Rapid Finite Fault Inversion for Megathrust Earthquakes

Roberto F. Benavente Bravo

A thesis submitted for the degree of Doctor of Philosophy (PhD) The Australian National University

March, 2016 ii

c Roberto Benavente 2016 Declaration

This thesis is an account of research undertaken between December 2011 and December 2015 at the Research School of Earth Sciences, College of Physical & Mathematical Sciences, The Australian National University, Canberra, Australia.

Except where acknowledged in the text, the material presented in this thesis is, to the best of my knowledge, original and has not been submitted in whole or part for a degree in any university.

Roberto Benavente July, 2016

iii iv Acknowledgements

Many people have contributed to this research both directly by providing me with some advice and knowledge and indirectly with moral support and encouragement. I will do my best to include everyone in this list. Firstly, I would like to thank my supervisor Phil Cummins whose guidance and support were fundamental for this research. He introduced me to the topic of finite fault inversion, which I find fascinating, and showed me how it relates to natural hazards. His acute scientific insight was crucial every time I was stuck in my work, assisting me in resolving the issues and finding new directions for my research when pertinent. He was also extremely supportive when I was struggling with personal issues which is something I sincerely appreciate.

Big thanks to Jan Dettmer who came to RSES when I was half way through my PhD. I have learned a lot from him, especially about inversion theory and statistics which turned out to be the core aspects of my research. His meticulous and rigorous approach to inverse problems substantially improved my first inversion methods and greatly motivated the introduction of the Bayesian framework I discuss in Chapter 5. Prof. Malcolm Sambridge was also very supportive and was always willing to share some insights about the inverse problem theory. I also benefited from discussions and advice from Hrvoje Tkalˇcićand Brian Kennett.

The calculations of the forward model in all the applications in this thesis were made using a Green Function database kindly provided by Luis Rivera, Hiroo Kanamori and Zacharie Duputel. I would particularly like to thank prof. Rivera who gave me useful information about the computation of the database. My short research stay at ETH was made possible through the eﬀorts of Andreas Fichtner who also instructed me how to use SES3D software. Stan Dosso reviewed a preliminary version of the manuscript presented in Chapter 5 providing helpful insights.

At RSES we are fortunate enough to have an excellent professional staﬀ. The IT support team is outstanding and I always got timely assistance when required. Maree Coldrick, Sherryl Kluver and Mary Hapel were always keen to help me with all the administrative issues I came across. The vast majority of software I

v vi

developed relies on open source applications such as Python, NumPy, SciPy, Obspy, Matplotlib and OpenQuake. Thank you to all those developers for such an amazing work. I am also grateful to all the students and post docs of the Earth Physics group at RSES. Not only have I solved many research problems thanks to them but also they helped in hard times and oﬀered me sincere friendship. I will miss you guys! My PhD was supported by the Chilean scholarship program “Becas Chile”, this whole experience would not be possible without their ﬁnancial support.

Finally, I want to thank my family for their unconditional support. My wife Carolina has been supportive throughout this adventure. I feel very lucky to share my life with such an amazing person. My parents encouraged me to work hard and pursue my dreams with passion. Thanks to all my family and friends in Chile, I truly think you are all part of this achievement. Abstract

The largest earthquakes take place at subduction zones, and their devastating impact in populated regions is often exacerbated by their ability to excite powerful tsunamis. Today, we understand that large subduction earthquakes, known as megathrust events, are caused by the sudden release of elastic strain energy stored at the plate boundaries where a localized, previously locked, section of the megathrust ruptures. The rupture process can propagate over hundreds of kilometres and slip on the fault can be tens of meters. Using ground motion data to image the spatio-temporal spread of slip over the fault surface is known as ﬁnite fault inversion (FFI). Over the past decade FFI has become almost routine, so that results produced by diﬀerent groups are available within several days or even hours after a large event. However, these results typically require manual processing of the data, and are not accompanied by appraisals of uncertainty. My PhD research has focused on obtaining slip models for such events in near real time. I divided my analysis into three main projects that are discussed in this thesis.

First, I evaluated the performance of a long period seismic wave, the W-phase, which arrives between P and S waves, in a classic FFI scheme for the Maule (2010, Mw = 8.8) and Tohoku (2011, Mw = 9.1) events. I found that, despite its long period, the W-phase can resolve ﬁrst order features of the rupture for both events. Since the W-phase is not very sensitive to 3D structure, the processing of data for the W-phase is generally simpler than it is for the body and surface waves that are commonly used for FFI. In addition, the W-phase is fast and can be obtained soon after the arrival of the P-wave.

Second, I improve the classic inversion scheme to increase robustness and rigour for rapid inversions. The most remarkable aspects of this inversion approach are that the faulting surface is constrained to follow the 3D subducting slab geometry and that the smoothness of the rupture is objectively determined. I used this approach for the recent Illapel event (2015, Mw = 8.3) and showed that a meaningful preliminary model can be obtained within 25 minutes from rupture onset. A reﬁned solution can be obtained 1 hour from the origin time, which is still useful for the management of the disaster.

vii viii

Finally, I have developed a novel linearized inversion method that allows slip uncertainties to be estimated during rapid finite fault inversion. This is an intrinsically complex problem as normally positivity constraints are imposed on finite fault models to ensure well behaved solutions. Uncertainties are typically unavailable for FFI results, but they can be crucial for meaningful interpretation of the slip models. To estimate them, I follow a probabilistic Bayesian framework but avoiding the computationally demanding Bayesian sampling. Instead, by using a coordinate transformation, the posterior distribution is approximated and obtained by linearized inversion. This inversion scheme was tested employing both simulated and real W-phase data, showing that meaningful uncertainty estimates can be inferred. Comparison with Bayesian sampling is also performed suggesting that the error of approximating the posterior is small. Including uncertainty estimates in early finite fault models will reduce the risk of working with misleading solutions.

The rigour, objectivity and robustness of the inversion techniques devised in this thesis can be a valuable contribution to the FFI community. Since I have utilized mostly open source software and a desktop computer to carry out this research, the tools I have developed can be easily used for early warning in most seismic observatories. I believe that, when facing such disastrous events, the methods developed here can be important to assist authorities with emergency response. Contents

Declaration iii

Acknowledgements v

Abstract vii

1 Introduction 1 1.1 Thesis Outline ...... 4 1.2 Publication Schedule ...... 6

2 Background 9 2.1 Basic Theory in Source Studies ...... 9 2.2 Point Source Approximation ...... 13 2.2.1 Rapid determination of the moment tensor: Application to Geoscience Australia (GA) and Centro Sismol´ogicoNacional (CSN, Chile) ...... 20 2.3 Finite Fault Problem ...... 22 2.4 The Multiple-time-window Method ...... 25

3 W-phase Finite Fault Inversion 29 3.1 Abstract ...... 29 3.2 The W-phase and Rapid Source Characterization ...... 30 3.3 Approach ...... 32 3.4 Models for the Maule and Tohoku Earthquakes ...... 37 3.4.1 Maule earthquake ...... 38 3.4.2 Tohoku earthquake ...... 39 3.5 Conclusion ...... 40

4 Automated Algorithm for Finite Fault Inversion in Megathrust Events 43 4.1 Abstract ...... 43 4.2 Introduction ...... 44 4.3 Inversion scheme ...... 45 4.4 Preliminary Rapid W-phase point source solution ...... 48

ix x Contents

4.5 Estimation of required parameters ...... 50 4.6 Inversion Results ...... 54 4.7 Conclusion and discussion ...... 58

5 Estimating Uncertainty from Rapid Finite Fault Models 67 5.1 Introduction ...... 67 5.2 Bayesian Inversion Framework ...... 70 5.2.1 Parametrization and Positivity Constraints ...... 71 5.2.2 Positivity by variable transformation ...... 73 5.2.3 Uncertainty estimation by linearization ...... 75 5.2.4 Inversion strategy ...... 76 5.3 Inversion for simulated data ...... 78 5.3.1 Eﬀects of smoothing in logarithmic space ...... 79 5.3.2 Uncertainty estimation for diﬀerent noise levels ...... 80 5.3.3 Comparison with nonlinear Bayesian sampling ...... 81 5.4 Application: Illapel great earthquake (2015, Mw=8.3), Chile . . . . . 84 5.5 Summary and Discussion ...... 85

6 Conclusion and Discussion 89 6.1 Discussion and future work ...... 90

A Computation of Hessian and gradient of the non-linear objective function 93

B Conﬁdence intervals for log-normal distribution 97 List of Figures

1.1 Tsunami travel times for the Sumatra 2004 event. The seismic source region is indicated by a red ellipse and contours show the arrival time of the tsunami wave in hours. The dark red region indicates tsunami arrival within 1 hour of the event. The boxes enclose the

number of fatalities in the region and Hmax is the maximum calculated tsunami height. Taken from Institute of Computational Mathematics and Mathematical Geophysics, Siberian Division Russian Academy of Sciences. http://tsun.sscc.ru/TTT rep.htm ...... 5

2.1 Force couples defined by the moment tensor. The first index refers to the orientation of the forces, and the second to the separation between the two forces. Taken from Shearer (2009)...... 15 2.2 Relation between faulting process and force couple. Here are shown the two fundamental strike-slip movements of a fault and their respective moment tensor components. Taken from Shearer (2009). . . 16 2.3 General faulting model for shear slip. The fault is described by two angles (φ and δ) and the slip direction is indicated by λ. Taken from Shearer (2009)...... 17 2.4 Scheme of the focal sphere representation for an earthquake focal mechanism. Assuming a thrusting fault, an imaginary sphere surrounding the source is shown where the first vertical P wave motions are projected (left). The projection of the lower hemisphere (left) is taken commonly as a representation of the focal mechanism. Taken from Lay and Wallace (1995)...... 18 2.5 Common faulting geometries and their correspondent beach balls. Taken from Shearer (2009)...... 19 2.6 Focal mechanism we obtain for three recent earthquakes. At the top we show our result (blue beach balls) obtained through W-phase inversion in our code. At the bottom, we include the GCMT solutions (red beach balls) for the same events as a reference. For all the events we find the same magnitude as GCMT...... 21

xi xii LIST OF FIGURES

2.7 Comparison of observed and modelled W-phase waveforms from ﬁnite fault (FFI) and point source (PS). The station SEED codes are shown on the x-axis...... 22

