INSAR TIME SERIES ANALYSIS OF SUBTLE TRANSIENT CRUSTAL DEFORMATION SIGNALS ASSOCIATED WITH THE 2010 SLOW SLIP EVENT AT KILAUEA, HAWAII

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1 INSAR TIME SERIES ANALYSIS OF SUBTLE TRANSIENT CRUSTAL DEFORMATION SIGNALS ASSOCIATED WITH THE 21 SLOW SLIP EVENT AT KILAUEA, HAWAII 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 Jingyi Chen February 214

2 c 214 by Jingyi Chen. All Rights Reserved. Re-distributed by Stanford University under license with the author. This work is licensed under a Creative Commons Attribution-Noncommercial 3. United States License. This dissertation is online at: ii

3 Jingyi Chen 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 Adviser 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. (Paul Segall) 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. (Sigrid Close) Approved for the University Committee on Graduate Studies iii

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5 Abstract We address here the use of Interferometric Synthetic Aperture Radar (InSAR) [Rosen et al., 2; Hanssen, 21] to measure and characterize subtle transient deformation events of the Earth s crust. We develop an imaging geodetic method to identify slow slip events (SSEs) that may go unrecognized because they occur in unexpected areas where instrumentation has not been installed. We illustrate our approach by studying Kilauea volcano and imaging the signature of an SSE that occurred in 21 [Poland et al., 21]. Kilauea is the youngest and most active volcano on the island of Hawaii. Nearly continuous eruptions along Kilauea s east rift zone over 3 years have built up a large area of accumulated lava on the volcano s south flank. Tectonic extension along the rift zone and gravitational spreading lead to the south flank of Kilauea slipping constantly seaward on a shallowly landward dipping basal decollement fault at rates of up to 1 cm/year [e.g., Owen et al., 1995; Owen et al., 2; Shirzaei et al., 213]. Since 22, a sequence of SSEs has been observed on Kilauea s south flank using continuous Global Positioning System (GPS) data [Cervelli et al., 22; Segall et al., 26; Brooks et al., 26; Wolfe et al., 27; Montgomery-Brown et al., 29, 213]. SSEs, viewed as fault activity somewhere between steady sliding and a catastrophic earthquake, release energy over a period of hours to months and can lead to crustal deformation on the order of centimeters. The mechanisms behind these SSEs are still poorly understood. High spatial resolution, accurate SSE displacement measurements can help us constrain the depth of slip and understand the SSEs potential relationship to catastrophic earthquakes and flank failure. The focus of this dissertation is using time series InSAR data to collect and anav

6 lyze subtle, transient deformation. InSAR time series are commonly used to obtain surface topography and surface motion [e.g., Schmidt and Bürgmann, 23; Lanari et al., 24; Hooper, 28]. The benefits of InSAR are fine spatial resolution and broad ground coverage, both compared to measurements using GPS or other geodetic network alone. We use 49 sets of TerraSAR-X data acquired between August, 29 and December, 21 to study the recent Kilauea SSE of February 1, 21. The TerraSAR-X satellite has a revisit cycle of 11 days, which is relatively short compared to most existing spaceborne radar systems. This shorter revisit cycle makes it possible to collect many measurements over a fixed period of time. Moreover, since a phase cycle in a TerraSAR-X interferogram corresponds to only 1.55 cm line of sight (LOS) deformation, the system is well-suited to monitoring ground deformation on the order of centimeters at Kilauea. The challenge in using X-band InSAR time series to study ground deformation at Kilauea is the very low signal to noise ratio (SNR) of the SSE deformation signal compared to atmospheric noise. We develop a small baseline subset InSAR time series analysis algorithm, which jointly inverts InSAR and GPS data to improve the accuracy of the displacement estimates. This algorithm is suitable for extracting both transient and secular ground deformation on the order of millimeters in the presence of atmospheric noise on the order of centimeters. We obtain high spatial resolution displacement estimates due to the 21 slow slip event as well as secular motion at Kilauea and demonstrate that the results are consistent with GPS time series over the same period. We also develop an L1-norm based sparse reconstruction algorithm to detect transient events in very noisy InSAR time series. This algorithm is well-suited to detecting unknown transient events using only InSAR time series, particularly when no auxiliary data such as GPS are available. We apply this algorithm to solve for the time of the SSE s occurrence and confirm that the largest jump detected in the TerraSAR-X InSAR time series is temporally and spatially correlated with the 21 Kilauea SSE. Because phase artifacts due to atmospheric propagation delays in InSAR images frequently degrade the interpretability of the phase signatures of terrain, we further analyze the impact of tropospheric artifacts in InSAR images. We show that trovi

7 pospheric noise is the primary error source in the X-band InSAR data we processed for the study of the 21 Kilauea slow slip event. We also address the impact of ionospheric delay artifacts in InSAR images, which are often seen in L-band interferograms. vii

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9 Acknowledgment I truly appreciate the opportunity of pursuing my doctoral degree at Stanford. I gratefully thank my advisor, Prof. Howard Zebker for his guidance and encouragement since I joined the radar and remote sensing group. He is the best advisor for me and a fun person to work with. I thank my co-advisor, Prof. Paul Segall for his insights about the Hawaii slow slip events. He is also my secondary research project advisor and his ideas always inspire me. I thank Prof. Sigrid Close for her suggestions on my thesis and our discussions about the ionospheric delay study. I also thank Prof. Simon Klemperer and Prof. John Pauly for serving as the members of my defense committee. I have been fortunate to work with my colleagues at Stanford. Past and present Radar Interferometry Group members include Hrefna Gunnarsdottir, Lauren Wye, Piyush Shanker Agram, Albert Chen, Cody Wortham, Jessica Reeves, Jaime Lien, Qiuhua Lin, Tao Chu, Clara Yoon and Jan Stepinski. None of the work is possible without the GPS and InSAR data over Kilauea. The Hawaii GPS time series data are provided by A. Miklius and M. P. Poland from the U. S. Geological Survey. The ALOS data are provided by the Japan Aerospace Exploration Agency and the TerraSAR-X data are provided by the TerraSAR-X team of German Aerospace Center (DLR). I present this thesis to my parents and my husband and thank them for their endless support. Jingyi Chen Stanford, California February 214 ix

10 Contents Abstract Acknowledgment v ix List of Abbrevations 1 1 Introduction Problem Definition Contributions Thesis Roadmap Crustal deformation at Kilauea Geologic setting Slow slip events at Kilauea Overview Slow slip event InSAR observations of surface motion InSAR background Synthetic aperture radar Interferometric synthetic aperture radar Small Baseline Subset (SBAS) Method SAR missions InSAR data processing chain x

11 4 Imaging deformation at Kilauea using InSAR Algorithm Combining InSAR data with two look angles Results Eastward motion at Kilauea Vertical motion at Kilauea Discussion Detecting transient events in time series Algorithm Synthetic Test Solution for SSE occurrence time Limitations Atmospheric errors in InSAR measurements Uncertainty in InSAR deformation measurements Tropospheric artifacts in InSAR data Estimating tropospheric delays in InSAR data using SBAS approach Discussion Ionospheric artifacts in InSAR data Iceland region results California region results Hawaii region Discussion Summary and Conclusions 83 A InSAR azimuth pixel shift 87 B Estimating ionospheric TECV using GPS data 95 C Mapping radio signal propagation paths 11 xi

12 List of Figures 2.1 Map of the eight major islands of Hawaii Map of the south flank of Kilauea Displacement due to the 21 SSE as recorded at 12 GPS sites GPS position time series at GPS sites KAEP and MANE Illustration of the geometry of a spaceborne imaging radar system Illustration of the full synthetic aperture length Illustration of a simplified SAR system InSAR imaging geometry (no ground deformation occurred between two SAR data acquisition times) InSAR imaging geometry (Small ground deformation occurred between two SAR data acquisition times) Illustration of InSAR LOS phase time series at a given ground pixel location Life spans of recent SAR missions that are capable of interferometric applications Time series InSAR data processing chain of the TerraSAR-X software package An TerraSAR-X interferogram with atmospheric artifacts d GPS d InSAR along ascending LOS direction as a function of log(α) InSAR data acquisition geometry with two LOS directions TerraSAR-X images coverage over Hawaii island xii

13 4.5 InSAR solution for the eastward displacement field due to the 21 SSE at Kilauea InSAR and GPS estimates of the eastward SSE displacement at 12 GPS sites InSAR solution for the eastward background velocity field at Kilauea InSAR and GPS estimates of the eastward background velocity InSAR eastward background velocity estimate along the yellow line in the eastward linear velocity map Eastward ground deformation time series as estimated from GPS and InSAR InSAR solution for the vertical displacement field due to the 21 SSE at Kilauea InSAR and GPS estimates of the vertical SSE displacement at 12 GPS sites InSAR and GPS estimates of the vertical SSE displacement at 12 GPS sites. Here we correct the error in InSAR vertical displacement estimate due to the SSE southward motion using GPS data InSAR solution for the vertical background velocity field at Kilauea InSAR and GPS estimate of the vertical background velocity at 12 GPS sites Vertical ground deformation time series as estimated from GPS and InSAR A ground motion model consists of a piecewise constant displacement superimposed on a continuous background displacement. Here we assume the continuous background motion is linear (constant-velocity model), which is a good approximation for the secular motion on Kilauea s south flank based on GPS time series observations shown in Figure Synthetic InSAR time series data (SNR 1) xiii

14 5.3 Ground motion solutions using L1-norm based sparse reconstruction for different values of β Ground motion solutions using L2-norm based Tikhonov regularization for different values of γ TerraSAR-X InSAR time series at the GPS station KAEP as derived from the L1-norm based sparse reconstruction TerraSAR-X InSAR time series at the GPS station KAEP as derived from the L2-norm based Tikhonov regularization Transient jump map as derived from Ascending TerraSAR-X data Transient jump map as derived from Descending TerraSAR-X data Relative tropospheric zenith delays as estimated from ascending InSAR data and GPS data Relative tropospheric zenith delays as estimated from descending In- SAR data and GPS data Standard deviation map of the tropospheric noise time series A comparison between the tropospheric delay field for the time interval from March 3, 21 to January 7, 21 and the 21 SSE displacement field along descending LOS direction An Iceland L-band interferogram with significant ionospheric artifacts Estimated TECV variation based on GPS carrier phase data from permanent GPS station BUDH in Iceland The projections of GPS and ALOS PALSAR signal propagation paths onto the thin-shell ionosphere in ellipsoidal coordinates Spatial TECV variation near the Iceland InSAR scene as estimated from GPS data A California L-band interferogram with no ionospheric artifacts Estimated TECV variation based on GPS carrier phase data from permanent GPS station ORES in California Spatial TECV variation near the California InSAR scene as estimated from GPS data xiv

15 6.12 A Hawaii L-band interferogram with no ionospheric artifacts Estimated TECV variation based on GPS carrier phase data from permanent GPS station GOPM in Hawaii Spatial TECV variation near the Hawaii InSAR scene as estimated from GPS data Spatial TECV variation near the Hawaii scene as estimated from GPS data A.1 Illustration of a satellite signal path through the ionosphere layer.. 88 A.2 Illustration of a signal path from a satellite to the ground A.3 Construction to relate GPS satellite positions over time with ALOS PALSAR satellite positions B.1 WAAS and Self-fitting TECV estimates from a permanent GPS station ORES in California C.1 Illustration of a signal path from a satellite at S(x s, y s, z s ) to the ground at G(x g, y g, z g ) xv

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17 List of Abbrevations ALOS JAXA s Advanced Land Observation Satellite ASI Italian Space Agency Cosmo-SkyMed ASI s Earth observation satellite system CSA Canadian Space Agency ERZ East Rift Zone D-InSAR Differential Interferometric Synthetic Aperture Radar DEM Digital Elevation Model DLR German Aerospace Center ECEF Earth-Centered, Earth-Fixed (a Cartesian coordinate system) Envisat ESA s Environmental satellite ERS-1 ESA s Earth Remote Sensing satellite 1 ERS-2 ESA s Earth Remote Sensing satellite 2 ESA European Space Agency FFT Fast Fourier Transform GPS Global Positioning System HFS Hilina Fault System HS Hilina Slump HVO Hawaiian Volcano Observatory InSAR Interferometric Synthetic Aperture Radar IRI International Reference Ionosphere JAXA Japan Aerospace Exploration Agency LOS Line Of Sight MAI Multi-Aperture Interferometry 1

18 2 MERIS MODIS NASA PRN Radar RADARSAT-1 RADARSAT-2 RMS SAR SBAS SEASAT SLC SNR SRTM SSC SSE SVD TEC TECU TECV TerraSAR-X WAAS Medium Resolution Imaging Spectrometer Moderate Resolution Imaging Spectroradiometer National Aeronautics and Space Administration Pseudo Random Number Radio Detection and Ranging CSA s Earth observation satellite CSA s Earth observation satellite Root Mean Square Synthetic Aperture Radar Small BAseline Subset method NASA s earth-orbiting satellite Single Look Complex (a SAR image format) Signal to Noise Ratio NASA s Shuttle Radar Topography Mission Single Look Slant range Complex (a SAR image format) Slow Slip Event Singular Value Decomposition Total Electron Content TEC units Vertical TEC DLR s radar Earth observation satellite Wide Area Augmentation System

19 Chapter 1 Introduction 1.1 Problem Definition Kilauea is the youngest and most active volcano on the Island of Hawaii. Frequent volcanic activity and associated earthquakes cause significant seaward movement of Kilauea s south flank. A surface expression of such motion is the Hilina Fault System (HFS), a normal fault system consisting of a series of south-facing fault scarps with up to 5 meter surface throws. During the most recent large earthquake (M > 7) in 1975, over 25 km of surface rupture occurred along the Hilina faults with up to a 1.5 meter vertical offset. Large earthquakes can trigger secondary flank failure along the HFS, which could potentially induce huge tsunami waves affecting the entire Pacific Rim [Moore et al., 1989]. Therefore, it is important to monitor and interpret crustal deformation at the HFS in order to reduce the risks from potential earthquakes, landslides and tsunamis. The Hawaiian Volcano Observatory (HVO), the University of Hawaii, and Stanford University jointly operate a network of continuous Global Positioning System (GPS) stations focused on monitoring crustal deformation on the south flank of Kilauea. Cervelli et al. [22] first discovered a transient southeastward event that occurred in November 2 using Kilauea s continuous GPS data. This event was an episode of aseismic fault slip called a slow slip event (SSE). Since the discovery of the 2 SSE, a sequence of slow slip events (SSEs) has been identified on Kilauea s south 3

