirgu, D. Keir, and A. Stork (2006),

Magma-maintained rift segmentation at continental rupture in the 2005 afar dyking

episode., Nature, 442(7100), 291 – 4, doi:10.1038/nature04978.

Yang, X.-M., P. M. Davis, and J. H. Dieterich (1988), Deformation from inflation of a

dipping finite prolate spheroid in an elastic half-space as a model for volcanic stressing.,

Journal of Geophysical Research, 93(B5), 4249 – 57.

Yoffe, E. H. (1960), The angular dislocation, Philosophical Magazine, 5(50), 161 – 175.

Yun, S., J. Ji, H. Zebker, and P. Segall (2005), On merging high- and low-resolution dems

from topsar and srtm using a prediction-error filter, IEEE Transactions on Geoscience

and Remote Sensing, 43(7), 1682 – 1690.

Yun, S., P. Segall, and H. Zebker (2006), Constraints on magma chamber geometry at sierra

negra volcano, galapagos islands, based on insar observations, Journal of Volcanology

and Geothermal Research, 150(1-3), 232 – 243.

BIBLIOGRAPHY 125

Yun, S., H. Zebker, P. Segall, A. Hooper, and M. Poland (in review), Interferogram forma-

tion in the presence of large deformation, Geophysical Research Letters.

Zebker, H. A., and K. Chen (2005), Accurate estimation of correlation in insar observa-

tions., IEEE Geoscience and Remote Sensing Letters, 2(2), 124 – 7.

Zebker, H. A., and R. M. Goldstein (1986), Topographic mapping from interferometric

synthetic aperture radar observations., Journal of Geophysical Research, 91(B5), 4993 –

9.

Zebker, H. A., and J. Villasenor (1992), Decorrelation in interferometric radar echoes.,

IEEE Transactions on Geoscience and Remote Sensing, 30(5), 950 – 9.

Zebker, H. A., S. N. Madsen, J. Martin, K. B. Wheeler, T. Miller, Y. L. Lou, G. Alberti,

S. Vetrella, and A. Cucci (1992), The topsar interferometric radar topographic mapping

instrument., IEEE Transactions on Geoscience and Remote Sensing, 30(5), 933 – 40.

Zebker, H. A., P. A. Rosen, and S. Hensley (1997), Atmospheric effects in interferometric

synthetic aperture radar surface deformation and topographic maps., Journal of Geo-

physical Research, 102(B4), 7547 – 63.

Zisk, S. H. (1972), Lunar topography: first radar-interferometer measurements of the

alphonsus-ptolemaeus-arzachel region., Science, 178(4064), 977 – 80.

A MECHANICAL MODEL OF THE LARGE-DEFORMATION A … · 2007. 9. 4. · Sierra Negra volcano in...

Documents

Transcript of A MECHANICAL MODEL OF THE LARGE-DEFORMATION A … · 2007. 9. 4. · Sierra Negra volcano in...