2.8 A common spatial discretization for the faulting surface. The source is modelled as a plane which is comprised of rectangular subfaults. Taken from Olson and Apsel (1982) ...... 24

2.9 Multiple-time-window scheme. The subfaults are represented by rectangles. The concentric circles represent hypothetical rupture fronts propagating outwards from the hypocentre with the same velocity. At the bottom is shown an example of a time function (triangules) used to parametrize each time window. Taken from Ide (2007). . . . 26

3.1 A seismic record of the W-phase. The observed displacements are shown by the black trace and the synthetics by the red trace. The time starts at the origin time (O.T.). Taken from Kanamori and Rivera (2008)...... 31

3.2 Synthetic example for the Maule event: input model (top) and output solution (bottom). The arrows indicate the rake distribution. From left to right the three patches of the input model have a slip of 60m, 40m and 20m respectively and were created at diﬀerent time windows. The size and the color of the arrows are used to indicate the magnitude of the slip. The half duration of each triangle and the time diﬀerence between them is 10 s and the maximum rupture velocity was taken as 1.5 km/s. The output model was obtained after polluting the synthetic traces with a 10% of Gaussian noise...... 34

3.3 Solution for the Tohoku event using one time window. The hypocentre is indicated by a blue star while the slip is shown by the color scale...... 35 LIST OF FIGURES xiii

3.4 Results for the Maule event. Waveform ﬁts are shown (left) for selected stations. Observed displacements and their respective synthetics are indicated by a continuous green line and a dashed blue line, respectively. Note that each trace is plotted in the W-phase time

window (tp, tp + 15∆, ∆: station-epicentre distance in degrees) and the epicentral distance is shown next to the station name. A reference model is plotted (center) from Koper et al. (2012). Diﬀerent slip regions are indicated by contours, the yellow rectangle marks the faulting area considered and the hypocentre is indicated by a white star. The solution we achieved is at the right side, plotted in the same way as the reference model (center)...... 38

3.5 Results for the Tohoku event. The details are the same as ﬁgure 3.4. However, in this case the reference solution (center) is from Ammon et al. (2011). The source rigidity structure we used for both the W- phase solution (right) and the reference solution is from Lay et al. (2011) ...... 39

3.6 Rake distributions for the Maule (top) and the Tohoku (bottom) events. The rake angle is indicated by the arrows and the color scale indicate the slip...... 41

4.1 Slip distributions obtained using diﬀerent bandpass ﬁlters. For all cases we use the largest dataset (stations up to 90o epicentral distance) and a maximum rupture velocity of 1.5 km/s. Periods considered are: 100-250 s (left), 150 - 500 s (center) and 200 - 1000 s (right)...... 47

4.2 Rapid WPPS for dataset with stations up to 30o (A), 50o (B), 75o (C) and 90o (D) of epicentral distance. Station locations (blue dots) considered in the inversion and the source mechanism (beachball, displayed at the centroid location) are shown for each case...... 49

4.3 Diagram of the algorithm employed in our FFI method. FDSN refers to the international Federation of Digital Seismograph Networks. . . . 50

4.4 The fault model used in the inversions. Each dot shows the location of a point source in our ﬁnite fault model. The color scale represents the depth of each subfault, the yellow star the hypocentre. The discretization is 25 km...... 52 xiv LIST OF FIGURES

4.5 Preferred slip distributions for each data set, see Table 4.2 for addi-

tional information about each model. ∆max indicates the maximum epicentral distance considered for a given model. In each solution the yellow box encloses the considered fault region, the star shows the hypocentre (i.e., nucleation point) and the red line indicates the trench. The contours enclose diﬀerent slip levels and the white arrows are the slip vectors...... 55 4.6 Observed (solid green) and predicted (dashed blue) W-phase waveforms for the data sets and models considered in Figure 4.5. The name and component of each station is shown in the label at the top left corner of each plot. ∆ corresponds to the epicentral distance of each stations. The x-axis shows the time in seconds from the theoretical P-wave arrival...... 56 4.7 Slip distributions obtained for a maximum rupture velocity of 1.5 km/s for the four data sets (see Table 4.2). Legends are the same as Figure 4.5...... 57 4.8 Slip distributions obtained for a maximum rupture velocity of 2.0 km/s for the four data sets (see Table 4.2). Legends are the same as Figure 4.5...... 57

o 4.9 Waveform ﬁts for data up to ∆max = 30 . Legends are the same as in Figure 4.6...... 59

o 4.10 Waveform ﬁts for data up to ∆max = 50 . Legends are the same as in Figure 4.6...... 60

o 4.11 Waveform ﬁts for data up to ∆max = 75 . Legends are the same as in Figure 4.6...... 61 4.12 Continuation of Figure 4.11...... 62

o 4.13 Waveform ﬁts for data up to ∆max = 90 . Legends are the same as in Figure 4.6...... 63 4.14 Continuation of Figure 4.13...... 64 4.15 Continuation of Figure 4.14...... 65 4.16 Continuation of Figure 4.15...... 65

5.1 Simulation geometry and true model for the simulations. Stations (blue), hypocenter (star) and fault geometry (left) are based on the 2014 Iquique earthquake. The true slip distribution (right) is chosen to be representative for an event of this size...... 79 LIST OF FIGURES xv

5.2 Simulation results for the log-space smoothing approach in terms of the MAP model. Results for 2% (left) and 5% (right) noise level are shown. Both results agree well with the true model (Fig. 5.1). . . . . 80 5.3 95% CIs of recovered models using log-space smoothing. Target model is shown in figure 5.1. CIs correspond to the models of figure 5.2, in the same order...... 82 5.4 Marginal PDFs for parameters of rake component 1. Each subplot shows the marginal PDF for the corresponding subfaults. The layout of the subplots is the same as the slip distributions shown in figures 5.1 and 5.3 so horizontal direction goes along strike and vertical direction along dip. The red curves were obtained using the analytic expression we proposed here and the blue histograms were obtained using Bayesian sampling of the posterior PDF...... 83 5.5 Marginal PDFs for parameters of rake component 2. Legends are same as figure 5.4 ...... 83 5.6 Inversion results for the Illapel event, stations up to 30o of epicentral distance. Legends are the same as figure 5.1, right. Left: MAP solution of the slip distribution. Right: 95% of CI width for slip on the fault...... 85 5.7 Inversion results for the Illapel event, stations up to 90o of epicentral distance. Legends are the same as figure 5.6...... 86 xvi LIST OF FIGURES Chapter 1

Introduction

Earthquakes are produced because some parts of the earth undergo brittle failure in response to an applied stress exceeding its yield strength. In particular, in large earthquakes, the faults exhibit stick-slip frictional behaviour. Brittle failure results in the generation of elastic waves, called seismic waves, which can be measured at the surface and analysed. These records, called seismograms, have been widely used to improve our understanding about both the Earth structure and the causes of earthquakes, or seismic sources. The stresses causing earthquakes can be produced by a variety of mechanisms, but relative motions of the tectonic plates comprising the earth’s surface, is the most important. Global seismicity is mainly concentrated in the plate boundaries and the largest earthquakes occur in subduction zones, that is, regions where one plate moves beneath another. This PhD research is focussed on the study of seismic sources in the particular case of large subduction earthquakes which are often called “megathrust earthquakes”.

The adequate characterization of the seismic source has been a fundamental topic in seismology since its origin as a quantitative science. The problem can be addressed at diﬀerent levels depending on how detailed a description is required. The most basic parameters which can be inferred from the seismograms are the location and the magnitude of an earthquake. However, seismic sources can be extremely complicated to describe. For instance, in megathrust earthquakes they can be thought as a rupture over a large section of the subduction plate boundary surface. As the rupture front propagates along the fault surface, seismic waves are radiated outward from the source. Thus, more parameters, such as geometry of the faulting, time dependency of the rupture, spatial slip distribution and rupture’s duration are needed to reach an accurate source description. Nevertheless, if the ruptured region has a characteristic length smaller than the radiated wavelength, it is possible to use a point source approximation for the source. Such an approximation allows us to avoid dealing with some of the most complex details of the source and focus on the average slip and fault geometry.

1 2 Introduction

An important advance in theoretical seismology was made by Burridge and Knopoff (1964), who showed the equivalence between a rupture or dislocation source, and a set of body forces applied at a point in the medium. The system of body forces replacing the dislocation is called an equivalent force representation. In particular, the authors found that if one considers an arrangement of four forces surrounding a point such that the net force and torque of the system are zero, the resulting radiation pattern in the far field is the same as that generated by a dislocation. Such a set of forces form a double couple. This approach has the advantage that the body forces are more easily integrated into the equations of motion. Moreover, as long as linear elasticity is an accurate description of the wave motion, such motion obeys the principle of superposition which is useful to construct more elaborated seismic sources. To date, many research groups compute point source solutions for different earthquakes, regularly providing a good characterization of the geometry of faulting and the magnitude for each event. However, these point source approaches cannot reveal more complex aspects of the rupture process such as the spatial and temporal slip distribution.