20 4 CHAPTER 1. INTRODUCTION flank using continuous GPS data [Segall et al., 26; Brooks et al., 26; Wolfe et al., 27; Montgomery-Brown et al., 29, 213]. SSEs are sometimes described as slow earthquakes or (nearly) silent earthquakes and can be interpreted as fault activity somewhere between steady sliding and a catastrophic earthquake. Accurate displacement measurements of SSEs can help us understand the SSEs potential relationship to catastrophic earthquakes and flank failure. While GPS data are widely used to study crustal deformation such as that associated with SSEs, GPS measurements suffer from low spatial resolution. The ground deformation at most locations of interest can only be inferred by interpolating GPS measurements from a few permanent GPS stations. In addition, GPS vertical ground displacement measurements are much noisier than the horizontal measurements due to the GPS data acquisition geometry. As results, the mechanisms behind these SSEs are still poorly understood. Interferometric Synthetic Aperture Radar (InSAR) techniques are commonly used to obtain surface topography and surface motion. InSAR techniques have the advantage of achieving finer spatial resolution and broader ground coverage than is possible with GPS geodetic networks alone. Further, while GPS measurements are more sensitive to horizontal motion, InSAR measurements are more sensitive to eastward and vertical motion. Combining both InSAR and GPS measurements can yield a more accurate estimate of the displacement due to SSEs. The challenge in using InSAR data to study SSE ground deformation at Kilauea is the very low signal to noise ratio (SNR), as the expected centimeter-level SSE phase signature is obscured by significant atmospheric artifacts on the order of centimeters or even greater. The focus of this dissertation is to develop new InSAR time series analysis techniques suitable for detecting episodic transient crustal deformation signals with very low signal to noise ratios (SNR 1) such as the SSEs at Kilauea. 1.2 Contributions The main contributions of this dissertation focus on the development of new InSAR time series analysis techniques for detecting episodic crustal deformation signals that are below the noise level of individual interferograms. Much of this work concerns the

21 1.2. CONTRIBUTIONS 5 application of these methods to the solution for crustal deformation due to the 21 slow slip event at Kilauea, Hawaii. A later chapter also describes the errors due to tropospheric and ionospheric delays in InSAR deformation measurements and their impact on InSAR image quality. Specifically, we have made the following contributions: We have developed an efficient and robust TerraSAR-X InSAR software package using a motion-compensation processing algorithm. We reduced errors introduced in multiple InSAR data processing steps by improving the resampling schemes and the geocoding algorithm. We have developed an InSAR time series analysis algorithm using a joint GPS- InSAR inversion approach to solve for a sub-centimeter transient jump superimposed on a continuous background motion in the presence of atmospheric noise on the order of centimeters. We apply this algorithm to study the 21 slow slip event as well as secular motion at Kilauea, Hawaii using TerraSAR-X InSAR data. The InSAR ground deformation time series we reconstruct are consistent with GPS measurements over the same time period. We have developed an InSAR time series analysis algorithm using a sparse reconstruction approach to detect the time of occurrence of the largest jump in a noisy InSAR time series. We successfully applied this algorithm to detect the time of occurrence of the 21 slow slip event at Kilauea using TerraSAR-X InSAR data. We have developed an algorithm to estimate tropospheric delays in InSAR images using a small baseline approach. The InSAR relative atmospheric zenith delay estimates are strongly correlated with independent GPS atmospheric zenith delay estimates for the same time period. We conclude that it is possible to use InSAR data to reconstruct high-resolution atmospheric noise maps. We have developed a method to measure the spatial ionospheric total electron content (TEC) variation at synthetic aperture length scales using multiplesatellite dual frequency GPS carrier phase data. We relate the gradient of the

22 6 CHAPTER 1. INTRODUCTION ionospheric TEC observed by GPS data to the misregistration of complex pixels seen in the L-band ALOS interferograms. We confirm the cause of the misregistration artifacts in InSAR images is due to dispersive ionospheric propagation rather than other decorrelation factors such as neutral atmospheric delays. 1.3 Thesis Roadmap In Chapter 2, we provide scientific background for the study of crustal deformation at Kilauea. We introduce the geologic setting of Kilauea, Hawaii. It is important to monitor crustal deformation on the south flank of Kilauea as it is closely related to potential catastrophic earthquakes, landslides and tsunamis. We review previous studies of a sequence of recent slow slip events (SSEs) observed on the south flank of Kilauea. As these SSEs have similar displacement patterns, we focus on one of the large events that occurred on 1-2 February 21. We summarize existing GPS observations of the 21 SSE and discuss the advantages of using InSAR data to study this event. In Chapter 3, we provide the technical background for interferometric synthetic aperture radar (InSAR) techniques. We first summarize the basics of synthetic aperture radar (SAR). We then explain how the phase difference between two SAR images over the same area relates to the actual ground deformation in the radar line of sight direction. We further describe how to monitor the temporal evolution of surface deformation using a set of coherent InSAR images. We also provide a list of the most important SAR missions and use data from the German Aerospace Center s (DLR) TerraSAR-X mission to demonstrate how InSAR data are processed. In Chapter 4, we present a small baseline subset InSAR time series analysis algorithm. This algorithm is well-suited to extracting both transient and secular ground deformation on the order of millimeters in the presence of atmospheric noise on the order of centimeters. We apply this algorithm to estimate the displacement field due to the 21 slow slip event as well as secular motion at Kilauea. We process 49 sets of X-band TerraSAR-X data over Kilauea and reconstruct the eastward and vertical displacement field at Kilauea with high spatial resolution. We demonstrate the InSAR ground deformation estimates are consistent with GPS time series over the

23 1.3. THESIS ROADMAP 7 same period. In Chapter 5, we propose an L1-norm based sparse reconstruction algorithm to detect transient events in very noisy InSAR time series. We employ a synthetic data set to demonstrate the difference between the L1-norm based sparse reconstruction algorithm and the traditional L2-norm based Tikhonov regularization. We apply this algorithm to solve for the time of the SSE s occurrence using the same TerraSAR-X data set as in Chapter 4. We confirm that the largest jump detected in the TerraSAR- X InSAR time series is temporally and spatially correlated with the 21 SSE. In Chapter 6, we discuss error sources in InSAR deformation measurements and error propagation in the SBAS InSAR time series analysis. We focus on estimating tropospheric delays in InSAR images, as it is the primary error source in the X-band InSAR data we processed for the study of the 21 Kilauea slow slip event. We also address the impact of ionospheric delay artifacts in InSAR images, which are often seen in L-band interferograms in high latitude regions. Finally, we make some concluding remarks and suggest areas of future work in Chapter 7.

24 8 CHAPTER 1. INTRODUCTION

25 Chapter 2 Crustal deformation at Kilauea, Hawaii In this chapter, we provide the scientific background for our crustal deformation study of Kilauea, Hawaii. We first introduce the geologic setting of Kilauea, a volcano on the island of Hawaii. We then review previous studies of a sequence of recent slow slip events (SSEs) observed on the south flank of Kilauea. We summarize existing GPS observations of the 21 SSE and state the motivation to study the 21 SSE using InSAR data. 2.1 Geologic setting The Hawaiian Islands are an archipelago of eight major islands, several atolls, many islets, and undersea seamounts that are part of the 36 mile long Hawaiian-Emperor seamount chain. The formation of the Hawaiian Islands is closely related to volcanic activity. An undersea magma source (a hotspot) in the northern Pacific Ocean near Hawaii has been slowly creating new volcanoes. As the Pacific plate moves to the northwest relative to the hotspot, these volcanoes move away from their magma source, become inactive and eventually erode below the sea over millions of years. Figure 2.1 shows a map of the eight major islands of Hawaii. The largest island is located at the southeast and is called Hawaii (the archipelago s namesake) or the Big Island. It is built from five separate shield volcanoes, a type of volcano built almost 9

26 1 CHAPTER 2. CRUSTAL DEFORMATION AT KILAUEA entirely of fluid lava flows. Due to the location of the hotspot, the only three currently active volcanoes (Hualalai, Mauna Loa and Kilauea) are located in the southern half of the Big Island. There is also a new active submarine volcano called the Loihi Seamount located south of the Big Island s coast. Niihau Kauai Oahu Molokai Lanai Maui Kahoolawe Mauna Loa Hawaii Kilauea Loihi Figure 2.1: Map of the eight major islands of Hawaii. The only two active volcanoes (Mauna Loa and Kilauea) are located in the southern half of the Big Island. Kilauea, the youngest and most active volcano on the Big Island is shown in Figure 2.2. It has a large summit caldera and two rift zones, in which a series of fissure vents allow lava to erupt with little explosive activity. Currently, the east rift zone is the most active region on the volcano. Kilauea has been erupting continuously from the Puu O o vent for the last 3 years. Due to tectonic extension along the two rift zones and gravitational spreading, the south flank of Kilauea constantly slips seaward on a shallowly landward-dipping basal decollement fault at rates of up to 1 cm/year [e.g.,

27 2.1. GEOLOGIC SETTING 11 Owen et al., 1995; Owen et al., 2; Shirzaei et al., 213]. The basal decollement fault, a gliding plane at the interface of the volcanic pile and the pre-volcanic sea floor, is located 7 to 8 km beneath the volcanos surface. Previous studies suggest that the majority of seismic activities occur in the vicinity of the decollement fault [Denlinger and Okubo, 1995; Got and Okubo, 23]. Rift Zone Kilauea East Koae Fault System Rift PuuOo Zone Southwest Hilina Fault System Figure 2.2: Map of Kilauea. The Koae Fault System and the Hilina Fault System are marked with blue lines. The two rift zones are marked in yellow. The south flank is defined as the portion of Kilauea to the south of Kilauea s caldera and its two rift zones. The most prominent structure on Kilauea s south flank is the Hilina Fault System (HFS), an active normal fault system consisting of a series of south-facing fault scarps with up to 5 meter surface throws [Parfitt and Peacock, 21]. The HFS is a surface

28 12 CHAPTER 2. CRUSTAL DEFORMATION AT KILAUEA expression of historic block movement of Kilauea s south flank and is closely related to volcanic activity and associated seismic activity. Seismic imaging data suggest that the Hilina faults extend possibly as deep as the basal decollement [Okubo et al., 1997]. However, the potential relationships and interactions between the HFS and the basal decollement is still unknown. Recurrent large earthquakes (M > 7) are expected every 8-26 years at Kilauea [Cannon and Bürgmann, 21]. Large earthquakes can trigger flank failure along the HFS [Moore et al., 1989], which can cause huge tsunami waves that could affect the entire Pacific Rim. The most recent large earthquake (M > 7) was the 1975 Kalapana Earthquake, accompanied by tsunami waves up to 15 meters in height. Over 25 km of surface rupture occurred along the Hilina faults with up to a 1.5 meter vertical offset [Cannon and Bürgmann, 21]. The earthquake and associated tsunami caused 2 deaths, 28 injuries, and about 4.1 million dollars in damages. A similar earthquake and associated tsunami today could potentially cause even larger losses. Although there has been no significant fault activity in this region since 1975, it is important to monitor and interpret crustal deformation at Kilauea in order to reduce the risks from potential earthquakes, landslides and tsunamis. The unique geological setting of the HFS is also of great interest to geoscientists studying volcanic activity and its relationship to earthquakes and landslides. 2.2 Slow slip events at Kilauea Overview Cervelli et al. [22] first recognized a transient southeastward displacement occurring on the south flank of Kilauea in November 2 using continuous Global Positioning System (GPS) data. The event lasted about 36 hours and resulted in a maximum 1.5 cm seaward displacement. This event was an episode of aseismic fault slip called a slow slip event (SSE). Slow slip events (SSEs) can be interpreted as fault activity somewhere between steady sliding and a catastrophic earthquake. Since the discovery of the 2 SSE at Kilauea, a sequence of SSEs has been identified on Kilauea s south flank. Segall et al. [26] reported three SSEs on 2-21

29 2.2. SLOW SLIP EVENTS AT KILAUEA 13 September 1998, 3-4 July 23, and January 25, all similar to the 2 event. The SSEs at Kilauea are accompanied by swarms of small earthquakes located at depths of 7-8 km [Segall et al., 26; Wolfe et al., 27], constraining the location of the SSEs to the basal decollement [Segall et al., 26]. Brooks et al. [26] analyzed 8 years of continuous GPS data from the Hilina slump (HS) and identified periodic SSEs between 1998 and 25. All the SSEs have similar durations and displacement patterns, suggesting that they have a common source. Montgomery-Brown et al. [29] compared displacement fields on Kilauea s south flank with displacement patterns in previously identified slow slip events. Matching displacement patterns were found for several new smaller candidate events between 1997 and 27. Brooks et al. [28] noted a dike intrusion happened 15 to 2 hours before the 27 slow slip event, suggesting that the intrusion triggered the 27 SSE. The most recent 21 and 212 SSEs [Poland et al., 21; Montgomery-Brown et al., 213] are similar to the previous large events in 1998, 2, 25 and 27. However, no corresponding dike intrusion was associated with either of these two events. Unlike many SSEs previously observed at plate boundary subduction zones, such as the Bungo Channel in southwestern Japan, Cascadia on the western coast of the USA and Canada, and Guerrero in southern Mexico [e.g., Hirose et al., 1999; Dragert et al., 24; Larson et al., 24], Montgomery-Brown et al. [213] reported that no associated tectonic tremor was detected during Kilauea s SSEs, suggesting that these SSEs reflect somewhat distinct slip processes than the SSEs at plate boundary subduction zones. It has been over 1 years since the discovery of Kilauea s SSEs. However, the mechanisms behind these SSEs are still poorly understood. High spatial resolution, accurate SSE displacement measurements can help us understand the SSEs potential relationship to catastrophic earthquakes and flank failure. It is the intent of this research to use time series InSAR data to collect and analyze these measurements Slow slip event In this section, we review existing GPS observations of one recent SSE, which started on February 1, 21. This event was first detected using continuous GPS data, lasted

30 14 CHAPTER 2. CRUSTAL DEFORMATION AT KILAUEA at least 36 hours, and resulted in a maximum 3-cm displacement superimposed on a long-term seaward background motion [Poland et al., 21]. The 21 SSE was accompanied by a swarm of earthquakes, similar to prior large SSEs at Kilauea. Latitude KOSM PGF1 PGF5 PGF6 OUTL MANE GOPM PGF3 KTPM PG2R PGF4 KAEP cm Longitude Figure 2.3: Map of Kilauea, Hawaii. The red arrows show the horizontal displacements for a period of 36 hours spanning the 21 SSE as recorded at the indicated GPS sites. Figure 2.3 shows the horizontal displacements due to the 21 SSE as recorded at the indicated GPS sites. Here we estimate the magnitude of the SSE from the daily GPS position time series at each GPS station. As an example, Figure 2.4 (left) shows the east, north and up components of the daily GPS position at the coastal GPS station KAEP from mid-29 until the end of 21 (in black). The transient SSE signal can be seen superimposed on an approximately linear seaward background motion. We fit two lines (in red) to the GPS data before and after the SSE. The magnitude of the SSE can be inferred by the jump immediately before and after the event. Note that the GPS vertical measurements are noisier than the horizontal measurements due to the GPS data acquisition geometry [Misra and Enge, 26]. Figure 2.4 (right) shows the east, north and up components of the daily GPS position at another station MANE over the same time period. The magnitude of the SSE is much smaller at this station compared to the coastal station KAEP.