Once the fault surface (usually approximated by a plane) is determined, it is possible to find the particular slip distribution within the surface, in a procedure known as finite fault inversion (FFI). To constrain the slip distribution Olson and Apsel (1982) and subsequetly Hartzell and Heaton (1983) implemented a methodology which can estimate slip over a finite fault using teleseismic waveforms. They considered a predetermined fault plane divided into a number of subfaults and solved for their individual seismic moments prescribing a maximum rupture velocity. While many improvements have been made to this approach (a review can be found in Ide, 2007), the philosophy of virtually all FFI methods remains similar: Discretize the source’s surface into a number of units that can be treated as point sources. A simultaneous inversion is then carried out for the moment of each unit. Then, the spatial and temporal distribution of the slip for the whole fault is recovered by mapping the point source solutions onto the fault plane. In spite of the straightforwardness of this principle, in practice to retrieve a slip model of an earthquake can be a very complex process whether employing linear or nonlinear methods. Linear approaches require a large number of parameters to adequately represent the rupture process. This leads unavoidably to a highly under-determined problem which often is addressed by adding additional constraints such as smoothness. Non linear approaches need to deal with the issue of the nonuniqueness of the solution. Both of these requiere either extensive computation or manual selection 3

of, for instance, model smoothness. Thus, it is still diﬃcult to get a slip model soon after large earthquakes, but the usefulness of a timely imaging of the source have led us to explore this problem.

In the last decade, a seismic wave called the W-phase has gained special importance as a technique for rapid point source inversions for large earthquakes. Firstly reported by Kanamori (1993), the W-phase is a long period wave arriving at the recording station together with the P wave. Because of its small amplitude, which prevents clipping in most near ﬁeld stations, and the time window used (prior to surface waves arrivals), the inversion of W-phase waveforms provides an eﬀective method for rapid determination of the moment tensor. Moreover, if the instrument response is deconvolved in the time domain, the clipping in the records typical for large events can be avoided. Thus far, W-phase inversions have been shown to be a robust and reliable method for point source inversion in large earthquakes,

Mw > 6.5 (Kanamori and Rivera, 2008; Duputel et al., 2012a), and it is a standard solution in the USGS catalogue. These characteristics have led us to evaluate the use of the W-phase in FFI.

One of the main criticism of the FFI relates to the variability of diﬀerent solutions for the same event (e.g., Beresnev, 2003). Also, while in common FFI schemes many assumptions about the rupture process need to be made, sensitivity to diﬀerent choices is not often discussed. Then, there is an inherent risk in taking a single model as a solution to the FFI: One may be interpreting aspects of the rupture pattern that in fact are not well resolved by the data which, in addition, can be noisy. In contrast, in the probabilistic Bayesian framework (e.g., Tarantola, 2005; Sivia and Skilling, 2006; Aster et al., 2005) the solution to the inverse problem is given by a probability distribution, known as the posterior distribution, which expresses the degree of knowledge about the model space given some prior information and data. When the inverse problems present strong non-linearities the best way of characterizing the posterior is randomly evaluating it in a procedure called Bayesian sampling. However, if the non-linearities are moderated it is possible to approximate the posterior (normally by a Gaussian distribution) and solve the inversion problem using iterative optimization schemes. Since Bayesian sampling is an extremely computationally demanding technique, it is not suitable yet for rapid FFI. Therefore, I devote the last part of my PhD exploring a method to obtain a rapid characterization of the posterior distribution in FFI. In particular, I study the case when the slip on the fault is required to be positive and the Gaussian approximation for the posterior can be poor. 4 Introduction

The devastating Sumatra earthquake (2004, Mw = 9.2) provides a good example of how rapid ﬁnite fault models can assist in the emergency response. At that time there were no reliable methods available to perform a rapid estimation of the extension of the fault or the peak/average slip on the fault. In fact, the magnitude of the event was underestimated in preliminary results. Figure 1.1 shows tsunami travel times for the event. In order to describe the observed tsunami travel times a seismic source of a length of ∼ 1000 km must be considered (red ellipse). The largest number of fatalities occurred near the hypocentre in Indonesia, where the waves struck within one hour of the event.However, because the length of the rupture was unusually long, many people in Sri Lanka and Thailand were also exposed to large tsunami waves 2 hours after the event. An adequate seismic source model and, of course, an adequate early warning infrastructure, would have been crucial in alerting population at a regional and teletsunami distances. Thus, I believe the development of inversion techniques and computer algorithms that can allow us to retrieve reliable source model timely is a crucial research task to pursue.

1.1 Thesis Outline

Throughout the next chapters I shall discuss relevant aspects of the theory of seismic sources and FFI, present the results of its application to recent megathrust earthquakes and discuss methodologies I have developed to perform rigorous and automated FFI. The main focus of these methodologies is the application of rapid inversion algorithms that can reveal preliminary characteristics of the rupture. This information can be valuable for disaster management and tsunami early warning at regional and teletsunami distances.

Chapter 2 presents a summary of the seismic source theory I rely on in this thesis. Basic concepts such as the point source, moment tensor and scalar moment are introduced in the context of the underlying elastodynamic theory. The ﬁnite fault problem is also discussed and the linearized parametrization scheme known as the Multiple Time Window Method (MTWM) is presented. In addition, the main features of the seismic phase known as the W-phase and its use in source inversion are also presented. Importantly, all the work exposed in this thesis relies on the W-phase as data and the MTWM as parametrization scheme.

In Chapter 3 the W-phase is employed to obtain rupture models for two §1.1 Thesis Outline 5

Figure 1.1: Tsunami travel times for the Sumatra 2004 event. The seismic source region is indicated by a red ellipse and contours show the arrival time of the tsunami wave in hours. The dark red region indicates tsunami arrival within 1 hour of the event. The boxes enclose the number of fatalities in the region and Hmax is the maximum calculated tsunami height. Taken from Institute of Computational Mathematics and Mathematical Geophysics, Siberian Division Russian Academy of Sciences. http://tsun.sscc.ru/TTT rep.htm 6 Introduction

megathrust events: Maule (2010, Mw = 8.8, Chile) and Tohoku (2011, Mw = 9.0, Japan). The main purpose of that study is to establish if the W-phase can provide meaningful information about the rupture kinematics. The data processing and inversion scheme are discussed as well as synthetic simulations. Comparison with alternative solutions from diﬀerent authors is also presented. The W-phase source models found for both events are reasonable and similar to other studies which utilize more complex processing techniques.

In Chapter 4 an automated FFI algorithm for megathrust events is presented. The algorithm is applied to the recent Illapel event (2015, Mw = 8.3, Chile). Again W-phase waveforms are used to constrain the slip distribution. Explanations about how to prescribe the fault surface using the slab geometry is provided. Special attention is given to the automated estimation of the optimal smoothness of the slip distribution. Computational time and evolution of the solution when more data are included in the inversion are explicitly addressed. I also discuss potential implementation issues and generalizations of the technique for use in near real-time. In Chapter 5 a new inversion approach is developed which can be used to estimate uncertainties in FFI with positivity constraints. I begin by providing a general background and introducing the Bayesian inversion framework. Then analytic expressions for the posterior distribution are derived and discussed. Next, the approach is utilized with simulated W-phase data and compared with a nonlinear Bayesian sampling method. I conclude by presenting results of this approach for the Illapel event showing that meaningful uncertainty estimates can be obtained.

Chapter 6 summarizes the main ﬁndings of this PhD thesis. General remarks and potential future research are discussed.

1.2 Publication Schedule

The material in Chapters 3 and 4 is already published in the journal Geophysical Research Letters. Chapter 5 will be submitted soon to a peer reviewed journal. I am the ﬁrst author in all the articles and I carried out the vast majority of the research, writing the algorithms and developing the required theory when necessary. In addition I have contributed to two articles by providing and modifying my point source and ﬁnite fault inversion programs to perform essential tasks in the collabo- rative research. A full list of the publications I was involved during my PhD is given below. Main author publications: §1.2 Publication Schedule 7

• Chapter 3: Roberto Benavente and Phil R. Cummins: “Simple and reliable ﬁnite fault solutions for large earthquakes using the W-phase: The Maule (Mw= 8.8) and Tohoku (Mw= 9.0) earthquakes” Geophysical Research Letters 40(14), 2013. DOI: 10.1002/grl.50648

• Chapter 4: Roberto Benavente, Phil R. Cummins and Jan Dettmer: “Rapid automated W-phase slip inversion for the Illapel great earthquake (2015, Mw = 8.3)” Geophysical Research Letters 43(5), 2016. DOI: 10.1002/2015GL067418

• Chapter 5: Roberto Benavente, Jan Dettmer, Phil R. Cummins and Malcolm Sambridge: “Bayesian uncertainty estimation in rapid ﬁnite fault inversion with positivity constraints” To be submitted.

Contributor publications:

• Jan Dettmer, Roberto Benavente, Phil R. Cummins and Malcolm Sambridge: “Trans-dimensional ﬁnite-fault inversion” Geophysical Journal International 199(2), 2014. DOI: 10.1093/gji/ggu280

• Sebastian Riquelme, Francisco Bravo, Diego Melgar, Roberto Benavente, Jianghui Geng, Sergio Barrientos and Jaime M. Campos: “W-phase source inversion using high-rate regional GPS data for large earthquakes” Geophysi- cal Research Letters 43(7), 2016. DOI: 10.1002/2016GL068302 8 Introduction Chapter 2

Background

Ground motion can be generated by a wide variety of seismic sources. However, our particular interest is to study megathrust earthquakes which often generate devastating tsunamis. Thus, we shall focus on the theory of internal sources (i.e. caused by the internal dynamics of the Earth). Particularly, we will use the model of an elastic dislocation fault source. We will think of an earthquake as produced by a rupture which is propagated over a fault’s surface. For brevity, we will refer to this kind of source as a seismic source.

In this Chapter we will introduce some of the key concepts which are fundamental to understand the most common difficulties that one can face when studying the seismic source. These are also important in developing the basic theory on which the methods we use rely. First, the most fundamental quantities we employ are defined to obtain the basic equation to predict ground motion given a seismic source. Then, this relation is used to explore two source descriptions, the point source and the finite fault, and show the basis of the inverse problem we have to solve. Finally, we explain the specific approach we take in our study and define the concepts related to it.

2.1 Basic Theory in Source Studies

The theory of seismic sources is fundamental in modern seismology and has been extensively discussed throughout years of development. Moreover, several classical textbooks discuss in detail its core concepts and results (e.g., Ben-Menahem and Singh, 2000; Aki and Richards, 2002; Lay and Wallace, 1995). A particularly detailed explanation about how the representation and uniqueness theorems in the context of linear elasticity lead to a description of the displacement ﬁeld given a seismic source is provided by Aki and Richards (2002). In this section, however, we will follow a more descriptive approach from Mura (1963) because our principal goal is to show how the problem can be addressed using basic concepts of linear

9 10 Background

elasticity. Also, our research is mainly directed to describing the source using existing methods instead of making fundamental theoretical improvements. A more general representation theorem for seismic sources was derived by Burridge and Knopoﬀ (1964).

The main (forward) problem we want to address can be formulated as follows: Having a seismic source completely determined, what is the displacement field generated in the elastic medium? Here, we will show that it is possible to infer the displacement field in the whole medium if we know exactly the spatial and temporal dependence of the seismic source. If we assume linear elasticity, that is, stresses and strains are linearly related, the displacement field obeys:

ρ u¨i = fi + (Cijkl uk,l),j, (2.1)

where ui = ui(~x,t) is the displacement field, ρ = ρ(~x) is the density of the medium, fi = fi(~x,t) is a body force and Cijkl = Cijkl(~x) represents the elastic constants of the medium. Also, we employ the Einstein summation convention and dot notation for time derivative. In our derivation we assume a homogeneous medium so the elastic constant does not change with position. Moreover, we will assume no body forces and therefore fi = 0. We will also assume an infinite medium, although in general, a similar reasoning can be applied for a finite medium by including appropriate boundary conditions. As we mentioned before, the seismic source will be modelled as a rupture in the medium which in turn can be seen as a 0 0 discontinuity in displacement vector ∆~u = ∆~u(xi, t ) defined over a faulting surface S. This discontinuity is called a dislocation. Hence, the mathematical problem we will address is: What is the displacement field ui(~x,t) which is generated by 0 0 0 a discontinuity ∆~u(xi, t ), xi ∈ S and governed by equation 2.1? The resulting expression will define our forward problem.

In order to obtain a useful expression to calculate the desired displacement ﬁeld, it is convenient to introduce the elastodynamic Green’s function of the medium. This function contains information about the response of the medium to a unit impulsive force applied in a given position, at a given position and time. We say that such a force fj and the resulting displacement ﬁeld ui are related by:

0 0 0 0 0 0 0 0 ui(~x,t) = fj(~x , t )Gij(~x − ~x , t − t ) =x ˆjδ(~x − ~x )δ(t − t )Gij(~x − ~x , t − t ), (2.2) §2.1 Basic Theory in Source Studies 11

where the second order tensor Gij is known as the Green’s function of the medium.

Using equation 2.1 it follows that Gij obeys the equation of motion:

0 0 0 0 ¨ 0 0 CijklGkm,lj(~x − ~x , t − t ) + δimδ(~x − ~x )δ(t − t ) = ρGim(~x − ~x , t − t ). (2.3)

Generally speaking, obtaining the Green’s function for a complex medium can be extremely diﬃcult and we will consider it as a separate problem. For now we will keep the focus on the description of the source.

Following Mura (1963), our starting point will be some symmetry considerations on Cijkl. It is well know that, due to the symmetry of the stress and strain tensors and thermodynamical arguments, the elastic tensor has the properties:

Cijkl = Cjikl = Cijlk = Clkij. (2.4)

Then, we can write the following identity:

0 0 0 0 0 0 0 0 Cijkl uk,l(~x , t ) Gim,j(~x − ~x , t − t ) = Cijkl ui,j(~x , t ) Gkm,l(~x − ~x , t − t ). (2.5)

The right hand side can be obtained easily after renaming the dummy indices i ↔ k, j ↔ l and using equation 2.4. The next step is to integrate equation 2.5 over the whole, inﬁnitely extended, material V with respect to the points ~x0 . Physically speaking, the primed coordinates will refer to the source space and the unprimed coordinates to the observation space. Prior to the integration, we have to notice that our material contains a discontinuity in ~u deﬁned over a surface S (this is our model of a seismic source):

0 0 0 0 0 0 0 0 ∆ui(~x , t ) = ui(~x , t )|S+ − ui(~x , t )|S− , ∀~x ∈ S(t ), (2.6)

S+ and S− denote the faces of S. Integrating the left hand side of equation 2.5 (LHS) and using Gauss’ theorem yields: Z Z 0 0 LHS = − Cijkl uk,lGim dSj + Cijkl uk,ljGim dV S(t0) V Z 0 0 0 0 0 = ρu¨i(~x , t )Gim(~x − ~x , t − t ) dV . (2.7) V

The surface integral in equation 2.7 in principle must be carried out over a surface at inﬁnity and positive and negative faces of S(t0). However, we assume that the displacements are zero far from the dislocation and that the strain and stress are continuous at S(t0), thus the surface integral vanishes. Using equation 2.1 in the 12 Background

volume integral we get easily equation 2.7. Integration of equation 2.7 over t0 ∈ ] − ∞, ∞[ gives: Z Z 0 0 0 ˙ 0 0 0 0 LHS dt = ρu˙ i(~x , t )Gim(~x − ~x , t − t ) dt dV , (2.8)

˙ where a static situation is assumed in the distant past and future, that is: Gim = 0 Gim = 0 for t = ±∞. The integration of the right hand side of equation 2.5 (RHS) is: Z Z 0 0 RHS = Cijkl uiGkm,l dSj + Cijkl uiGkm,lj dV S(t0) V Z Z 0 0 0 ¨ 0 0 0 = Cijkl ∆ui(~x , t )Gkm,l dSj + ρGimui(~x , t ) dV S(t0) V Z 0 0 0 0 0 − δimδ(~x − x, t − t )ui(~x , t ) dV . (2.9) V

Again, the displacements are assumed to be zero at inﬁnity, but in this case the integrand in the surface integral is discontinuous (equation 2.6) so we can express this integral in terms of the jump ∆u in the displacement. For the volume integral we use equation 2.3 to get equation 2.9. Integration of equation 2.9 over t0 ∈]−∞, ∞[ gives: Z Z Z 0 0 0 0 ˙ 0 0 0 RHS dt = Cijkl ∆u(~x , t )Gkm,l0 dt dSj + ρGimu˙ i(~x , t ) dt dV

− δimui(~x,t). (2.10)

− + The normal vector of the dislocation’s surface is pointing from S to S and Gkm,l0 = 0 0 ∂Gkm/∂xl. Again we assume a static situation for t = ±∞. Now, the desired expression to compute the displacements in the medium can be achieved easily by equality of equations 2.8 and 2.10. This yields: Z 0 0 0 0 0 0 um(~x,t) = Cijkl ∆ui(~x , t )Gkm,l0 (~x − ~x , t − t ) dt dSj. (2.11)

Although we have made some assumptions for simplicity’s sake such as inﬁnitely extended and homogeneous medium and that the dependence of G on ~x0 and ~t0 is only via ~x − ~x0 and t − t0, equation 2.11 is quite general. A complete derivation of 2.11, including comments on its validity is found in Aki and Richards (2002) Chapters 2 and 3.

The expression 2.11 is fundamental in seismic source studies and will be the §2.2 Point Source Approximation 13

basis for our research. In principle, this equation implies that if we had a perfect knowledge of the source, the elastic constant and the Green’s function of the Earth, we would be able to obtain the displacement ﬁeld at any point of the Earth. It should be noticed that our problem is inverse: given an observed displacement ﬁeld we want to obtain the slip distribution ∆ui over the faulting surface. An important approximation is usually made in order to simplify the calculations while dealing with equation 2.11. This consists in considering the source as a purely shear dislocation, that is, dSj · ∆uj = 0 over the whole fault. Although we will not mention explicitly this approximation it will be a basic assumption when calculating our source models.

2.2 Point Source Approximation

Probably the most important concepts in source studies are introduced and under- stood in terms of what we call the “point source approximation”. Although it can be viewed in several ways, we will say that this approximation is basically a way of simplifying equation 2.11 with the aim of obtaining an insightful interpretation for a simple source. This procedure results in a natural deﬁnition of key concepts in seismology such as moment tensor, seismic magnitude and force couples that we explain in this section. As in Section 2.1 our purpose here is to give an overview of the physical quantities we utilize in our work. For more exhaustive explanation and rigorous derivations it is recommended that the reader consult, for example, Ben- Menahem and Singh (2000); Aki and Richards (2002); Lay and Wallace (1995); Madariaga (2007).