31 2.3. INSAR OBSERVATIONS OF SURFACE MOTION 15 east (cm) north (cm) up (cm) time time KAEP time east (cm) north (cm) up (cm) time time MANE time Figure 2.4: Daily GPS position observed at the GPS station KAEP (left) and MANE (right) from mid-29 until the end of 21 (in black). We fit two lines (in red) to the data before and after the slow slip event. The magnitude of the SSE can be inferred by the jump immediately before and after the event. 2.3 InSAR observations of surface motion GPS measurements are widely used to study crustal deformation such as that associated with SSEs. Similar to GPS techniques, Interferometric Synthetic Aperture Radar (InSAR) techniques are also commonly used to obtain surface topography and surface motion [e.g., Massonnet et al., 1993; Fialko et al., 22]. InSAR techniques have the advantage of achieving finer spatial resolution and broader ground coverage than is possible with the GPS geodetic network alone. Further, while GPS measurements are more sensitive to horizontal motion, InSAR measurements are more sensitive to eastward and vertical motion, as shown in Section 4.2. Combining both InSAR and GPS measurements can yield a more accurate estimate of the displacement due to SSEs, which can help to constrain the depth of slip and quantify the

32 16 CHAPTER 2. CRUSTAL DEFORMATION AT KILAUEA potential hazards that SSEs may trigger. In this dissertation, we focus on developing new InSAR time series techniques to solve for the displacement field due to the 21 SSE and secular background motion at Kilauea. The method we developed can be used to isolate a centimeter-level transient jump superimposed on any continuous background motion, including but not limited to deformation due to other SSEs at Kilauea. We begin by describing the InSAR technique, its advantages and limitations, in Chapter 3.

33 Chapter 3 InSAR background In this chapter, we provide the technical background for interferometric synthetic aperture radar (InSAR) techniques. We first summarize the basics of synthetic aperture radar (SAR). We then explain how the phase difference between two SAR images over the same area relates to the actual ground deformation in the radar line of sight direction. We further describe how to monitor the temporal evolution of surface deformation using a set of coherent InSAR images. We also provide a list of the most important SAR missions and use data from the German Aerospace Center s (DLR) TerraSAR-X mission to demonstrate how InSAR data are processed. 3.1 Synthetic aperture radar The word radar is an acronym for radio detection and ranging. A radar instrument determines distance by measuring the 2-way travel time of a radio wave between the radar antenna and a target. Figure 3.1 illustrates the geometry of a spaceborne imaging radar system. The radar instrument is mounted on a satellite at height h moving along the satellite track (azimuth direction) with a velocity v. Periodically, the radar transmitter emits a pulse, which is often a linearly-swept frequency radio wave called a chirp. The radar pulse propagates along the line of sight (LOS) direction and illuminates a swath on the ground of width w. The radar antenna later receives echoes reflected from scatterers on the illuminated surface. The echo arrival time depends on the travel distance between the radar instrument and a given scatterer. 17

34 18 CHAPTER 3. INSAR BACKGROUND Radar Antenna v Satellite Track h LOS direction Azimuth Direction w Swath Range Direction Figure 3.1: Illustration of the geometry of a spaceborne imaging radar system. The radar instrument is mounted on a satellite at height h moving along the satellite track with a velocity v. We define the range direction as the across track direction and azimuth as the along track direction. The power of the received echo can be estimated as P r using the radar equation: P r = P tg t A r A scat σ (4π) 2 R 4 (3.1) where P t is the transmitter power, G t is the gain of the transmitting antenna, A r is the effective area of the receiving antenna, A scat is the area of the scatterer, σ is the normalized radar cross section of the target, and R is the distance between the transmitter and the target. The signal to noise ratio (SNR) of an imaging radar system depends on the energy of the transmitted pulse, that is, the product of the radar peak power and the pulse length. To increase the SNR, we want to transmit very long pulses. However, the slant range resolution, which is the minimum distance along the LOS direction at which two scatterers are distinguishable, is also proportional to the pulse length. To achieve fine resolution, we want to transmit very short pulses. One way to address this dilemma

35 3.1. SYNTHETIC APERTURE RADAR 19 is to design a filter to optimally discriminate the radar echo from the background noise. Such an optimal filter (in the least squares sense) is called a matched filter, which has a response conjugate to the transmitted radar chirp signal [Curlander and McDonough, 1991, Chapter 3]. Compressing received radar echos using a matched filter is also called range modulation or range coding, which can greatly improve the performance of a radar system. As the satellite travels along its orbital track, the radar transmitter emits pulses at fixed time intervals. A series of return echoes builds up an image of the 2D surface, where the brightness of the image corresponds to the radar reflectivity of scatterers on the ground. For a real aperture radar system, the azimuth resolution is determined by the azimuth beamwidth w, as illustrated in Figure 3.1. Based on antenna theory, the azimuth beamwidth w can be written as: w = Rλ l (3.2) where R is the range to the ground from the radar satellite, λ is the radar wavelength, and l is the antenna length. Equation (3.2) shows that the azimuth resolution of a real aperture radar system is limited by the size of the antenna that can be mounted on the radar platform. Supposing we have an L-band (λ =.24 m) real aperture system with R = 8 km and l = 1 m, the corresponding azimuth beamwidth w = 19.2 km. Usually, the resolution of spaceborne real aperture radar systems is too coarse for most practical uses. Synthetic aperture radar (SAR) [Curlander and McDonough, 1991; Cumming and Wong, 25] is a more complex type of imaging radar system. As the radar satellite moves along the satellite track, echoes reflected from a ground target are received by the radar antenna at different azimuth positions as shown in Figure 3.2. SAR distinguishes target locations along the azimuth direction using the Doppler effect induced by the relative motion between the radar antenna and the ground. The received echoes are summed in the SAR processor after correction for the ideal phase history of the target. The resulting azimuth resolution of this system depends on the

36 2 CHAPTER 3. INSAR BACKGROUND a full synthetic aperture moving antenna target Figure 3.2: Illustration of the full synthetic aperture length. v x γ z R y Figure 3.3: Illustration of a simplified SAR system. We define x as the along track direction, y as the across track direction and z as the vertical direction. The range between the radar satellite at (,, z) and a given pixel P at (x, y, ) is R = x 2 + y 2 + z 2. The radar satellite velocity is v = (v x,, ). γ is the radar satellite look angle and θ is the radar incidence angle. x θ P length of the full synthetic aperture rather than the actual size of the radar antenna. Figure 3.3 shows a simplified SAR system. Here γ is the radar satellite look angle and θ is the radar incidence angle. We define x as the along track direction, y as the across track direction and z as the vertical direction. Let the radar satellite move in the azimuth (along track) direction with a velocity vector v = (v x,, ). In this illustration, we neglect the Earth s curvature because it introduces an effective velocity change much smaller than v x. The relative movement between the satellite at

37 3.2. INTERFEROMETRIC SYNTHETIC APERTURE RADAR 21 (,, z) and a given pixel P at (x, y, ) leads to the well-known expression for Doppler frequency f d : f d = 2 λ v x x R (3.3) where λ is the signal wavelength and R is the range between the radar satellite and the given pixel P. Based on Equation (3.3), the SAR processor resolves an azimuth location x in the image as a function of f d : x = f dλr 2v x (3.4) The phase history of an azimuth point scatterer forms a chirp signal similar to what we used for the range modulation [Cumming and Wong, 25, Chapter 4]. A corresponding azimuth matched filter is expressed as a function of Doppler frequency. To process the azimuth information and obtain the finest possible resolution, we can simply apply an azimuth matched filter identically to the range case. In practice, range and azimuth compression can be efficiently implemented using a Fast Fourier transform (FFT). 3.2 Interferometric synthetic aperture radar Interferometric synthetic aperture radar (InSAR) [Rosen et al., 2; Hanssen, 21] techniques use two or more SAR images over the same region to obtain surface topography or surface motion. In this section, we explain how an InSAR phase measurement relates to actual ground deformation. Figure 3.4 illustrates the InSAR imaging geometry. At time t 1, a radar satellite emits a pulse at S 1, then receives an echo reflected from a ground pixel A and measures the phase φ 1 of the received echo. Note that all scatterers within the associated ground resolution element contribute to φ 1. As a result, the phase φ 1 is a statistical quantity that is uniformly distributed over interval (, 2π) and we cannot directly use φ 1 to infer the distance r 1 between S 1 and A. Later at time t 2, the satellite emits

38 22 CHAPTER 3. INSAR BACKGROUND S 1 S 2 r 2 z r 1 A Ground Figure 3.4: Illustration of InSAR imaging geometry. The distance between the satellite at S 1 and a ground pixel A is r 1 and the distance between the satellite at S 2 and the ground pixel A is r 2. The topographic height of the pixel A is z. Here we assume r 1 r 2 << r 1 (the parallel-ray approximation) and no ground deformation occurs at pixel A between the two SAR data acquisition times. another pulse at S 2 and makes a phase measurement φ 2. If the scattering property of the ground resolution element has not changed since t 1, all scatterers within the resolution element contribute to φ 2 the same way as they contribute to φ 1. Under the assumption that r 1 r 2 << r 1 (the parallel-ray approximation), the phase difference between φ 1 and φ 2 can be used to infer the topographic height z of the pixel A [Hanssen, 21, Section 3.2]. If we know the topographic height z, we can further measure any small ground deformation occurring at pixel A between t 1 and t 2. Figure 3.5 illustrates the InSAR imaging geometry in this case. At time t 1, a radar satellite measures the phase φ 1 between the satellite and a ground pixel A along the LOS direction. Later at time t 2, the ground pixel A moves to A and the satellite makes another phase measurement φ 2 between the satellite and the ground pixel. After removing the known phase φ due to the surface topography, the unwrapped InSAR phase φ = φ 2 φ 1 φ is proportional to the ground deformation d between t 1 and t 2 along the satellite LOS

39 3.2. INTERFEROMETRIC SYNTHETIC APERTURE RADAR 23 LOS direction ϕ 1 ϕ 2 A Ground at t 1 Ground at t 2 A' Δϕ Figure 3.5: Illustration of InSAR imaging geometry. At time t 1, a ground pixel of interest is at point A and a radar satellite measures the phase φ 1 between the satellite and the ground pixel along the LOS direction. Later at time t 2, the ground pixel moves to A and the satellite makes another measurement φ 2 between the satellite and the ground pixel. The phase difference φ is proportional to the ground deformation between t 1 and t 2 along the LOS direction. direction as: φ = φ 2 φ 1 φ = 4π λ d (3.5) where λ is the radar wavelength. Equation (3.5) assumes that there is no error in the InSAR phase measurement. In Chapter 6, we will discuss in depth various error sources in InSAR deformation measurements and their impact on InSAR image quality. Note that InSAR techniques only measure one-dimensional LOS motion. However, deformation is better characterized in three dimensions: east, north and up. Given an LOS direction unit vector e = [e 1, e 2, e 3 ], we can project the deformation in east,

40 24 CHAPTER 3. INSAR BACKGROUND north and up coordinates along the LOS direction as: d = e 1 d east + e 2 d north + e 3 d up (3.6) Because radar satellites are usually polar orbiting, the north component of the LOS unit vector e 2 is often negligible relative to the east and vertical components. When InSAR measurements along two or more LOS directions are available, we can combine multiple LOS deformation measurements over the same region to separate the east and vertical ground motions, given that the term e 2 d north is negligible. 3.3 Small Baseline Subset (SBAS) Method Berardino et al. [22] proposed the Small BAseline Subset (SBAS) algorithm for monitoring the temporal evolution of surface deformation. In this section, we review how we can use this algorithm to solve for the InSAR phase time series at a ground pixel location. Figure 3.6: An LOS phase history we want to solve for at a given ground pixel location. The phase in a single interferogram formed from SAR data acquired at t m and t n measures φ i.

41 3.3. SMALL BASELINE SUBSET (SBAS) METHOD 25 Suppose we want to solve for the unknown LOS phase history φ(t) at a ground pixel as shown in Figure 3.6. The i th interferogram formed from two sets of SAR data acquired at t m and t n measures the unwrapped phase φ i at this pixel. Assuming the topography-related phase has been removed, we can write φ i as: n 1 φ i = (t l+1 t l )v l (3.7) l=m where v l is the unknown mean LOS velocity (in radians/day) between SAR acquisition times t l and t l+1. An interferogram can measure φ i in Equation (3.7) with negligible topographic or spatial decorrelation error only if the baseline (the spatial separation between the satellite locations at two SAR acquisition times) is small [Zebker and Villasenor, 1992]. The interferogram correlation becomes zero when the baseline component perpendicular to the LOS direction (perpendicular baseline) is larger than the critical baseline B c, defined as: B c = Rλ 2 cos θδy (3.8) where R is the range between the radar satellite and the ground pixel, λ is the signal wavelength, θ is the radar incidence angle. The ground range resolution δy is the minimum separation between two distinguishable targets in the range direction, defined as: δy = cτ 2 sin θ (3.9) where c is the speed of light and τ is the radar pulse length. Note that an interferogram with a perpendicular baseline smaller than B c may still suffer from decorrelation artifacts due to other factors such as temporal decorrelation due to, for instance, the presence of vegetation. In practice, we process all possible InSAR pairs with perpendicular baselines smaller than B c and discard images with low correlation.