Here, we make two basic approximations to treat equation 2.11, namely:

1. The radiation produced by each source element dS~ belonging to S(t0) is in phase with each other.

2. At any time t0, |~x − ~x0| >> l0, where l0 is a characteristic length of the fault. Then, we are interested in the radiation at large distances compared to the source dimensions. 14 Background

Therefore, equation 2.11 can be rewritten: Z 0 0 0 0 0 um(~x,t) = Cijkl ∆ui(~x , t )Gkm,l0 (~x − ~x , t − t ) dSj dt Z Z 0 0 0 0 0 ≈ Gkm,l0 (~x − ~x0, t − t ) Cijkl ∆ui(~x , t ) dSj dt t0 S(t0) Z Z 0 0 0 0 0 = Gkm,l0 (~x − ~x0, t − t ) mkl(~x , t ) dS dt t0 S(t0) 0 = Gkm,l0 (~x − ~x0, t) ∗ Mkl, (2.12)

0 where ~x0 is the location of the point source and we have deﬁned:

0 0 0 0 0 mkl(~x , t ) = Cijkl(~x ) ∆ui(~x , t )x ˆj (2.13) and: Z Mkl = mkl dS. (2.14) S(t0) The symbol “∗” denotes time convolution. At first sight, it may appear that equation 2.12 does not represent a huge improvement from equation 2.11. However, from the definitions 2.13 and 2.14 one can see that in equation 2.12 the effects of the source and the path connecting the source and the measurement point (~x,t) are separated. In fact, the quantity Mkl summarizes the information about our simple source model and it is called the

“seismic moment tensor”. Consequently, the quantity mkl is know as the “moment tensor density”.

The connection between the moment tensor and the faulting plane can also be seen in terms of force couples. In fact, if we compare equation 2.2 with equation 2.12, it is possible to conclude:

0 ∂Gkm(~x − ~x0, t) um(~x,t) = 0 Mkl ∂xl h i M ≈ G (~x − ~x0 , t) − G (~x − ~x0 + d,~ t) kl dˆ km 0 km 0 d l 0 0 ~ = Gkm(~x − ~x0, t)fk − Gkm(~x − ~x0 + d, t)fk. (2.15)

Here, d~ is a small arbitrary vector. So, if

Mkl = fkdl, (2.16) then we can rewrite 2.12 in terms of two unit impulsive forces according to 2.2. We §2.2 Point Source Approximation 15

Figure 2.1: Force couples deﬁned by the moment tensor. The ﬁrst index refers to the orientation of the forces, and the second to the separation between the two forces. Taken from Shearer (2009).

notice that we can think of each force as applied at a small distance from ~x0, but acting in opposite direction. This set of forces is termed “force couple” and we will see that it is intuitively connected with the geometry of faulting. It is important to notice that equation 2.16 relates a property of the seismic source with a body force applied in the absence of a dislocation. Burridge and Knopoﬀ (1964) proved that in terms of radiated seismic waves both representations of the source are equivalent. That is one of the most fundamental results in seismology.

The nine force couples deﬁned by the moment tensor are shown in Figure 2.1. While seismic sources of varied geometries can be constructed by appropriately combining these couples two conditions are useful to assume. First, as we can connect the diagonal elements of Mij with volumetric changes in the source, a plausible condition for many earthquake sources would be M11 + M22 + M33 = 0. That implies that no volumetric changes have occurred in the source. Of course, one can lift this condition to allow for more general sources, but in real inversion problems it is hard to constrain using observed data all the diagonal terms of Mij in shallow earthquakes (Duputel et al., 2012a) so this condition can be very helpful. 16 Background

Figure 2.2: Relation between faulting process and force couple. Here are shown the two fundamental strike-slip movements of a fault and their respective moment tensor components. Taken from Shearer (2009).

Another important condition arises from the fundamental physical fact that if there are no external forces or torques acting over the medium then the linear and angular moment must be conserved. From Figure 2.1 one can easily see that the oﬀ-diagonal terms of Mij will produce a net angular moment, but if we impose the condition Mij = Mji all these contributions will cancel out. That means that instead of modelling an earthquake using a single force couple, a double couple seems to be more physically consistent. In fact, Burridge and Knopoﬀ (1964) show that for a simple dislocation model a double force couple exhibits exactly the same radiation pattern. Therefore, a symmetry condition on Mij is not only useful but also realistic for most of the sources generated by internal processes of the Earth.

Force couples and faulting process can be associated in a straightforward manner. Let us model a shear faulting as the relative motion between two blocks as shown by Figure 2.2. Clearly this type of motion is a simplified version of our original model in which the strength of the dislocation ∆u(x0, t0) may change over the fault. However, one can assume that it is possible take an effective slipu ¯ as the average slip within the faulting surface. Also, this simplified version of the failure has proven to be extremely useful in quantifying source properties. It is important to note that owing to the symmetry of the moment tensor the two fault planes shown in Figure 2.2 are associated with the same double couple model.

Other geometries for the faulting planes can be also described by the moment §2.2 Point Source Approximation 17

Figure 2.3: General faulting model for shear slip. The fault is described by two angles (φ and δ) and the slip direction is indicated by λ. Taken from Shearer (2009). tensor. In general we can think of a fault source described by a plane that can be fully described by two angles, and one more is needed to account for the direction of the slip. This is illustrated in Figure 2.3. To orient the fault plane we use the strike φ (the angle between the projection of the plane on the surface and the North) and the dip δ (the inclination of the plane referred to the horizontal). The direction of relative motion of the blocks is given by the rake λ (the angle formed by the slip vector and the strike). It is common to call the lower block “Foot wall” and the upper block “Hanging wall”. As a matter of fact, our research is mainly focused in the case in which the hanging wall moves upwards (reverse faulting) and the dip angle is lower than 45o. This geometry is know as “thrusting faulting”. Given a set of φ, δ and λ representing slip on a fault, it is possible to ﬁnd a moment tensor which describes the same failure while, as we discussed before, a moment tensor is associated with two faulting planes. The relation between φ, δ and λ with each moment tensor component is discussed in detail by Aki and Richards (2002), BOX 4.4.

A convenient method to display the faulting geometry or “focal mechanism” is the use of the “focal sphere”. The underlying idea here is that the focal mechanism will determine whether the ﬁrst vertical motion at the surface will be upwards (compressional) or downwards (dilatational) for rays emanating downwards from the source. This involves the assumption of a simpliﬁed earth structure, which is given by the Green’s function in our original formulation (see equation 2.12) but 18 Background

Figure 2.4: Scheme of the focal sphere representation for an earthquake focal mechanism. Assuming a thrusting fault, an imaginary sphere surrounding the source is shown where the ﬁrst vertical P wave motions are projected (left). The projection of the lower hemisphere (left) is taken commonly as a representation of the focal mechanism. Taken from Lay and Wallace (1995).

it is still providing an excellent insight into the seismic source’s geometry. Figure 2.4 shows how a given faulting mechanism (left) produces a characteristic pattern of compressional and dilatational motion at the surface. When such a pattern is drawn using a stereographic projection of the lower hemisphere, as shown by Figure 2.4 (right), a beach ball-like ﬁgure is produced. In this kind of diagram the region representing a compressional P wave motion is shaded so one can easily distinguish between diﬀerent focal mechanisms. In Figure 2.5 we show how the main faulting geometries look in this focal sphere representation.

Thus far, we have discussed how the point source approximation can be used to extract information about the geometry of the faulting process. From equation 2.13 it can be seen that the moment tensor representation also provides information about the magnitude of the mean slip over the fault. In terms of the moment tensor components we can deﬁne the scalar moment M0 (Silver and Jordan, 1982) as:

!(1/2) 1 X 2 M0 = √ Mij . (2.17) 2 ij

The scalar moment is a fundamental quantity in seismology and was introduced by Aki (1966) who related it to the fault’s area A, the average dislocation D¯ and §2.2 Point Source Approximation 19

Figure 2.5: Common faulting geometries and their correspondent beach balls. Taken from Shearer (2009).

the rigidity µ of the source for a simple faulting model via:

¯ M0 = µDA. (2.18)

Subsequently, Kanamori (1977) used M0 to define the “Moment magnitude” Mw of an earthquake as: 2 M = (log M − 9.1), (2.19) w 3 0 where M0 is given in Nm. This magnitude, among other advantages, provides a direct quantification of the strength of the seismic source and it was developed to be consistent with other scales defined previously. To date, it has been adopted as a standard way to measure an earthquake’s size and it is calculated soon after an event by several agencies on a routine basis. 20 Background

2.2.1 Rapid determination of the moment tensor: Appli- cation to Geoscience Australia (GA) and Centro Sis- mol´ogicoNacional (CSN, Chile)

To determine the moment tensor of an earthquake is now a fundamental problem in seismology. Assuming that we know the Green’s function of the medium and we have collected sufficient records of the ground displacement we can invert 2.12 for the moment tensor components. While this approach seems to be quite straightforward, a number of issues appear when we are interested in a rapid characterization of the source, especially for large events. Essentially, the data needed to characterize properly such events must contain long period information that is more readily determined by surface waves and free oscillation. However, these phases can take many minutes to arrive at the seismograpic station and in very large events a couple of hours of data may be needed to capture the real size of the event. This is clearly an important limitation for technical governmental agencies that need to immediately inform to the emergency responders about the potential impact associated with an event (e.g., tsunamigenecity, expected damage in the urban areas, etc). Here we will briefly describe work that we have been carrying out with Geoscience Australia1 (GA) that can help them to promptly estimate a moment tensor solution for large earthquakes. In addition we mention how the rapid point source inversion code we wrote is being used at Centro SismológicoNacional (CSN, Chile)

One main focus of our research so far is the rapid characterization of the seismic source. As part of the work that we have developed in this PhD research we have implemented software that can potentially be used in a real-time context to determine point source solutions. Basically, we are using a novel technique developed by Kanamori and Rivera (2008) and Duputel et al. (2012b) which exploits the advantages of using a seismic wave called the “W-phase” (Kanamori, 1993) in the inversion. The W-phase is particularly suited to perform rapid real-time inversions for the moment tensor and we describe some of the main features of the W-phase in Chapter 3. At this stage our code is able to compute reasonable results for past events and the system is being tested for operational purposes. Details about the inversion approach are given in Chapter 4.

We show the results we obtained for three recent large earthquakes in Figure 2.6. Blue beach balls are used to display our solutions and the Global centroid

1http://www.ga.gov.au/ §2.2 Point Source Approximation 21

Figure 2.6: Focal mechanism we obtain for three recent earthquakes. At the top we show our result (blue beach balls) obtained through W-phase inversion in our code. At the bottom, we include the GCMT solutions (red beach balls) for the same events as a reference. For all the events we ﬁnd the same magnitude as GCMT. moment tensor 2 (GCMT) solution are shown by red beach balls. The 2010 Maule

Mw = 8.8 and 2011 Tohoku Mw = 9.0 events are examples of megathrust events that excited powerful tsunamis over the Chilean and Japanese coasts respectively. We examine in detail both events in Chapter 3 employing a more sophisticated technique. Although our solution for the Maule event presents the strike slightly rotated compared with the GCMT, the thrust geometry is clear. For the Tohoku event the solutions are almost identical. The last event we include here is the 2012

Sumatra Mw = 8.6 which is the largest strike-slip event ever recorded. Also, a degree of complexity has been observed in this event requiring slightly more complex techniques to explain the observations more satisfactorily (Duputel et al., 2012c). However, we can recover a fairly good solution where the strike-slip geometry of the source is evident.

One of the main advantages of our implementation of the point source inversion algorithm originally written by Duputel et al. (2012b) is that it can be easily adapted to particular purposes. This is mainly because the code is written in Python and employs the extensive scientiﬁc library that it is available for it. In Chile the CSN has an especial interest in speeding up source inversion using regional data. Because of this, they have adapted the code to be used with continues GPS

2http://www.globalcmt.org/ 22 Background

Figure 2.7: Comparison of observed and modelled W-phase waveforms from ﬁnite fault (FFI) and point source (PS). The station SEED codes are shown on the x-axis.

(cGPS) data. When cGPS sensors are available, a solution can be obtained within 4 to 5 min from the origin time Riquelme et al. (2016). Techniques like this can be extremely important for tsunami early warnings since the information about the source can be retrieved so quickly that there is some time for the authorities to issue appropriate alerts. One of the main purposes of this PhD is to employ the W-phase in finite fault studies which represent a more detailed description of the source (see section 2.3). Such description allows us to better explain the observations (in this case W-phase waveforms) than we can do with point source approximations. In figure 2.7 we show a comparison between observed W-phase data and synthetic data obtained from finite fault and point source modelling. It can be seen that in most cases the finite fault synthetics follow more closely the observed waveforms.

2.3 Finite Fault Problem

In Section 2.1 we presented some results from seismic source theory. In particular, we obtained an expression (see equation 2.11) to compute the displacement ﬁeld given the details of a faulting source. Then, in Section 2.2 we made some approximations to equation 2.11 in order to derive some useful expressions and gain a degree of geometrical insight into the seismic source. We discussed how a simple fault model can be constrained using records of ground motion. Here we introduce a technique that can be used to determine more complex aspects of the §2.3 Finite Fault Problem 23

source. In order to get explicit expressions that we can invert for a source model it is necessary to postulate a parametrization for the problem. Below, in Section 2.4 is shown the parametrization we use for our application which leads to a linear inverse problem.

As we have discussed, an earthquake can be model as the radiated seismic energy from a faulting source. However, large earthquakes are associated with complex faulting processes that are not adequately described by the point source approximation. A finite fault inversion (or finite slip inversion) is a numerical estimation of the spatial and temporal evolution of slip over a fault surface that best fits the observation. Usually, seismic waveforms are used as data, but in some cases geodetic and/or tsunami waveforms are also used. The construction of accurate slip models of large earthquakes is very important in seismology. For example, seismic hazard can be assessed by comparing ruptured areas of historical earthquakes in similar regions (e.g., Vigny et al., 2011). Moreover, a rapid finite source model can be used to evaluate tsunamigenecity, by estimating the seafloor displacement caused by fault slip, particularly in the shallowmost part of the fault (Hayes et al., 2011). As soon as the appropriate theoretical and computational tools were developed, the finite slip problem became an important topic in seismology.

The first attempt to describe the finite character of the source of an event using ground motion data is attributed to Trifunac (1974). In that study near field strong motion seismic records were used to objectively constrain a slip model for the Feb. 9, 1971 San Fernando earthquake. Later, Olson and Apsel (1982) used strong motion data to find the slip distribution of the Oct. 15, 1979 Imperial Valley earthquake. This study treated carefully the issue of the under-determination arising from the parametrization by solving the problem using different least square methods and comparing them. It also formally introduced the parametrization method called “Multiple-time-window” that we discuss in Section 2.4. A similar approach was followed by Hartzell and Heaton (1983) in their study of the same event, this time employing teleseismic data in addition to strong motion data to perform the inversion. They also carried out a joint inversion of teleseismic and strong motion data, a technique which is now widely used to obtain models that are consistent with different types of data. Improved and optimized methods to compute finite fault models of several earthquakes have been developed over the years and the range of types of data used has increased, but the underlying approach is essentially the same: Discretize the fundamental relation 2.11 and invert for the slip in the resulting expression. We refer the reader to Ide (2007) for a detailed review of the methods 24 Background

Figure 2.8: A common spatial discretization for the faulting surface. The source is modelled as a plane which is comprised of rectangular subfaults. Taken from Olson and Apsel (1982) and strategies typically adopted in solving the finite slip problem. Equation 2.11 is the basis of all finite fault studies. The general method employed is to use some parametrization of ∆u(~x0, t0), a set of data u(x, t) and the Green’s functions according to the chosen parametrization to solve the inverse problem of recovering ∆u(~x0, t0). The slip ∆u(~x0, t0) can be discretized in space and time and in the next section we give an example of such a parametrization. The spatial discretization is generally accomplished by dividing the fault plane into smaller subfaults as is shown in Figure 2.8. The temporal dicretization can be realized by allowing for the subfaults to slip at predetermined times or solving directly for the slip duration (rise time) of each subfault. While the second approach decreases the number of parameters required in the inversion, it results in a nonlinear problem. We stress that whichever method is used to parametrize 2.11, the resulting discrete version of the problem is just an approximation of the original one and care must be exercised in quantifying the relationship of the uncertainties to the parametrization. A critical review of the finite fault inversion scheme pointing out these type of issues in the methodology is found in Beresnev (2003).

Commonly, when linear approaches are taken to address the ﬁnite fault problem one usually needs to employ a high number parameters to obtain realistic models which, in turn, results in a highly unstable problem. That means that small changes §2.4 The Multiple-time-window Method 25

in the data may lead to large variations in the solution. Of course, real data contain noise so the practice of naively inverting the data does not guarantee that a reliable solution will be obtained. To address this issue a technique called regularization is typically used. Basically, it involves imposing some conditions on the solution in the form of equations appended to the linear system. Because the model has to fit the data and satisfy the regularization conditions simultaneously, in general the result will do both approximately. Two regularization conditions are commonly used. The first is to require that each subfault has a moment close to zero. In that way subfaults that only weakly contribute to the general fit can be forced to have slip equal to zero. The other condition is to require the solution to be smooth, meaning that neighbouring subfaults will be forced to have a similar slip. This condition is desirable as we do not expect abrupt changes in the rupture pattern. It is important to notice that in both cases the strength of the conditions can be controlled by a scalar parameter as we show in Chapter 3.

2.4 The Multiple-time-window Method

In this section we examine a particular approach to parametrize the finite slip problem. The underlying concept is that we can account for variations in the rise time of each subfault by allowing it to slip a number times. Owing to this fact, this method is known as a multiple-time-window parametrization. The principal merit of this approach is that it keeps the inversion linear, avoiding the complicated non-uniqueness issue arising in non-linear problems, and moreover the parametrization provides an intuitive insight into the source variables. Probably for these reasons it was the first method used to take into account some degree of variation in the initiation of the rupture (Olson and Apsel, 1982) and is now one of the most popular methods to perform finite fault inversion (Ide, 2007). It is also a good example to show how finite fault inversion can work in practice. We make use of this technique throughout this thesis as its straightforwardness is extremely important while inverting the source parameters in a nearly real-time context. Following Olson and Apsel (1982), our starting point is equation 2.11: Z 0 0 0 0 0 0 um(~x,t) = Cijkl ∆ui(~x , t )Gkm,l0 (~x − ~x , t − t ) dt dSj, where the integral is taken over the faulting surface and time. According to what 0 0 we wave discussed we will discretize ∆ui(~x , t ) spatially and temporally. We shall assume that each of the spatio-temporal units has a constant slip value. The spatial discretization we adopt is given by rectangular subfaults as is shown in figure 2.8 26 Background

Figure 2.9: Multiple-time-window scheme. The subfaults are represented by rectangles. The concentric circles represent hypothetical rupture fronts propagating outwards from the hypocentre with the same velocity. At the bottom is shown an example of a time function (triangules) used to parametrize each time window. Taken from Ide (2007).

comprising J subfaults. A rupture front will initiate at a given subfault (hypocentral m subfault) and propagate outwards with a constant velocity vr . Once this hypothetical rupture front reaches a subfault, the slip can start at that subfault. After that, that subfault is allowed to slip Nt times at time increments of th. This temporal discretization is sketched in Figure 2.9. Therefore, our discrete version of the slip distribution is given by:

J N X Xt ∆~u(~x,t) = Bj(~x) ~sjkPk(~x,t) (2.20) j=1 k=1  1, if ~x is in the j-th cell Bj(~x) = (2.21) 0, otherwise

Pk(x, t) = F (t − T (~x) − (k − 1)∆t). (2.22)

As can be seen Bj is basically a boxcar function that is used to eﬀectively divide the faulting plane spatially. The function Pk contains all the time dependency of the source and is deﬁned by F which is known as the source time function. For the multiple-time-window method the source time function usually adopts the form of §2.4 The Multiple-time-window Method 27

a triangle (see Figure 2.9) of half duration th or a boxcar function of total duration th. ∆t is the time elapsed between the arrival of two rupture fronts to the subfault and is commonly taken as ∆t = th. The time T (~x) indicates the moment when the m subfault can start to slip. In practice, T (~x) = |~x − ~x0|/vr , where ~x0 is the location of the hypocentre. That means that slip can take place only after a hypothetical m rupture front with velocity vr reaches a subfault at ~x. In most of the applications th is very small compared with the duration of the rupture process, so if just a few m triangles are used to characterize the source, vr can be considered eﬀectively as m a maximum rupture velocity. In general vr corresponds to the maximum rupture velocity considered in the model. If one wants to solve for the rake of each subfault,

~sjk can be decomposed into two orthogonal components to treat them independently in the inversion. This increases the number of unknown parameters from JNt to

2JNt.