42 26 CHAPTER 3. INSAR BACKGROUND Given N SAR images forming M interferograms with small baselines, we can define a matrix representation of the M equation SBAS system as: Bv = Φ (3.1) where v = [ v 1... v N 1 ] T is the vector of unknown mean velocities between each consecutive SAR acquisition and Φ = [ φ 1... φ M ] T is the vector of known values of the M interferograms at the given pixel. The M (N 1) matrix B is the SBAS matrix. If the i th interferogram measures the InSAR phase φ i between SAR data acquisition times t m and t n as defined in Equation (3.7) at the pixel of interest, the i th row of B has t l = (t l+1 t l ) in the l th entry for l = m,..., (n 1) and zeros in the remaining entries. The vector v in Equation (3.1) can be approximated as: v = B Φ (3.11) where B is any generalized inverse of B. In this dissertation, we choose B as the Moore-Penrose pseudoinverse, which can be efficiently computed using the singular value decomposition (SVD). We define the SVD of B as: B = UΣW T (3.12) where Rank(B) = r, U = [u (1)... u (r) ], W = [w (1)... w (r) ] and Σ = diag(σ 1,..., σ r ) with σ 1... σ r >. u (i) are the left singular vectors of B, w (i) are the right singular vectors of B, and σ i are the singular values of B. The Moore-Penrose pseudoinverse B is: B = W Σ 1 U T (3.13) Solving Equation (3.11) gives the minimum-norm least squares approximate solution of the Equation (3.1). An additional integration of v over time yields the LOS phase history φ(t) at the pixel of interest. Note that the minimum norm constraint

43 3.4. SAR MISSIONS 27 for the velocity v implies that there are no large discontinuities in the final solution φ(t). Moreover, we can easily incorporate additional ground deformation model in the presented SBAS formulation. For example, by assuming a constant velocity ground deformation model, we can write the SBAS system as: BP v c = Φ (3.14) where the M (N 1) matrix B and the M 1 vector Φ are defined in Equation (3.1). P is an (N 1) 1 vector of ones, [ ] T. The scalar v c represents the unknown constant velocity of the background motion in the LOS direction. It is also possible to use higher-order polynomial models such as a cubic deformation model. In this model, we can write the SBAS system as: BMp = Φ (3.15) where the 3 1 unknown vector p = [v c a a]. Here v c represents the mean velocity, a represents the mean acceleration and a represents the mean acceleration variation. The (N 1) 3 matrix M can be written as: 1 t 1 2 t t 1 2 t 2 2 M = t N 1 2 t N (3.16) where t l = (t l+1 t l ) for l = 1,..., (N 1) and t l is the l th SAR acquisition time. 3.4 SAR missions Figure 3.7 shows the life spans of recent SAR missions that are capable of interferometric applications. The solid lines indicate past operational service periods and the dashed lines indicate expected future operational service periods. In this dissertation, we analyze SAR data from the X-band TerraSAR-X satellite

44 28 CHAPTER 3. INSAR BACKGROUND Figure 3.7: Life spans of recent SAR missions that are capable of interferometric applications. The solid lines indicate past operational service periods and the dashed lines indicate expected future operational service periods (By courtesy of Lin Liu). and the L-band ALOS satellite. TerraSAR-X is a German Earth observation satellite which has been operational since January, 28. We use 49 sets of TerraSAR-X data acquired between August, 29 and December, 21 to study the 21 slow slip event and secular deformation at Kilauea, Hawaii. The TerraSAR-X satellite has a revisit cycle of 11 days, which is relatively short compared to most existing spaceborne radar systems. This shorter revisit cycle makes it possible to collect more measurements over a fixed period of time. Moreover, since a phase cycle in a TerraSAR-X interferogram corresponds to only 1.55 cm line of sight (LOS) deformation, the system is well-suited to monitoring ground deformation on the order of centimeters at Kilauea. Also, as ionospheric delay is proportional to the square of the radar wave-

45 3.5. INSAR DATA PROCESSING CHAIN 29 length, X-band spaceborne radar systems (λ 3 cm) suffer from fewer ionospheric artifacts than many presently operating C-band (λ 6 cm) and L-band (λ 24 cm) spaceborne radar systems. In Chapter 6, we also use the same TerraSAR-X data set to study centimeter-level atmospheric noise during the SAR data acquisition time. The Advanced Land Observation Satellite (ALOS) is a Japanese satellite launched in 26. PALSAR is a L-band SAR instrument mounted on the ALOS satellite. It was operational for 5 years with a revisit cycle of 46 days and a wavelength of 24 cm. Because InSAR images with longer wavelengths suffer from more ionospheric artifacts, we use ALOS data to study the impact of the ionospheric delays in InSAR images in Chapter InSAR data processing chain We also developed an efficient and robust TerraSAR-X InSAR software package using a motion-compensation algorithm [Zebker et al., 21]. In this section, we use this software package as an example to demonstrate InSAR data processing. Figure 3.8 illustrates the InSAR data processing chain. The input data are raw TerraSAR-X single look slant range complex (SSC) image files. We first reformat the raw SSC files to single look complex (SLC) slant range SAR images. The header files are also parsed and stored in a radar parameter database. We then resample all the SLC images to common radar coordinates. As each interferogram is acquired in a slightly different geometry, SAR images are stretched and shifted with respect to each other. The resampling step is necessary in order to coregister all SAR images. We next form interferograms by a point-wise complex multiplication of each pair of SAR images. Multiple looks are computed in the range and azimuth directions in order to reduce the image size and improve coherence. During this step, interferograms with low coherence are discarded. The InSAR phase we compute is related to both surface topography and surface deformation between the two SAR data acquisition times. Since we are interested in measuring surface deformation, the topographyrelated phase needs to be removed. We use digital elevation model (DEM) data from the Shuttle Radar Topography Mission (SRTM) along with radar satellite position files to estimate the expected topography-related phase in the interferogram. This

46 3 CHAPTER 3. INSAR BACKGROUND Figure 3.8: Time series InSAR data processing chain of the TerraSAR-X software package. phase is then removed from the interferogram during the topocorrect step. relate the InSAR phase to the LOS surface deformation as described in Equation (3.5), a phase unwrapping step [Chen and Zebker, 2, 22] is needed to solve for the unknown integer ambiguity in the InSAR phase measurement. Note that radar images are presented in radar coordinates and the two axes are the range and azimuth directions. In order to compare InSAR measurements with other ground deformation measurements such as GPS time series, we resample the unwrapped interferograms into ellipsoidal coordinates (latitude, longitude) in the geocoding step before we apply the SBAS algorithm to the InSAR data. To

47 Chapter 4 Imaging deformation at Kilauea using InSAR data In this chapter, we present an SBAS InSAR time series analysis algorithm to extract both transient and secular ground deformation on the order of sub-centimeters in the presence of atmospheric noise on the order of centimeters. We apply this algorithm to study the 21 slow slip event as well as secular motion at Kilauea, Hawaii. The InSAR ground deformation time series we reconstruct are consistent with GPS time series over the same period. 4.1 Algorithm The SBAS algorithm has been successfully used to monitor ground motion in many Earth science applications [e.g., Schmidt and Bürgmann, 23; Lanari et al., 24; Hooper, 28]. The challenge in using this algorithm to study ground deformation at Kilauea is the very low signal to noise ratio (SNR) of the SSE deformation compared to neutral atmospheric delays. For example, Figure 4.1 is a single interferogram formed from two sets of ascending TerraSAR-X data acquired on January 7, 21 and March 3, 21 over the Kilauea area. A phase cycle of 2π in the interferogram corresponds to 1.55 cm LOS deformation. Phase values are wrapped. Based on GPS data, the February 1, 21 Kilauea SSE leads to an approximately 7mm maximum deformation signal along the TerraSAR-X ascending LOS direction and we expect 31

48 32 CHAPTER 4. IMAGING DEFORMATION AT KILAUEA USING INSAR to observe a phase signature due to the SSE in the red circled area. However, since atmospheric noise is typically on the order of centimeters or even greater in the Kilauea area [Foster et al., 26], the expected SSE phase signature is obscured by significant visual artifacts. Azimuth Range Figure 4.1: A single interferogram formed from ascending TerraSAR-X data acquired on January 7, 21 and March 3, 21. The horizontal axis corresponds to the radar range direction and the vertical axis corresponds to the radar azimuth direction. A phase cycle 2π corresponds to a 1.55 cm LOS deformation. We expect to observe a phase signature due to the SSE in the red circled area, but the expected SSE phase signature is obscured by significant visual artifacts due to atmospheric noise. As the SSE signal is below the noise level, traditional SBAS results may not accurately represent the ground motion if some effort is not made to handle or suppress the atmospheric noise signature. We employ a simple ground deformation model consisting of an offset at the time of the SSE superimposed on a constant-velocity background motion. To prevent overfitting in particularly noisy regions, we formulate our problem as a joint GPS-InSAR inversion and solve for the phase history at all interferogram pixels simultaneously with a spatial smoothness constraint on the offset estimate. Smoothing of the background velocity is not necessary as the linear background motion model tolerates random noise well. In the rest of this section, we demonstrate how this algorithm works. First, we

49 4.1. ALGORITHM 33 define the pixel at i th row and j th column in a 2D interferogram as the k th pixel, where k = [(i 1)n r + j] and n r is the number of rows in the interferogram. Given N SAR images forming M small baseline interferograms, we can solve for the unknown constant background velocity v (k) (in radians per day) and jump δ (k) (in radians) due to the SSE in the LOS direction at the k th pixel by minimizing the squared error as: G (k) m (k) Φ (k) 2 2 (4.1) Here the M 2 matrix G (k) = [ s c ]. The linear SBAS vector s = BP, where the M (N 1) matrix B and (N 1) 1 vector P are given by Equation (3.14). The M 1 vector c has a one in the l th entry if the l th interferogram spans the SSE. Otherwise, the l th entry of c equals zero. The 2 1 vector m (k) = [ v (k) δ (k) ] T unknown and the M 1 vector Φ (k) = [ φ (k) 1... φ (k) M ] T contains the phase values of the M interferograms at the k th pixel. Solving the subtle SSE signal at every pixel independently using the least squares method above leads to a solution that overfits the atmospheric noise. To prevent overfitting, we introduce a spatial smoothness constraint on the offset estimate and solve for the phase history at all interferogram pixels simultaneously as a regularized least squares problem by minimizing: is Gm Φ α Dm 2 2 (4.2) Here G is block diagonal, where the k th diagonal block entry equals G (k) in Equation (4.1). The number of diagonal block entries in G equals the number of pixels n p in one interferogram. The unknown model vector m = [ m (1)... m (np) ] T and the data vector Φ = [ Φ (1)... Φ (np) ] T, where the k th entry in m and Φ are also defined in Equation (4.1). The matrix D is constructed such that Dm 2 2 = D rδ D aδ 2 2, where D r and D a are matrices that compute discrete approximations to the derivatives in the range and azimuth directions. The vector δ consists of only the offset part of the solution vector m, that is, δ = [ δ (1), δ (2),..., δ (np) ] T. Note that the regularization term does not apply to the velocity solution v. The regularization parameter Dm 2 2 penalizes large spatial gradients in the re-

50 34 CHAPTER 4. IMAGING DEFORMATION AT KILAUEA USING INSAR constructed offset solutions. When the damping parameter α is too small, the solution suffers from overfitting. When α is too large, the solution suffers from excessive smoothing. We determine the optimal damping parameter α heuristically using GPS data from two GPS stations KAEP and MANE. We select KAEP and MANE as our reference because InSAR techniques can only measure relative motions between pixels and the SSE signal is much more significant at KAEP than at MANE as discussed in Section We denote the magnitude of the SSE in east, north and up coordinates as d K = [ E 1 N 1 U 1 ] at KAEP and d M = [ E 2 N 2 U 2 ] at MANE. Based on Equation (3.6) in Section 3.2, we project the difference between d K and d M to the LOS direction as: d GPS = e 1 (E 1 E 2 ) + e 2 (N 1 N 2 ) + e 3 (U 1 U 2 ) (4.3) where e = [ e 1, e 2, e 3 ] is the InSAR LOS direction unit vector. Minimizing Equation (4.2) for a given α, we can compute an InSAR estimate of the magnitude of the SSE in the LOS direction, which we denote as δ (K) at KAEP and δ (M) at MANE. We define d InSAR as: d InSAR = δ (K) δ (M) (4.4) We next compute d GPS d InSAR for difference values α. As an example, Figure 4.2 shows d GPS d InSAR along ascending LOS direction as a function of log(α). The optimal α is the one that minimizes d GPS d InSAR. We process the ascending and descending InSAR data separately and then extract the east and vertical ground deformation by combining the ascending and descending LOS deformation measurements, as described in the following Section 4.2. The noisy GPS vertical measurements lead to errors in the final InSAR solution. To improve the accuracy of our algorithm, we do not directly compute d GPS using (U 1 U 2 ) measured by GPS data in Equation (4.3). Instead, we guess many possible values U between (U 1 U 2 ɛ) and (U 1 U 2 +ɛ), where ɛ is the GPS vertical measurement uncertainty. For each possible U, we compute d GPS = e 1 (E 1 E 2 )+e 2 (N 1 N 2 )+ e 3 U and the corresponding InSAR east and vertical ground deformation solution.

51 4.2. COMBINING INSAR DATA WITH TWO LOOK ANGLES 35 8 difference (mm) log(α) Figure 4.2: d GPS d InSAR along ascending LOS direction as a function of log(α). Here d GPS is defined in Equation (4.3) and d InSAR is defined in Equation (4.4). We select the optimal U as the one that minimizes the root mean square (RMS) error of the InSAR SSE east component estimate at all available GPS stations in the area of interest. 4.2 Combining InSAR data with two look angles A radar satellite usually obtains data over a single area from both ascending and descending passes as shown in Figure 4.3. Therefore, InSAR measurements from two LOS directions are available. In this section, we use the TerraSAR-X data over the Kilauea region as an example to illustrate how to infer the east and vertical components of the ground deformation by combining ascending and descending LOS measurements. The ascending LOS unit vector of the TerraSAR-X data over Kilauea region is approximately [.5318,.996, -.841] and the descending LOS unit vector is approximately [-.5137,.935, ]. Recall that in Section 3.2, we show that given an LOS direction unit vector e = [e 1, e 2, e 3 ], we can project the deformation in east, north and up coordinates along the LOS direction as d LOS = e 1 d east +e 2 d north +e 3 d up.

52 36 CHAPTER 4. IMAGING DEFORMATION AT KILAUEA USING INSAR Figure 4.3: InSAR data acquisition geometry (top view) with both ascending and descending radar passes over Kilauea s south flank. Therefore, we have: d ascd =.5318 d east d north.841 d up + n ascd d descd =.5137 d east d north.8529 d up + n descd (4.5) Here n ascd represents noise in the ascending LOS ground deformation measurement d ascd and n descd represents noise in the descending LOS ground deformation measurement d descd. Combining the ascending and descending LOS measurements leads to: d east =.9631 d ascd.9496 d descd.71 d north.9631 n ascd n descd d up =.581 d ascd.65 d descd d north n ascd +.65 n descd (4.6)

53 4.3. RESULTS 37 We can estimate east and vertical components of the ground deformation as: d east.9631 d ascd.9496 d descd d up.581 d ascd.65 d descd (4.7) Here we neglect the terms that are related with d north, n ascd and n descd in Equation (4.6). Suppose d north equals 1 cm. Ignoring the term related with d north in the east ground deformation estimate as defined in Equation (4.7) leads to less than.1 mm error and ignoring the term related with d north in the vertical ground deformation estimate as defined in Equation (4.7) leads to approximately 1 mm error. 4.3 Results We processed 24 sets of TerraSAR-X ascending scenes and 25 sets of TerraSAR-X descending scenes acquired between August, 29 and December, 21. The resulting 576 TerraSAR-X interferograms cover the south flank of Kilauea, Hawaii as illustrated in Figure 4.4. In this section, we present the InSAR SBAS solutions of the ground deformation at Kilauea. Because InSAR techniques can only measure relative motions between pixels, we calibrate InSAR ground deformation results by a constant shift, which we derive from GPS data. Figure 4.4: Map of Hawaii island. The red box shows the InSAR images coverage.

54 38 CHAPTER 4. IMAGING DEFORMATION AT KILAUEA USING INSAR Eastward motion at Kilauea The InSAR SSE estimate at a given pixel k is the solution for the jump δ (k) in our model m (k) as defined in Equation (4.1). We compute a map of the eastward displacement field due to the 21 SSE at Kilauea, Hawaii using TerraSAR-X InSAR data, as shown in Figure 4.5. The white arrows illustrate the magnitude of the SSE eastward displacement at each of the 12 GPS stations (labeled in yellow) as estimated using InSAR. As a comparison, the dark yellow arrows illustrate the magnitude of the SSE horizontal displacement at these locations as estimated from GPS. Here we compute the SSE eastward displacement as the difference between the GPS 1- day average displacement before and after the SSE. We plot the InSAR and GPS eastward SSE displacement estimates at these 12 GPS sites in Figure 4.6. If we take the GPS SSE estimate as ground truth, then the RMS error of the InSAR eastward SSE estimate at these 12 GPS sites is about 1.3 mm. As a comparison, the eastward displacement due to the 21 SSE is on the order of centimeters. In spite of the fact that atmospheric noise in the Kilauea region is on the order of centimeters, our algorithm achieves millimeter-level accuracy in the east SSE estimate. The InSAR background velocity estimate at a given pixel k is the solution for v (k) in our model m (k) as defined in Equation (4.1). We compute a map of the InSAR eastward background velocity field at Kilauea, Hawaii, as shown in Figure 4.7. The white arrows illustrate the magnitude of the eastward background velocity as estimated from InSAR data at each of the 12 GPS stations, while the dark yellow arrows illustrate the magnitude of the horizontal background velocity as estimated from GPS data at these locations. We plot the InSAR and GPS eastward background velocity estimates at the 12 GPS sites in Figure 4.8. If we take the GPS measurements as ground truth, then the RMS error of the InSAR background velocity estimate at these 12 GPS sites is about 6.5 mm/year. As a comparison, the median eastward background velocity on the south flank of Kilauea is about 2 cm/year. We next extract the InSAR eastward background velocity estimate along the yellow line illustrated in Figure 4.9 (top). The eastward velocity profile in Figure 4.9 (bottom) suggest that the coastal region on the south side of the pali (cliff in Hawaiian) moves at a higher

55 4.3. RESULTS OUTL KOSM MANE GOPM KTPM PG2R Latitude PGF1 PGF5 PGF6 PGF3 PGF4 KAEP cm GPS Horizontal 1cm InSAR East Longitude cm Figure 4.5: Map of the eastward displacement field due to the 21 SSE at Kilauea, Hawaii as derived from TerraSAR-X InSAR data. The white arrows illustrate the magnitude of the SSE eastward displacement at each of the 12 GPS stations (labeled in yellow) as estimated from InSAR data. As a comparison, the dark yellow arrows illustrate the magnitude of the SSE horizontal displacement at these locations as estimated from GPS data. centimeters East SSE displacement estimates at 12 GPS stations GPS InSAR GOPM KAEP KOSM KTPM MANE OUTL PG2R PGF1 PGF3 PGF4 PGF5 Figure 4.6: InSAR and GPS estimates of the eastward SSE displacement at 12 GPS sites. PGF6

56 4 CHAPTER 4. IMAGING DEFORMATION AT KILAUEA USING INSAR 19.4 OUTL KOSM MANE GOPM KTPM PG2R Latitude PGF1 PGF5 PGF6 PGF3 PGF4 KAEP cm/year GPS Horizontal 2cm/year InSAR East Longitude cm/year Figure 4.7: Map of the eastward background velocity field at Kilauea, Hawaii as derived from TerraSAR-X InSAR data. The white arrows illustrate the magnitude of the eastward background velocity as estimated from InSAR data at each of the 12 GPS stations, while the dark yellow arrows illustrate the magnitude of the horizontal background velocity as estimated from GPS data at these locations. cm/year East linear velocity estimates at 12 GPS stations GPS InSAR GOPM KAEP KOSM KTPM MANE OUTL PG2R PGF1 PGF3 PGF4 PGF5 PGF6 Figure 4.8: InSAR and GPS estimates of the eastward background velocity at 12 GPS sites.

57 4.3. RESULTS Eastward linear velocity (cm/year) Latitude Figure 4.9: InSAR eastward background velocity estimate along the yellow line in the eastward linear velocity map. rate than the region on the north side. Note that we do not observe a jump in the InSAR SSE displacement estimate along the same profile. This is because the spatial smoothness constraint we employ on the jump estimate does not preserve the discontinuity in the solution. We reconstruct the east ground deformation history using the InSAR SBAS solution. We compare the InSAR and GPS eastward ground deformation time series at 12 GPS stations as shown in Figure 4.1. The differences between the InSAR and GPS measurements at most GPS sites are on the order of millimeters. The estimation error may be due to the following reasons. First, although the spatial smoothness

58 42 CHAPTER 4. IMAGING DEFORMATION AT KILAUEA USING INSAR.4 deformation at GOPM deformation at KAEP deformation at KOSM GPS time series InSAR time series GPS time series InSAR time series GPS time series InSAR time series.4 deformation at KTPM deformation at MANE deformation at OUTL East displacement (meters) GPS time series InSAR time series deformation at PG2R GPS time series InSAR time series deformation at PGF1 GPS time series InSAR time series deformation at PGF3.2.4 GPS time series InSAR time series GPS time series InSAR time series GPS time series InSAR time series.4 deformation at PGF4 deformation at PGF5 deformation at PGF GPS time series InSAR time series GPS time series InSAR time series Date GPS time series InSAR time series Figure 4.1: The eastward ground deformation time series estimated from GPS data (in black) and InSAR data (in red) at 12 GPS stations on the south flank of Kilauea. The jump in the time series corresponds to the 21 slow slip event. constraint on the SSE displacement estimate prevents overfitting in noisy regions, the residual atmospheric noise is also spatially correlated and can not be removed completely using our algorithm. Furthermore, the coastal GPS stations are further away from the reference location MANE. As a result, the atmospheric noise at these GPS sites is less correlated with the atmospheric noise at the reference location, which leads to larger InSAR SBAS estimation errors at locations such as the GPS site PGF6. Second, we assume the background motion is linear in the Kilauea region. This assumption may not be valid over a long period of time, especially near the Ki-

59 4.3. RESULTS 43 lauea caldera due to additional nonlinear volcano deformation. Third, decorrelation of X-band interferometry over vegetated areas is another cause of estimation errors at pixel locations such as the GPS site KTPM Vertical motion at Kilauea 19.4 OUTL KOSM MANE GOPM PG2R KTPM Latitude PGF1 PGF5 PGF6 PGF3 PGF4 KAEP Longitude cm Figure 4.11: Map of the vertical displacement field due to the 21 SSE at Kilauea, Hawaii as derived from TerraSAR-X InSAR data. To calibrate the InSAR results, we assume that there is no vertical displacement due to the SSE at the GPS site MANE. Figure 4.11 shows a map of the vertical displacement field due to the 21 SSE at Kilauea, Hawaii as derived from TerraSAR-X InSAR data. Because the GPS vertical time series are very noisy, we do not take the GPS vertical SSE estimate as ground truth. To calibrate the InSAR results, we assume that there is no vertical SSE displacement at the GPS site MANE. This assumption is consistent with GPS observations at MANE as shown in Figure 2.4. We plot the InSAR and GPS vertical SSE displacement estimates at 12 GPS sites in Figure Both GPS and InSAR measurements suggest that the magnitude of the 21 SSE vertical component is much smaller than the SSE east component. The relative errors between the GPS and InSAR vertical SSE estimates are larger than those for the eastward estimates,

60 44 CHAPTER 4. IMAGING DEFORMATION AT KILAUEA USING INSAR centimeters vertical SSE displacement estimates at 12 GPS stations GPS InSAR GOPM KAEP KOSM KTPM MANE OUTL PG2R PGF1 PGF3 PGF4 PGF5 Figure 4.12: InSAR and GPS estimates of the vertical SSE displacement at 12 GPS sites. centimeters vertical SSE displacement estimates at 12 GPS stations GPS InSAR GOPM KAEP KOSM KTPM MANE OUTL PG2R PGF1 PGF3 PGF4 PGF5 Figure 4.13: InSAR and GPS estimates of the vertical SSE displacement at 12 GPS sites. Here we correct the error in InSAR vertical displacement estimate due to the SSE southward motion using GPS data. with an RMS error at the 12 GPS sites of about 5 millimeters. The RMS error may be due to the following reasons. First, the uncertainty in the GPS vertical measurement is about 5 millimeters [Montgomery-Brown et al., 29] and the noise in the GPS data prevents a better comparison with InSAR data. Second, residual atmospheric noise PGF6 PGF6

61 4.3. RESULTS 45 causes errors in the InSAR estimate. Note that Equation (4.6) in Section 4.2 suggests that atmospheric noise causes different artifacts in InSAR eastward and vertical SSE estimates. Third, Equation (4.6) indicates that neglecting the centimeter level SSE southward displacement can lead to millimeter level errors in the InSAR upward SSE estimate. We can correct this error using GPS data at the 12 GPS sites, as shown in Figure However, the low resolution GPS data is not sufficient for correcting the InSAR vertical displacement field over the entire south flank of Kilauea KOSM OUTL MANE GOPM PG2R KTPM Latitude PGF1 PGF5 PGF6 PGF3 PGF4 KAEP Longitude cm/year Figure 4.14: Map of the vertical background velocity field at Kilauea, Hawaii as derived from TerraSAR-X InSAR data. To calibrate the InSAR results, we assume that there is no vertical background motion at the GPS site MANE. Figure 4.14 shows a map of the vertical background velocity field at Kilauea, Hawaii. We also assume that there is no vertical background motion at the GPS site MANE in order to calibrate the InSAR results. We plot the InSAR and GPS estimates of the vertical background velocity at 12 GPS sites as shown in Figure The GPS and InSAR ground deformation measurements are consistent at all 12 GPS sites and the root mean square difference of the GPS and InSAR estimate is about 3.8 mm/year. As a comparison, the median vertical background velocity relative to the GPS site MANE is about 1 cm/year.

62 46 CHAPTER 4. IMAGING DEFORMATION AT KILAUEA USING INSAR 4 vertical linear velocity estimates at 12 GPS stations cm/year GOPM KAEP KOSM KTPM MANE OUTL PG2R PGF1 PGF3 PGF4 GPS InSAR PGF5 PGF6 Figure 4.15: The InSAR and GPS estimate of the vertical background velocity at 12 GPS sites. We reconstruct the vertical ground deformation history using the InSAR SBAS solution. We then compare the InSAR and GPS vertical ground motion time series at the 12 GPS stations as shown in Figure While there is no clear jump in the vertical direction due to the 21 SSE, the InSAR and GPS estimates of the long term vertical motion are consistent with sub-centimeter accuracy. 4.4 Discussion The algorithm we proposed in Section 4.1 can be modified to isolate a transient jump superimposed on any continuous background motion, including but not limited to cubic deformation model as described in Equation (3.15). While this algorithm solves for ground deformation time series with high accuracy and fine spatial resolution, there are also some limitations. First, the spatial smoothness constraint we employ on the jump estimate does not preserve the discontinuity in the SSE solution. As a result, we can not determine if the SSE displacements concentrated at the Hilina and Holei palis using the method we propose here. Second, InSAR techniques can only measure relative motions between pixels. In order to calibrate the missing constant in the InSAR ground deformation solution, it is necessary to know the magnitude

63 4.4. DISCUSSION 47.4 deformation at GOPM deformation at KAEP deformation at KOSM GPS time series InSAR time series GPS time series InSAR time series GPS time series InSAR time series Vertical displacment (meters) deformation at KTPM GPS time series InSAR time series deformation at PG2R deformation at MANE GPS time series InSAR time series deformation at PGF1 deformation at OUTL GPS time series InSAR time series deformation at PGF3.2.4 GPS time series InSAR time series GPS time series InSAR time series GPS time series InSAR time series.4 deformation at PGF4 deformation at PGF5 deformation at PGF GPS time series InSAR time series GPS time series InSAR time series Date GPS time series InSAR time series Figure 4.16: The vertical ground deformation time series estimated from GPS data (in black) and InSAR data (in red) at 12 GPS stations on the south flank of Kilauea. There is no clear jump in the vertical direction due to the 21 SSE. of the ground deformation at a reference pixel location as prior information. Third, the magnitude of the transient displacement at another pixel location is also needed in order to determine the optimal weighting for the spatial smoothness constraint. We choose GPS data from two GPS stations KAEP and MANE as our reference in this study. In case that GPS data are not available, other ground deformation measurements can be used as prior information. Note that only ground deformation magnitude at two pixel locations is needed in order to reconstruct the 2D InSAR

64 48 CHAPTER 4. IMAGING DEFORMATION AT KILAUEA USING INSAR ground deformation map. Last, we can not infer the north component of the ground deformation using the SBAS algorithm we proposed in this chapter. Multi-Aperture Interferometry (MAI) [Bechor and Zebker, 26] techniques extract the along-track component of the ground deformation based on split-beam processing of the InSAR data. As InSAR satellites usually operate in polar orbits, it is possible to use MAI to extract the north component of the ground deformation.