Using the parametrization deﬁned by equations 2.20-2.22, equation 2.11 can be rewritten: Z X fg f f ˜f 0 ˜f 0 0 um(~x,t) = si Aj Cijkl F (t )Gkm,l0 (~x,t − t ) dt . (2.23) f,g Here, the functions F˜f and G˜f correspond to F and G respectively, but evaluated f with the source location of the f-th subfault and Aj is the surface vector of the f-th subfault. Equation 2.23 can be further simpliﬁed by noting that the factors outside the integral are in fact the moment tensor (see equation 2.14) of the f-th subfault at the g-th time window. As we mention earlier the moment tensor contains information about the strength and geometry of the source, but the geometry of each subfault is already known, so we only need to determine the scalar moment tensor of each subfault. This is analytically show by Aki and Richards (2002), BOX 4.4, where the moment tensor is written in the form:

¯ Mij = M0Mij, (2.24)

¯ where Mij is a normalized unit moment tensor containing the geometry of the source in terms of the strike, dip and rake angles. Therefore, 2.23 is in fact:

X fg ¯ f ˜fg ˜fg um(~x,t) = M0 Mkl F ∗ Gkm,l0 . (2.25) f,g

Equation 2.25 is indeed the discrete version of equation 2.11 that we use in our application in the next chapter. This expression enables us to turn the problem into a system of linear equations to be inverted. By, using an explicit expression for the source time function the convolution can be carried out and the only unknowns in 28 Background

the right hand side would be the scalar moments of the subfaults. Then, the slip distribution can be recovered using equation 2.18 on each subfault. By mapping each moment back into the fault plane, an image of the source and rupture process is obtained from the data which is, in the end, our ﬁnal purpose. Chapter 3

W-phase Finite Fault Inversion

In the previous chapter, we reviewed ﬁnite fault inversion and provided explicit expressions for the forward problem using the multiple-time-window method. We also remarked on the diﬃculties of imaging the source soon after large earthquakes. In this chapter we will show an application of the multiple-time-window method to two recent large events: the Maule (Mw = 8.8) and Tohoku (Mw = 9.0) earthquakes. Our approach may be used to obtain a timely image of the rupture. Most of the content of this chapter has been adapted from our already published work Benavente and Cummins (2013).

3.1 Abstract

We explore the ability of W-phase waveform inversions to recover a first-order coseismic slip distribution for large earthquakes. To date, W-phase inversions for point sources provide fast and accurate moment tensor solutions for moderate to large events. We have applied W-phase finite fault inversion to seismic waveforms recorded following the 2010 Maule earthquake (Mw=8.8) and the 2011 Tohoku earthquake (Mw=9.0). Firstly, a W-phase point source inversion was performed to assist us in selecting the data for the finite fault solution. Then, we use a simple linear multiple-time-window method accounting for changes in the rupture velocity with smoothing and moment minimization constraints to infer slip and rake variations over the fault. Our results describe well the main features of the slip pattern previously found for both events. This suggests that fast slip inversions may be carried out relying purely on W-phase records.

29 30 W-phase Finite Fault Inversion

3.2 The W-phase and Rapid Source Characteri- zation

Details of earthquake sources, such as fault orientation and the spatial variation of slip, are potentially useful in the first phases of disaster response. Understanding the proximity of fault rupture to population centers can be important for rapid casualty estimates (Jaiswal and Wald, 2011). Understanding the spatial distribution of slip is especially important for large megathrust earthquakes, since tsunamigenicity is very sensitive to the amount of slip concentrated at shallow depth (Satake and Tanioka, 1999; Hill et al., 2012). These considerations have motivated recent work on the use of GPS data to estimate earthquake slip distributions in near real-time (Ohta et al., 2012; Crowell et al., 2012). These studies have demonstrated that dense GPS networks deployed in the near field of large earthquakes are able to obtain reliable estimates of slip distribution within minutes of an earthquake’s occurrence, and are not prone to clipping, which plagues seismometer recordings in the near field.

However, because not all megathrust earthquakes occur adjacent to dense GPS networks providing near-real time positioning, it is important to consider alternative means of obtaining rapid and reliable earthquake slip distributions. Seismic waves have long been used in finite fault inversions to obtain slip distributions for large earthquakes (see, e.g., Hartzell and Heaton, 1983), and several studies have considered the rapid application of such techniques (Mendoza and Hartzell, 2013; Mendoza, 1996; Ammon et al., 2006; Dreger et al., 2005). These studies either use regional seismic waveforms or teleseismic body and surface waves. However, these methods often require sophisticated processing and manual review, and can be sensitive to 3D earth structure. In this chapter, we consider use of the W-phase (Kanamori, 1993) in finite fault inversion as an alternative that can potentially overcome some of the difficulties with using seismic body and surface waves in rapid finite fault inversions.

Figure 3.1 illustrates a seismic record of the 2001 Peru earthquake (Mw = 8.4) and indicates the long period signal that has been called the “W-phase” (Kanamori, 1993). It can be seen that the W-phase has a small amplitude and arrives at the station somewhere between the P and the S wave arrivals. It is also important to note that after the S wave arrival the amplitude of the displacements increase considerably, specially with the arrival of the surface waves. For large earthquakes moment tensor solutions can make use of surface waves to constrain long period information about the source (Kanamori and Given, 1981) but for distant stations §3.2 The W-phase and Rapid Source Characterization 31

Figure 3.1: A seismic record of the W-phase. The observed displacements are shown by the black trace and the synthetics by the red trace. The time starts at the origin time (O.T.). Taken from Kanamori and Rivera (2008). one has to wait longer to get the data than if P waves were used. Moreover, records from stations near to the source may be clipped due to the maximum allowed displacement for the instrument being exceeded. In contrast, it is clear from Figure 3.1 that the W-phase contains long period information of the source, it is fast and it has a small amplitude that can avoid the clipping even at regional stations. These are some of the reasons that motivated Kanamori and Rivera (2008) to develop a methodology to use the W-phase in real time, point-source moment tensor determination. Later studies (e.g., Duputel et al., 2012a, 2011) have shown the robustness of the method which has been adopted by several institutions (e.g., USGS, Paciﬁc Tsunami Warning Center (PTWC), Institut de Physique du Globe de Strasbourg (IPGS) ). Additionally, Duputel et al. (2012c) have shown that the W-phase can be used to perform multiple-point-source inversion of one event in order to describe more complex aspects of the source. Therefore, we consider the usefulness of making ﬁnite fault inversion for large earthquakes employing the W-phase data alone.

To recover the coseismic slip diﬀerent types of data can be used (geodetic, teleseismic, tsunami records, etc.), although far ﬁeld seismic records play an important role because they are widely available within minutes after an earthquake occurs. The use of teleseismic data to constrain the spatial slip distribution was introduced by Hartzell and Heaton (1983) . They considered a predetermined fault plane divided into a number of subfaults and solved for their moment using constant rupture velocity (see Section 2.4). While many improvements have been made to this approach (a review can be found in Ide (2007)), the philosophy of the most recent methods remains similar: Discretize the source’s surface into a number of units that can be treated as point sources. A simultaneous inversion is then carried out for 32 W-phase Finite Fault Inversion

the moment of each unit. Then, the spatial and temporal distribution of the slip for the whole fault is recovered by mapping the point source solutions in the fault plane.

In the remaining sections we show how this methodology can be used to obtain a rupture image for large earthquakes exploiting the advantages of using W-phase waveforms in rapid source characterization. To this end, we applied ﬁnite slip inversion for two recent large megathrust earthquakes: the 2010 Maule

(Mw = 8.8) and the 2011 Tohoku (Mw = 9.0). Both of them excited devastating tsunami waves and have numerous ﬁnite fault solutions available (e.g., Lay et al., 2010; Wang et al., 2012; Koper et al., 2012; Ammon et al., 2011; Lay et al., 2011). The Maule earthquake occurred oﬀshore central Chile. The W-phase point source solution (http://earthquake.usgs.gov/earthquakes/eqinthenews/2010 /us2010tfan/neic tfan wmt.php), hereafter referred to as WPPS, yields a moment 22 o o of M0 = 2.0 · 10 Nm, a fault plane with strike γ = 16 and dip δ = 14 , and a rake angle of λ = 104o; for a centroid located at 35.83oS, 72.67oW and depth 35 km using 28 stations. For the Tohoku earthquake the WPPS is reported in detail by 22 Duputel et al. (2011). They found a total moment of M0 = 4.26 · 10 Nm, a fault plane with γ = 196o and δ = 12o, and λ = 85o; for a centroid located at 37.92oN, 143.11oE and depth 19.5 km using 69 stations. In both earthquakes WPPS agrees fairly well with the GCMT and USGS moment tensor solutions.