65 Chapter 5 Detecting transient jumps in noisy InSAR time series In many applications, we are interested in detecting unknown transient events using only InSAR time series, particularly when no auxiliary data such as GPS are available. In this chapter, we present such an algorithm that preserves transient jump signals in very noisy InSAR time series based on a sparse reconstruction approach. We apply this algorithm to solve for the time of an SSE s occurrence using X-band TerraSAR- X data over Kilauea. We show that the largest jump detected in the TerraSAR-X InSAR time series is temporally and spatially correlated with the 21 Kilauea SSE. 5.1 Algorithm In Chapter 4, we proposed an SBAS InSAR time series analysis algorithm to isolate a centimeter-level transient jump due to the 21 Kilauea SSE. This is an L2-norm based Tikhonov regularization algorithm with a spatial smoothness constraint on the transient jump estimate. This algorithm requires prior knowledge of the SSE s time of occurrence. It is also necessary to know the magnitude of the SSE horizontal deformation at two reference locations in order to choose an optimal regularization parameter and solve for the SSE displacement field over the entire south flank of Kilauea. In our study, the 21 SSE has already been detected using continuous GPS 49

66 5 CHAPTER 5. DETECTING TRANSIENT EVENTS IN TIME SERIES data. The horizontal displacements due to the SSE at two GPS stations KAEP and MANE are accurately known as prior information. However, in many applications, we are interested in detecting unknown transient events using only InSAR time series, particularly when no auxiliary data such as GPS are available. In this section, we present such an algorithm that preserves transient jump signals in very noisy InSAR time series. Our model assumes the ground motion consists of a piecewise constant displacement superimposed on a linear background motion as shown in Figure 5.1. The background motion is continuous, including but not limited to constant-velocity background motion. The piecewise constant motion represents jumps in the time series, which we seek to find using a sparse reconstruction. Figure 5.1: A ground motion model consists of a piecewise constant displacement superimposed on a continuous background displacement. Here we assume the continuous background motion is linear (constant-velocity model), which is a good approximation for the secular motion on Kilauea s south flank based on GPS time series observations shown in Figure 2.4. In the rest of this section, we demonstrate how this algorithm works. Given N SAR images forming M small baseline interferograms, the matrix representation of the M InSAR phase measurements at a given pixel k can be written as: BP v (k) + Ax (k) = Φ (k) (5.1) where Φ (k) = [ φ (k) 1... φ (k) M ] T is the vector of known phase values of the M unwrapped interferograms at the pixel k. The M (N 1) matrix B and (N 1) 1 vector P are given by Equation (3.14). The scalar v (k) represents the unknown constant velocity of the background motion in the LOS direction at the pixel k. The M N matrix A corresponds to the SAR scene indices used to form the interferograms.

67 5.1. ALGORITHM 51 Given that the i th interferogram is formed using SAR data acquired at the m th and n th SAR data acquisition times, the i th row in A has a one in the m th entry, a minus one in the n th entry and zeros in the remaining N 2 entries. The N 1 vector x (k) = [x (k) 1... x (k) N ] represents the piecewise constant motion at pixel k illustrated in Figure 5.1. We assume x (k) 1 equals zero. (x (k) i x (k) i 1) indicates that there is a transient jump between (i 1) th and i th SAR data acquisition times. We are interest in deriving a solution x (k) with no more than r jumps corresponding to r largest transient events. To find such a solution, we formulate this problem as a cardinality problem: minimize Ax (k) + BP v (k) Φ (k) 2 subject to card(d t x (k) ) r (5.2) where the (N 1) N matrix D t computes discrete approximations to the time derivatives. The cardinality function card(d t x (k) ) equals the total number of nonzeroes in the vector D t x (k). Computing the optimal solution for Equation (5.2) is NP-hard [Garey and Johnson, 199]. However, given that the transient events in the time series are sparse, Equation (5.2) can be approximately solved as an L1-norm regularization problem [Boyd and Vandenberghe, 24]: minimize subject to Ax (k) + BP v (k) Φ (k) 2 D t x (k) 1 β (5.3) When β equals zero, the optimal solution contains no jumps. As β increases, the number of jumps in the the optimal x (k) increases. In practice, we adjust β so that the optimal solution x (k) contains r jumps. This method is also called total variation reconstruction and Dt x (k) 1 is defined as the total variation of signal x (k). An equivalent form of Equation (5.3) is: minimize Ax (k) + BP v (k) Φ (k) 2 + β Dt x (k) 1 (5.4)

68 52 CHAPTER 5. DETECTING TRANSIENT EVENTS IN TIME SERIES For comparison, we can also solve for x (k) and v (k) using an L2-norm based Tikhonov regularization as: minimize subject to Ax (k) + BP v (k) Φ (k) 2 D t x (k) 2 γ (5.5) An equivalent form of Equation (5.5) is: minimize Ax (k) + BP v (k) Φ (k) 2 + γ Dt x (k) 2 (5.6) From a statistical perspective, the difference between the L1-norm and L2-norm based methods is the distribution of the error penalty function. Increasing the error penalty causes more errors to be driven to zero in the L1-norm based method, while with Tikhonov regularization, all errors are reduced but still non-zero. In other words, L1 regularization leads to sparser solutions that correlate with the largest jumps in InSAR time series, while the L2 regularization tend to smooth both the transient jump signals and atmospheric noise at the same time. 5.2 Synthetic Test In this section, we demonstrate the difference between the L1-norm and L2-norm based regularization using synthetic time series data with low signal to noise ratio (SNR 1), shown in Figure 5.2. A 1 cm transient event (Figure 5.2 (a)) is superimposed on a linear background motion of about 4.3 cm/year (Figure 5.2 (b)). The magnitudes of the transient event and the background velocity are comparable to the 21 SSE and the secular linear motion at Kilauea. We sample the synthetic InSAR time series at the same time as the TerraSAR-X satellite acquired data in its ascending orbit between August 29 and December 21. The corresponding N data samples are marked as blue dots. Random Gaussian noise N (, σ 2 ) (Figure 5.2 (c)) is added to the time series, where σ equals 1 cm. As the SNR is low, it is very hard to identify the transient jump in the noisy InSAR time series (Figure 5.2 (d)).

69 5.2. SYNTHETIC TEST 53 1 (a) True SSE 6 (b) Background motion cm.5 cm Date (c) Random noise Date (d) Combined motion with noise 1 cm 1 1 cm Date Date Figure 5.2: Synthetic InSAR time series data (SNR 1). (a) A 1 cm transient jump. (b) Linear background motion of about 4.3 cm/year. (c) Random Gaussian noise N (, σ 2 ), where σ = 1 cm. (d) Combined transient and secular motion with noise. We define the synthetic time series shown in Figure 5.2 (d) as an N 1 vector z and generate a M 1 synthetic InSAR data set ˆΦ (k) as: ˆΦ (k) = Az (5.7) where M is the total number of interferograms and the M N SAR matrix A is defined in Equation (5.1). We apply the the L1-norm based sparse reconstruction algorithm to the synthetic InSAR data ˆΦ (k). Figure 5.3 shows the corresponding L1 solutions for the ground motion for different values of β. The L1 regularization leads to a sparse solution containing no transient jump in most time intervals. When β is too small, the solution contains one jump at the time of the transient event but the magnitude of the event is smaller than the true event. When β is optimal, the solution exactly reconstructs

70 54 CHAPTER 5. DETECTING TRANSIENT EVENTS IN TIME SERIES the piecewise constant motion ˆx, but underestimates the linear motion due to the presence of the noise. When β is too large, the solution contains more than one jump. However, the step signal associated with the transient event is still well preserved. Because no auxiliary data such as GPS are available as reference data, we cannot accurately determine the optimal value for β. In practice, we adjust β so that the solution only contains r jumps. We can use this solution to determine the occurrence time of the largest r jumps in a noisy InSAR time series. 6 (a) True ground motion (b) L1 norm based solution (β too small) 6 cm 4 2 cm Date (c) L1 norm based solution (optimal β) 6 cm Date Date (d) L1 norm based solution (β too large) 6 cm Date Figure 5.3: Ground motion solutions using L1-norm based sparse reconstruction for different values of β. (a) True ground motion. (b) L1-norm based solution with a relatively small β. (c) L1-norm based solution with the optimal β. (d) L1-norm based solution with a relatively large β. The parameter β in every case here is defined in Equation (5.3). We also apply the L2-norm based Tikhonov regularization algorithm to the same synthetic InSAR data set ˆΦ (k). Figure 5.4 shows the corresponding L2 solutions for the ground motion for different values of γ. The L2 regularization leads to a nonsparse solution containing transient jumps in most time intervals. The magnitude of γ determines the total smoothness of ˆx L2 in the least squares sense. As γ decreases,

71 5.3. SOLUTION FOR SSE OCCURRENCE TIME 55 the L2 regularization tend to smooth both the transient jump signals and atmospheric noise at the same time. The L2 solutions are also more sensitive to the presence of noise and there is no optimal solution that exactly reconstructs the piecewise constant motion ˆx. cm (a) True ground motion cm (b) L2 norm based solution (small γ) Date (c) L2 norm based solution (medium γ) Date (d) L2 norm based solution (large γ) 6 4 cm 2 cm Date Date Figure 5.4: Ground motion solutions using L2-norm based Tikhonov regularization for different values of γ. (a) True ground motion. (b) L2-norm based solution with a relatively small γ. (c) L2-norm based solution with a medium γ. (d) L2-norm based solution with a relatively large γ. The parameter γ in every case here is defined in Equation (5.5). 5.3 Solution for SSE occurrence time In this section, we apply the L1-norm based sparse reconstruction algorithm to solve for the time of occurrence of the 21 SSE using the TerraSAR-X data analyzed in Chapter 4. As we do not impose any spatial smoothness constraint in this algorithm, the LOS motion at each InSAR pixel can be solved independently. We choose GPS station MANE as the reference pixel and solve for the relative LOS motion at the GPS station KAEP, where the 21 SSE signal is much more significant than at the

72 56 CHAPTER 5. DETECTING TRANSIENT EVENTS IN TIME SERIES reference MANE..5.5 deform ation (m eters) -.5 G PS tim e series G PS tim e series InSAR tim e series InSAR tim e series tim e tim e (a) Ascending LOS time series (b) Descending LOS time series Figure 5.5: TerraSAR-X InSAR time series (in red) as derived from the L1-norm based sparse reconstruction, showing the ground deformation at the GPS station KAEP relative to MANE along the ascending and descending LOS directions. For comparison, we also project the relative motion between KAEP and MANE as observed using GPS to both LOS directions (in black). Figure 5.5 shows the TerraSAR-X InSAR time series (in red) as derived from the L1-norm based sparse reconstruction, showing the ground deformation at the GPS station KAEP relative to MANE along the ascending and descending LOS directions. We adjust the parameter β in Equation (5.3) so that the L1 solution only contains one jump. We circle the transient jump detected in the InSAR time series in blue. Note that the 21 SSE leads to a positive jump in the ascending time series and a negative jump in the descending time series, which suggests that the eastward SSE component dominates in the SSE LOS motion according to Equation (4.5). In both cases, the transient jump occurs in the time interval spanning the 21 SSE. Therefore, we successfully solve for the time of occurrence of the SSE using the sparse reconstruction algorithm. For comparison, we also project the relative motion between GPS stations KAEP and MANE as observed using GPS data to both LOS directions (in black). While the LOS GPS time series predict similar long term trends as the InSAR time series, it is hard to identify the transient event associated with the 21 SSE using the LOS GPS time series as the uncertainty of the GPS vertical measurements is large. We also plot ascending and descending TerraSAR-X InSAR time series as derived

73 5.4. LIMITATIONS deform ation (m eters) -.5 G PS tim e series InSAR tim e series tim e (a) Ascending LOS time series G PS tim e series InSAR tim e series tim e (b) Descending LOS time series Figure 5.6: TerraSAR-X InSAR time series (in red) as derived from the L2-norm based Tikhonov regularization, showing the transient and secular linear motion at the GPS station KAEP relative to MANE along the ascending and descending LOS directions. For comparison, we also project the relative motion between KAEP and MANE as observed using GPS to both LOS directions (in black). from the L2-norm based Tikhonov regularization along with the LOS GPS time series in Figure 5.6. Note that different γ values lead to non-sparse solutions with different smoothness in the least squares sense, and none of the solutions preserve sharp jumps in InSAR time series. Again, we circle the signature associated with the SSE in the InSAR time series in blue. However, we cannot solve for the occurrence time of the SSE without ambiguity using either of the L2 solutions, as both InSAR time series suggest that there might be more than one transient event. 5.4 Limitations In Section 5.3, we demonstrate that the L1-norm based sparse reconstruction algorithm can be used to detect the 21 SSE signature in the noisy X-band InSAR time series. Because we do not use auxiliary data to determine the optimal L1 solution, the magnitude of the predicted SSE may be incorrect. Nonetheless, we can still use this solution to determine the time of occurrence of the SSE. We apply this algorithm to every pixel k in the interferograms and reconstruct a map of the transient jumps for each time interval. Figure 5.7 shows the resulting map of the transient jump along the ascending LOS direction for the time interval spanning the 21 SSE (top)

74 58 CHAPTER 5. DETECTING TRANSIENT EVENTS IN TIME SERIES Figure 5.7: Transient jump map along the ascending LOS direction (top) for the time interval spanning the 21 SSE (bottom) for another time interval not spanning the 21 SSE. and another time interval not spanning the 21 SSE (bottom). We observe that the L1 solution correctly detects a transient jump associated with the 21 SSE over the coastal region circled in red and indicates that no jump occurred in the other time interval. According to Figure 2.3, the magnitude of the SSE is the largest in

75 5.4. LIMITATIONS 59 Figure 5.8: Transient jump map along the descending LOS direction (Top) for the time interval spanning the 21 SSE (Bottom) for a time interval not spanning the 21 SSE. the red circled region, however, the SSE occurred over a much broader region, so the L1 algorithm failed to reconstruct the entire SSE displacement field. Although the sparse reconstruction algorithm is less sensitive to atmospheric noise compared to the Tikhonov regularization technique, more samples would be required in order

76 6 CHAPTER 5. DETECTING TRANSIENT EVENTS IN TIME SERIES to reconstruct a signal with lower SNR. Candès et al. [26]; Donoho and Tanner [29] discussed hard limits on the degree to which the L1 method can reconstruct the signal with overwhelming probability. The full depth of this topic is beyond the scope of this dissertation. For the TerraSAR-X data we processed, the variation of atmospheric noise during the descending data acquisition time is smaller than during the ascending data acquisition time, as shown in Figure 6.3. This is due to the difference in the ascending and descending data acquisition times. As a result, it is more likely to detect the SSE signature using the descending data set. Figure 5.8 shows the map of the transient jump along the descending LOS direction for the time interval spanning the 21 SSE (top) and another time interval not spanning the 21 SSE (bottom). We observe that the L1 solution detects a transient jump associated with the 21 SSE over a larger area, which again confirms the theory agrees with the observations.

77 Chapter 6 Atmospheric errors in InSAR deformation measurements In this chapter, we discuss error sources in InSAR deformation measurements and error propagation in the SBAS InSAR time series analysis. We focus on estimating tropospheric delay in InSAR images, as it is the primary error source in the TerraSAR- X data we processed for the study of the 21 Kilauea slow slip event. We also address the impact of ionospheric delay artifacts in InSAR images, which are often seen in L-band interferograms in high latitude regions. 6.1 Uncertainty in InSAR deformation measurements In Chapter 3, we relate the phase measured by a single InSAR pair to ground motion along the LOS direction. Considering errors introduced during SAR data acquisition and InSAR data processing, we can rewrite the InSAR phase measurement φ as [Hooper et al., 24]: φ = φ def + φ err = φ def + φ atmos + φ orb + φ topo + φ n (6.1) 61

78 62 CHAPTER 6. ATMOSPHERIC ERRORS IN INSAR MEASUREMENTS where φ def is the phase due to the true ground deformation in the LOS direction, φ atmos is the phase due to the tropospheric and ionospheric delays, φ orb is the phase due to satellite orbit errors, φ topo is the phase due to error in the surface topography, and φ n is a residual noise term which includes scattering variability and thermal noise. Recall that given N SAR images forming M small baseline InSAR measurements, the matrix representation of an M equation SBAS system can be written as: Bv = Φ = Φ def + Φ err (6.2) where B is an M (N 1) SBAS matrix as defined in Section 3.3 and v = [ v 1... v N 1 ] T is the vector of unknown mean velocities between each time-adjacent SAR acquisition. The M 1 vector Φ def is the expected ground deformation of the M interferograms at the given pixel. The M 1 vector Φ err represents errors in the M InSAR phase measurements, as defined in Equation 6.1. We approximate v in Equation 6.2 as B Φ, where B is the Moore-Penrose pseudoinverse of B. Since the error-free velocity estimation is v = B Φ def, the error in the velocity estimates can be written as: e = B Φ err (6.3) Equation 6.3 shows how errors are propagated in the SBAS system. Satellite orbital errors φ orb can be estimated and removed as a ramp across the interferogram [Shirzaei and Walter, 211]. Topographic errors φ topo can often be reduced by accurately coregistering interferograms and the digital elevation model (DEM). In the following sections, we focus on estimating the atmospheric delay in InSAR images, as it is often the dominant error source in InSAR measurements. 6.2 Tropospheric artifacts in InSAR data Phase artifacts in InSAR images are often attributed to neutral tropospheric delays [Zebker et al., 1997; Hanssen et al., 1999]. Because the Earth s troposphere is non-

79 6.2. TROPOSPHERIC ARTIFACTS IN INSAR DATA 63 dispersive at appropriate frequencies, radar signals that are operating at different frequencies are subject to the same tropospheric delays. Figure 4.1 in Section 4.1 is a typical X-band interferogram with significant tropospheric noise. A phase cycle of 2π in the interferogram corresponds to λ/2 = 1.55 cm deformation, where λ is the radar signal wavelength. We observe that the tropospheric noise occurs with variation on the order of centimeters or even greater across the interferogram. As a result, the expected centimeter-level crustal deformation signature is obscured by tropospheric noise. In order to obtain accurate InSAR deformation measurements, some effort is needed to handle or suppress the atmospheric noise signature. Onn and Zebker [26] introduced a method to correct for atmospheric phase artifacts in a radar interferogram using spatially interpolated zenith wet delay data obtained from a network of GPS receivers in the region imaged by the radar. Li et al. [25, 26] used Medium Resolution Imaging Spectrometer (MERIS), Moderate Resolution Imaging Spectroradiometer (MODIS) and GPS data to estimate the water vapor field in order to correct interferograms that are corrupted by atmospheric artifacts. Foster et al. [26] employed a high-resolution weather model to predict tropospheric delays for the acquisition times of SAR images. However, estimating tropospheric delays using auxiliary data such as GPS, MERIS/MODIS or weather model usually produces a tropospheric noise model with resolution much coarser than InSAR image resolution, and the model uncertainty can be relatively large for studying centimeter-level crustal deformations. By contrast, many scholars proposed algorithms to estimate tropospheric delays during SAR data acquisition times directly from InSAR data. Emardson et al. [23] mitigated tropospheric effects by averaging N independent interferograms, because the wet component of the neutral atmosphere is uncorrelated at time scales longer than 1 day. This stacking approach is limited by the number of interferograms that are available over the time of interest. Lin et al. [21]; Lauknes [211]; Hooper et al. [212] assumed that tropospheric delays in InSAR data are topographically correlated. However, their assumption that tropospheric delay is proportional to surface elevation may not be valid for turbulent tropospheric processes. In the rest of this section, we propose an SBAS algorithm to estimate tropospheric

80 64 CHAPTER 6. ATMOSPHERIC ERRORS IN INSAR MEASUREMENTS delays in InSAR data. This algorithm is based on the assumption that errors in InSAR deformation estimates are primarily due to tropospheric noise. Note that we do not impose any topography-related model for the atmospheric noise distribution. We use this algorithm to quantify the tropospheric noise variation in TerraSAR-X data we processed over the Kilauea region Estimating tropospheric delays in InSAR data using SBAS approach We define the LOS tropospheric phase delays (in radians) at a pixel of interest k and N SAR data acquisition times as an N 1 vector x (k) = [ x (k) 1... x (k) N ] T, and the LOS tropospheric phase delays (in radians) at the InSAR reference pixel r and N SAR data acquisition times as an N 1 vector x (r) = [ x (r) 1... x (r) N ] T. The relative LOS tropospheric phase delays at the pixel k are then defined as x (k) = x (k) x (r). We write the InSAR data residual vector Φ (k) res at the pixel k as: Φ (k) res = Φ (k) G (k) m (k) (6.4) Here G (k), m (k) and Φ (k) are defined in Equation (4.1). The size of Φ (k) res is M 1, where M is the total number of interferograms. The i th entry in Φ (k) res corresponds to the difference between the observed phase value at the k th pixel of the i th interferogram and the fitted phase value provided by the linear plus offset model as described in Section 4.1. Suppose the i th interferogram is formed using two sets of SAR data acquired at t m and t n. Since SBAS data fitting errors are mainly due to tropospheric noise, the i th entry in Φ (k) res approximately represents the relative LOS tropospheric phase delay difference ( x (k) m x (k) n ). We can solve for the relative LOS tropospheric phase delays at N SAR data acquisition times using M interferograms (M > N). The matrix representation of the system is: A x (k) = Φ (k) res (6.5)

81 6.2. TROPOSPHERIC ARTIFACTS IN INSAR DATA 65 Here A is an M N matrix corresponding to the SAR scene indices used to form the interferograms as defined in Equation (5.1). The vector x (k) in Equation (6.5) can be estimated using the SVD as in Section 3.3. We then compute the relative tropospheric zenith delays (in meters) as: x (k) = 4π λ x(k) e 3 (6.6) where λ is the radar wavelength in meters and e 3 is the vertical component of the LOS direction unit vector e = [e 1, e 2, e 3 ]. In order to validate the algorithm we present here, the GPS data for the same time period are also analyzed to estimate tropospheric zenith delays using the GIPSY- OASIS software package. We define the GPS tropospheric zenith delay estimates at a GPS station of interest s and N SAR acquisition times as y (s) = [ y (s) 1... y (s) N ] T and the GPS zenith delay estimates at the reference GPS station r and N SAR acquisition times as y (r) = [ y (r) 1... y (r) N ] T. The relative zenith delays at the GPS station S are then defined as y (s) = y (s) y (r), which we compare directly with InSAR relative zenith delay estimates. Figure 6.1 shows the relative tropospheric zenith delays as estimated from ascending InSAR data (in red) and GPS data (in black) at 11 Kilauea GPS stations. Here we select the GPS station MANE as the reference station for processing InSAR and GPS data. Similarly, Figure 6.2 shows the relative tropospheric zenith delays as estimated from descending InSAR data (in red) and GPS data (in black) at these stations. The relative tropospheric zenith delays estimated from GPS and InSAR data are strongly correlated, which confirms that the ground deformation signal and tropospheric noise in InSAR data are successfully separated using the SBAS InSAR time series analysis. We also compute the standard deviation of the relative tropospheric noise time series vector x (k) in Equation (6.6) for every InSAR pixel k. Figure 6.3 shows the resulting tropospheric noise temporal standard deviation map as derived from ascending and descending TerraSAR-X interferograms. We observe that the variation of tropospheric noise during the ascending data acquisition times is larger than dur-

82 66 CHAPTER 6. ATMOSPHERIC ERRORS IN INSAR MEASUREMENTS 6 GOPM 11 KAEP 4 KOSM GPS InSAR 2 GPS InSAR 5 GPS InSAR 6 KTPM 4 OUTL 6 PG2R Relative zenith delay (cm) PGF1 1 2 GPS InSAR PGF3 3 GPS InSAR PGF4 GPS InSAR PGF5 GPS InSAR 1 PGF6 GPS InSAR GPS InSAR GPS GPS InSAR 2 InSAR Date Figure 6.1: Relative tropospheric zenith delays as estimated from ascending InSAR data (in red) and GPS data (in black) at 11 Kilauea GPS stations. Here we select the GPS station MANE as the reference station for processing both InSAR and GPS data. ing the descending data acquisition times. This is because the ascending data were acquired at local time 6:22 pm in the late afternoon and the descending data were acquired at local time 6:2 am in the early morning. The expected tropospheric phase signature is more significant around sunset than around sunrise. Both ascending and descending InSAR data suggest that the relative tropospheric delay variation at Ki-

83 6.2. TROPOSPHERIC ARTIFACTS IN INSAR DATA 67 6 GOPM 12 KAEP 4 KOSM GPS InSAR 3 GPS InSAR 5 GPS InSAR 6 KTPM 3 OUTL 7 PG2R Relative zenith delay (cm) PGF1 GPS InSAR PGF3 GPS InSAR PGF4 GPS InSAR PGF5 GPS InSAR 1 11 PGF6 GPS InSAR GPS InSAR GPS InSAR Date Figure 6.2: Relative tropospheric zenith delays as estimated from descending InSAR data (in red) and GPS data (in black) at 11 Kilauea GPS stations. Here we select the GPS station MANE as the reference station processing for both InSAR and GPS data. GPS InSAR lauea is on the order of centimeters, consistent with our assumption that centimeter level noise in the InSAR data is mainly due to tropospheric noise variation. Note that the coastal region is further away from the reference pixel location at GPS site MANE. As a result, tropospheric noise in the coastal region is less correlated with tropospheric noise at the reference location, which leads to larger variation in the relative tropospheric noise time series.

84 68 CHAPTER 6. ATMOSPHERIC ERRORS IN INSAR MEASUREMENTS (a) (cm) Azimuth Azimuth (b) Range Range.5 (cm) Figure 6.3: (a) Standard deviation of the relative tropospheric noise time series at each pixel as derived from ascending TerraSAR-X interferograms. (b) Standard deviation of the relative tropospheric noise time series at each pixel as derived from descending TerraSAR-X interferograms. Both images are presented in radar coordinates. The horizontal axis corresponds to the radar range direction and the vertical axis corresponds to the radar azimuth direction Discussion In Section 6.2.1, we developed an algorithm to estimate the tropospheric delay variation relative to a given reference pixel using the InSAR SBAS approach. We verified that the InSAR relative atmospheric zenith delay estimates are strongly correlated

85 6.2. TROPOSPHERIC ARTIFACTS IN INSAR DATA 69 2π Azimuth Range -2π Phase (rad) 2π Azimuth Range -2π Phase (rad) Figure 6.4: A comparison between the tropospheric delay field for the time interval from March 3, 21 to January 7, 21 and the 21 SSE displacement field along descending LOS direction. (Top) Descending LOS tropospheric delay for the time interval from March 3, 21 to January 7, 21. This map is computed using the SBAS algorithm described in Section (bottom) Descending LOS displacement field due to the 21 Kilauea SSE. This map is computed using the joint GPS-InSAR inversion algorithm described in Section 4.1. with independent GPS atmospheric zenith delay estimates for the same time period. We conclude that it is possible use InSAR data to reconstruct high-resolution atmospheric noise maps at each SAR acquisition time, which helps us to visualize the atmospheric noise distribution in InSAR data. As a example, Figure 6.4 (top) shows

86 7 CHAPTER 6. ATMOSPHERIC ERRORS IN INSAR MEASUREMENTS the descending LOS tropospheric delay for the time interval from March 3, 21 to January 7, 21. This map is computed using the SBAS algorithm described in Section Note that phase values are wrapped and a color cycle corresponds to 4π or 3.1 cm LOS atmospheric delay. For comparison, Figure 6.4 (bottom) shows the 21 SSE displacement field along the descending LOS direction. This map is computed using the joint GPS-InSAR inversion algorithm described in Section 4.1. We see that the tropospheric phase correlates well with the pali (cliff in Hawaiian) and is much larger than the SSE signature. It is possible to correct InSAR images using the atmospheric noise estimates we obtain from InSAR measurements and further improve the InSAR ground deformation estimates. If a particular set of SAR data is highly corrupted by atmospheric noise, we can discard this data set or assign it a smaller weight when necessary. 6.3 Ionospheric artifacts in InSAR data Phase artifacts in InSAR images can be attributed to ionospheric delays in addition to neutral tropospheric delays. In Appendix A, we show that the most pronounced signature of ionospheric delays in InSAR data is an azimuth pixel shift due to the average azimuth change in the ionospheric total electron content (TEC) over one synthetic aperture length. The azimuth shift in interferograms causes poor correlation observed as azimuth streaks in InSAR images and degrades the interpretability of the phase signatures of the terrain. Gray et al. [2]; Wegmuller et al. [26]; Strozzi et al. [28] reported such azimuth streaks in L-band interferograms, and Meyer and Nicoll [28b] discussed the theoretical background for ionospheric effects on InSAR. Pi et al. [1997, 211] presented a technique of imaging the ionospheric inhomogeneities using polarimetric measurements of spaceborne SAR and verified the SAR observations using global GPS ionospheric TEC maps derived from the GEONET array in Japan and the International GPS Service network. By contrast, our study focuses not on the large-scale or long-term TEC variations derived from GPS data. Instead, we attempt to directly relate the misregistration of azimuth pixels seen in radar interferograms to simultaneous, independent measurements of the ionospheric TEC gradient over one synthetic

87 6.3. IONOSPHERIC ARTIFACTS IN INSAR DATA 71 radar length. Specifically, we estimate the ionospheric vertical TEC (TECV) gradient over one synthetic radar length using dual frequency GPS carrier phase data as described in Appendix B. We show that the azimuth pixel misregistration in L- band interferograms, considered alongside estimates of the TECV from simultaneous nearby GPS observations, suggest the same order of magnitude of ionospheric variation. Thus we conclude that azimuth streaks observed in L-band interferograms are likely due to ionospheric variability rather than other decorrelation factors such as neutral tropospheric delays [Zebker et al., 1997]. In the rest of this section, we present simultaneously acquired L-band InSAR and GPS TECV observations over a high-latitude Iceland region, a mid-latitude California region and a low-latitude Hawaii region. Note that equatorial and auroral ionospheres are far more likely to cause errors in the InSAR data. We choose L-band ALOS PAL- SAR satellite data (λ 24 cm) rather than many presently operating C-band (λ 6 cm) and X-band (λ 3 cm) spaceborne radar data for this study as the ionospheric delay is directly proportional to the ionospheric TEC and inversely proportional to the square of the signal frequency Iceland region results Figure 6.5 (left) shows an Iceland interferogram centered at N, W, which is formed by two sets of ALOS PALSAR data acquired on September 2, 27 and October 18, 27 at 23:16 UTC. The most prominent azimuth streak artifact with severe phase decorrelation is circled in red. We estimate the misregistration of azimuth offset in this interferogram using amplitude cross-correlation and a low pass filter. Figure 6.5 (right) shows the resulting azimuth misregistration measured in pixels. We observe streaks in the azimuth direction with a maximum misregistration of ±3 pixels, which is spatially correlated with the phase decorrelation artifacts in the interferogram. We next compute the ionospheric TEC using GPS carrier phase data from a permanent GPS station BUDH. The ionospheric variability depends on the time of day, and the fact that the highest TEC is seen when the satellites are at the lowest elevation because of the path length. We project the GPS TEC measurements to the zenith

88 72 CHAPTER 6. ATMOSPHERIC ERRORS IN INSAR MEASUREMENTS Range (pixels) 3 Azimuth Figure 6.5: (left) An Iceland interferogram centered at N, W, which is formed by two sets of ALOS PALSAR data on September 2, 27 and October 18, 27 at 23:16 UTC (By courtesy of Albert Chen). The image is presented in radar coordinates. The horizontal axis corresponds to the radar range direction and the vertical axis corresponds to the radar azimuth direction. (right) The azimuth misregistration measured in pixels corresponding to the Iceland interferogram. Here the length of one azimuth pixel is about 3.6 m. direction to compensate the path length difference and the resulting GPS TECV estimates from different satellite-receiver pairs are shown in Figure 6.6. On each day, the TECV estimates from different satellite-receiver pairs converge to a similar temporal variation, but contain short-term irregularities. This suggests that the magnitude of the spatial ionospheric variations along some GPS signal propagation paths are large, which may lead to azimuth offset misregistration in the interferogram. To find the nearest GPS TECV observations to the ALOS PALSAR scene, we project the propagation paths of ALOS PALSAR and GPS signals onto the thin-shell ionosphere as described in Appendix C. We map the intersections of all propagation

89 6.3. IONOSPHERIC ARTIFACTS IN INSAR DATA 73 TECV BUDH,Iceland,9/2/27 GPS PRN number TECV BUDH,Iceland,1/18/27 GPS PRN number UTC (a) between 22.5 and 23.5 UTC on September 2, UTC (b) between 22.5 and 23.5 UTC on October 18, 27 Figure 6.6: Estimated TECV variation based on GPS carrier phase data from permanent GPS station BUDH in Iceland. The unit of the vertical TEC estimates is TEC units (TECU), or 1 16 electron/m 2. The GPS satellites are identified by the GPS receiver by means of pseudorandom noise (PRN) numbers and each color represents TECV measurements from a visible GPS satellite at GPS station BUDH. latitude BUDH,Iceland,9/2/27,Satellite map GPS 3 GPS 9 GPS 16 GPS 19 GPS 25 ALOS latitude BUDH,Iceland,1/18/27,Satellite map GPS 4 GPS 11 GPS 12 GPS 16 GPS 19 GPS 22 ALOS longitude (a) longitude (b) Figure 6.7: The projections of GPS and ALOS PALSAR signal propagation paths onto the thin-shell ionosphere in ellipsoidal coordinates (latitude, longitude). The black boxes show the radar image edges, projected from the ALOS PALSAR satellite. Each curve represents the projection of a propagation path from a moving GPS satellite to the ground receiver between 22.5 and 23.5 UTC. paths with the thin-shell ionosphere in ellipsoidal coordinates (latitude, longitude) as shown in Figure 6.7. The black boxes show the location of the edges of the radar image, projected from the ALOS PALSAR satellite. Each curve represents

90 74 CHAPTER 6. ATMOSPHERIC ERRORS IN INSAR MEASUREMENTS the projection of a propagation path from a moving GPS satellite to the ground receiver between 22.5 and 23.5 UTC. As we can see in Figure 6.7, the nearest GPS observations to the ALOS PALSAR scene are from the PRN 9 satellite on September 2, 27 and the PRN 16 satellite on October 18, 27. We first analyze the spatial TECV gradient of PRN 9 on September 2, 27 to calculate the expected azimuth pixel shift as derived in Appendix A. Noting that the data from PRN 16 are smooth and slowly-varying over the time interval of interest, we assume that this particular satellite s measurement is a good representation of the unknown long-term temporal trend in the TECV. We then subtract off this assumed temporal trend to yield an estimate of the spatial component of the TECV variation as measured by any of the other satellites. This is shown for PRN 9 in Figure 6.8 (a), where we have removed the assumed temporal component and plotted only the underlying spatial TECV estimate along the satellite path. We also plot the location of PRN 9 in terms of the angle α in Equation (A.16) as a function of time in the same figure. Following Equation (A.18) in Appendix A, the change in the spatial TECV as α changes by.16 is equivalent to the change in the spatial TECV as the ALOS PALSAR satellite travels one synthetic aperture length ( x 2 km). More specifically, in Figure 6.8 (a), as the angle α changes by.16 around the SAR data acquisition time 23:16 UTC, the corresponding TECV change is TECV =.782 TECU, where TECU is the unit of vertical TEC, or 1 16 electron/m 2. Given that the ALOS satellite signal frequency f = 1275 MHz, one synthetic aperture length x 2 km, the distance between the satellite and a ground pixel R 8 km and the obliquity factor cos β cos(34.3 ), we can calculate the expected azimuth shift in Equation (A.15) as: x shift = 4.3R TECV f 2 cos β x = TECV cos βf 2 = 9.68 meters (6.7) The azimuth pixel spacing is 3.6 m, thus the spatial TECV gradient predicted by PRN 9 can lead to an azimuth shift of 2.69 pixels in the simultaneous SAR data. Recall that we measured offsets of ±3 pixels, so that our offset prediction matches the observed data well.

91 6.3. IONOSPHERIC ARTIFACTS IN INSAR DATA 75 1 BUDH,Iceland,9/2/ BUDH,Iceland,1/18/27 4 spatial TECV variation α(deg) 2 spatial TECV variation α(deg) UTC UTC (a) on September 2, 27, measured from the GPS PRN 9 satellite (b) on October 18, 27, measured from the GPS PRN 16 satellite Figure 6.8: Spatial TECV variation near the Iceland InSAR scene as estimated from GPS data. The dashed lines represent the satellite location variation in term of the change in α angle on both days. Similarly, we analyze the spatial TECV gradient of PRN 16 on October 18, 27. As in the previous case, we assume PRN 11 in Figure 6.6 (b) represents the long-term temporal TECV variation to estimate the underlying spatial TECV component of any of the other satellites. This is shown for PRN 16 in Figure 6.8 (b). We also plot the α angle of PRN 16 as a function of time in the same figure. As the angle α changes by.16 around the SAR data acquisition time 23:16 UTC, the corresponding TECV change is TECV =.988 TECU, where TECU is the unit of vertical TEC, or 1 16 electron/m 2. The expected azimuth shift in Equation (A.15) is: x shift = 4.3R TECV f 2 cos β x = TECV cos βf 2 = 1.22 meters (6.8) The azimuth pixel spacing is 3.6 m, thus the spatial TECV gradient predicted by GPS PRN 16 GPS satellite only results in less than one pixel shift in the interferogram and does not cause the azimuth misregistration in the interferogram. The spatial TECV gradient at two separate SAR data acquisition times again suggest a maximum 3 pixel shift in the Iceland interferogram, which is consistent with the actual InSAR observation.

92 76 CHAPTER 6. ATMOSPHERIC ERRORS IN INSAR MEASUREMENTS California region results Figure 6.9 (left) shows an interferogram acquired over California centered at N, W, which is formed by two sets of ALOS PALSAR data on June 9, 27 and September 9, 27 at 6:3 UTC. Figure 6.9 (right) shows the azimuth misregistration measured in pixels corresponding to the interferogram. There is no significant pixel misregistration and no obvious decorrelation in the interferogram follows the azimuth offset map. Range (pixels) 3 Azimuth Figure 6.9: (left) A California interferogram centered at N, W, which is formed by two sets of ALOS PALSAR data on June 9, 27 and September 9, 27 at 6:3 UTC. The image is presented in radar coordinates. The horizontal axis corresponds to the radar range direction and the vertical axis corresponds to the radar azimuth direction. (right) The azimuth misregistration measured in pixels corresponding to the California interferogram. Here the length of one azimuth pixel is about 3.6 m. We further analyze the GPS carrier phase data from a permanent GPS station ORES near the California ALOS PALSAR scene. The resulting TECV estimates

93 6.3. IONOSPHERIC ARTIFACTS IN INSAR DATA 77 TECV ORES,California,6/9/27 GPS PRN number 4 16 TECV ORES,California,9/9/27 GPS PRN number UTC (a) between 6 and 7 UTC on June 9, UTC (b) between 6 and 7 UTC on September 9, 27 Figure 6.1: Estimated TECV variation based on GPS carrier phase data from permanent GPS station ORES in California. The unit of the vertical TEC estimates is TEC units (TECU), or 1 16 electron/m 2. The GPS satellites are identified by the GPS receiver by means of pseudorandom noise (PRN) numbers and each color represents TECV measurements from a visible GPS satellite at GPS station ORES. spatial TECV variation ORES,6/9/27 GPS PRN number 4 16 Spatial TECV Variation ORES,9/9/27 GPS PRN number UTC (a) on June 9, 27, measured from all visible GPS satellites UTC (b) on September 9, 27, measured from all visible GPS satellites Figure 6.11: Spatial TECV variation near the California InSAR scene as estimated from GPS data. from different satellite-receiver pairs are shown in Figure 6.1. We approximate the common temporal TECV variation on each SAR acquisition day as the mean of all available TECV estimates. We then subtract off this assumed temporal trend to yield an estimate of the spatial component of the TECV variation as measured by each satellite. This is shown in Figure We observe that none contain irregularities similar to those observed in the Iceland results. Further, the magnitude of the spatial

94 78 CHAPTER 6. ATMOSPHERIC ERRORS IN INSAR MEASUREMENTS variation from every available satellite predicts no significant azimuth pixel shift, agreeing with nearby InSAR observations as well. While this null result may be unsatisfying, it nonetheless shows that the theory agrees with the observations Hawaii region Range (pixels) 3 Azimuth Figure 6.12: (left) A Hawaii interferogram centered at 19.2 N, W, which is formed by two sets of ALOS PALSAR data on May 5, 27 and June 2, 27 at 6:3 UTC. The image is presented in radar coordinates. The horizontal axis corresponds to the radar range direction and the vertical axis corresponds to the radar azimuth direction. (right) The azimuth misregistration measured in pixels corresponding to the Hawaii interferogram. Here the length of one azimuth pixel is about 3.6 m. As a further verification of our method, we also compare the simultaneous GPS and InSAR observations from a Hawaii region. Figure 6.12 (left) shows an interferogram acquired over Hawaii centered at 19.3 N, W, formed by two sets of ALOS PALSAR data acquired on May 5, 27 and June 2, 27 at 8:58 UTC. Figure 6.12 (right) shows the azimuth misregistration measured in pixels corresponding to