3.3 Approach

As usual in finite source inversions, the first step in our procedure is the data selection. Because of the large number of parameters involved in the inversion, it is crucial to identify and remove corrupt data (e.g dead channels, wrong instrument response information) prior to the inversion. To address this issue, we first perform a point source inversion, following Kanamori and Rivera (2008). The W-phase inversion can be performed using the three displacement components, but the horizontal components are often noisy in the W-phase frequency band. Indeed, after a rigorous noise analysis for the Tohoku event Duputel et al. (2011) reached a final low-noise dataset comprised mostly of vertical component waveforms. Thus, for both events we retrieve LHZ (long period, vertical component) channels for stations in a distance range of 5 − 90o from the epicentre. We take the centroid’s location (point source solution’s location), the half duration th and the time delay td from the WPPS in the USGS website for the Maule event and from Duputel et al. (2011) for the Tohoku event. We deconvolve the traces in the time domain and §3.3 Approach 33

bandpass them using a Butterworth filter in the band 200-1000 s. After trimming the data in the typical W-phase time window (tp, tp + 15∆, ∆: station-epicentre distance in degrees) we remove the stations presenting anomalous traces based on the median criterion described by Duputel et al. (2012a) . Next, we invert for the moment tensor components (Using the same database of Green Functions as Kanamori and Rivera (2008)). We reject all the traces with a high individual misfit, and a final inversion is performed.

Apart from the data selection it is necessary to set the subfault parameters. We carried out a number of synthetic checkerboard tests to ﬁnd out the optimal subfault size that can be used in this approach. We found that subfaults as small as 15 km by 15 km can be resolved. Nevertheless, solutions are extremely unstable (i.e., sensitive to the noise) and therefore we favour a subfault size of 30 km by 30 km , which better tolerates the introduction of noise (see Figure 3.2). We used the fault geometry provided by the point source solutions. In both earthquakes, the actual fault plane can be easily distinguished from the point source solution using the subduction geometry. In general, the WPPSs we obtain do not diﬀer greatly from the USGS and the Global Centroid Moment Tensor (GCMT) solutions and we achieve similar results using a fault geometry from any of them.

A rupture velocity must be determined (or guessed) in order to invert for the moment of each subfault (Hartzell and Heaton, 1983; Mendoza and Hartzell, 1988). We first used a simple source time function consisting of one triangle, and used a constant rupture velocity, but the results for the Tohoku event were not compatible with most of the finite slip solutions we are aware of. After performing several inversions using different values of vr we found that for the Tohoku event the best fit is in the range 1.5-3 km/s, while for the Maule event 2-3 km/s is best. In these ranges the W-phase appears not to be extremely sensitive to the rupture velocity. Nevertheless, the results were symmetric patterns with two regions of concentrated slip far away from the hypocentre. Figure 3.3 shows the solution we obtained for the Tohoku event using 2.1 km/s of rupture velocity. The model we got is in disagreement with most of the results in the literature that locate most of the slip above the hypocentre, near to the trench. These symmetric patterns appear to be a result of the an oversimplified rupture model (Hartzell and Langer, 1993) . Therefore, we prefer to include possible variations in the rupture velocity in our approach using a slightly more complex source time function.

We adopt a fairly simple multiple-time-window method, which is described in 34 W-phase Finite Fault Inversion

Figure 3.2: Synthetic example for the Maule event: input model (top) and output solution (bottom). The arrows indicate the rake distribution. From left to right the three patches of the input model have a slip of 60m, 40m and 20m respectively and were created at diﬀerent time windows. The size and the color of the arrows are used to indicate the magnitude of the slip. The half duration of each triangle and the time diﬀerence between them is 10 s and the maximum rupture velocity was taken as 1.5 km/s. The output model was obtained after polluting the synthetic traces with a 10% of Gaussian noise. §3.3 Approach 35

Figure 3.3: Solution for the Tohoku event using one time window. The hypocentre is indicated by a blue star while the slip is shown by the color scale.

Section 2.4, in order to account for variations in the rupture velocity. The maximum m rupture velocity vr is chosen in accordance with the values discussed in the literature for each event. For each subfault we use a source time function comprised of Nt triangles and half duration th depending on the event. The triangles are overlapped by the same amount th so each subfault is allowed to slip Nt times at succes- m sive time increments of th after a rupture front with velocity vr reaches the subfault.

Because of the bandpass ﬁltering applied to the data, they are insensitive to variations in frequency content at periods less than 200 s, which is far longer than typical earthquake rise times. On the other hand, the phase shift associated with a delay of th, even when it is as small as the 10 and 15 s used for the Maule and Tohoku earthquakes, respectively, is detectable in the waveforms given the good signal-to-noise ratio that is typical in W-phase inversions. This sensitivity of the W-phase waveform inversions to phase shift is consistent with our tests using constant rupture velocity, as well as with the ability of the W-phase to resolve spatial variations in slip as indicated by our resolution tests (see ﬁgure 3.2). In practice, this time shift enables the W-phase to distinguish variations in rupture velocity as well as spatial variations in cumulative slip.

Finally, as usual in ﬁnite slip inversions, we implemented a regularization approach as described in Section 2.3 to deal with the typical instability arising from the underdetermined matrices involved in the inversion problem. In our application, we impose smoothing and moment minimization constraints on the solution. Re- quiring a solution to be smooth means that we prefer solutions in which the moment 36 W-phase Finite Fault Inversion

λ1 misﬁt [%] 0 11.7885 1 · 10−30 11.7885 1 · 10−28 11.7885 1 · 10−26 11.7885 1 · 10−25 11.7983 1 · 10−24 11.9683 1 · 10−23 12.7861 1 · 10−22 33.7638

Table 3.1: Example of the search for the smoothing parameters for the Maule event. In this case, λ2 = 0 and the misfit is the L1 norm, percent misfit. of two neighbouring subfaults is similar. Minimizing the moment implies that subfaults which do not contribute much to the general fit must be forced to be zero. In both cases the stabilization of the solution is achieved by appending a set equations to the original system. Noting that equation 2.25 can be rewritten in a matrix form if u(x, t) is a vector containing the concatenated observed traces and the Green’s function are written accordingly, the inversion problem appears as follows:

 A   B       λ1I  · x =  0  , (3.1) λ2F 0

where A is a matrix with the synthetics of each subfault, B is a vector with the observed seismograms, x is a vector containing the unknown moment of each subfault, I is the identity matrix, F is a finite difference Laplacian operator applied over each time window and λ1 and λ2 are the moment minimization and smoothing weights, respectively. These weights are determined such that the overall misfit of the solution does not increase greatly. In our parameter vector x we include two elements for each subfault to account for variations in rake.

For the estimation of appropriate values for the smoothing parameters, λ1 and

λ2, we perform an iterative search, by making several inversions and looking at the misﬁt. As an example, we will describe this process for the Maule event. The

first step consists in increasing λ1 while keeping λ2 = 0. Our goal is to find an order of magnitude for which λ1 starts to affect the misfit. Table 3.1 shows the results in that part of the process. Looking at the misfits we find that 10−24 is an −24 adequate order of magnitude to locate λ1. Then, keeping λ1 = 10 we repeat the process for λ2 and the result is shown by Table 3.2. We conclude that the order of magnitude of λ2 must be the same as λ1 but, since changes in λ2 affected less §3.4 Models for the Maule and Tohoku Earthquakes 37

λ2 misﬁt [%] 0 11.9683 1 · 10−30 11.9683 1 · 10−28 11.9683 1 · 10−26 11.9683 1 · 10−25 11.9258 1 · 10−24 11.9771 1 · 10−23 12.2084 1 · 10−22 12.8839 1 · 10−21 15.4297

−24 Table 3.2: Same as Table 3.1, but in this case λ1 is fix to 1 · 10 and λ2 is varied. to the misfit, we use a higher value for it. After a third search, we find that the −24 −24 values λ1 = 2 · 10 and λ2 = 4 · 10 regularize well the solution while keeping the misfit as low as 12.13%.

3.4 Models for the Maule and Tohoku Earth- quakes

In this section we present the results of applying our finite fault inversion approach to the Maule and Tohoku earthquakes. Typically teleseismic finite fault inversion solutions differ from one to another, especially when they are computed soon after the event. Because of that, we are more interested in testing whether our solutions can reproduce roughly the spatial accumulated slip distribution rather than match every detail in a particular solution.

It should also be noted that the results of the W-phase inversion are the seismic moments of the individual point sources comprising the ﬁnite fault solution, and the Green’s functions used in the inversion are calculated using the PREM model (Dziewonski and Anderson, 1981). PREM is a good representation for globally averaged earth structure, which should be appropriate for the very long wavelengths associated with the W-phase. In order to compare our results with published ﬁnite fault models, we have converted seismic moment to slip, and in doing so we have accounted for small-scale variations with depth of rigidity structure near the source by using local crustal velocity models instead of PREM. 38 W-phase Finite Fault Inversion

Figure 3.4: Results for the Maule event. Waveform ﬁts are shown (left) for selected stations. Observed displacements and their respective synthetics are indicated by a continuous green line and a dashed blue line, respectively. Note that each trace is plotted in the W-phase time window (tp, tp + 15∆, ∆: station-epicentre distance in degrees) and the epicentral distance is shown next to the station name. A reference model is plotted (center) from Koper et al. (2012). Diﬀerent slip regions are indicated by contours, the yellow rectangle marks the faulting area considered and the hypocentre is indicated by a white star. The solution we achieved is at the right side, plotted in the same way as the reference model (center).

3.4.1 Maule earthquake

In this event we use 31 LHZ channels with an maximum azimuthal gap of 52o. We base the 1D rigidity structure at the source on the CRUST 2.0 averaged m continental model (http://igppweb.ucsd.edu/˜gabi/crust2.html). For vr , th and Nt we choose 2.5 km/s, 10 s and 3, respectively. Strike φ, dip δ and the initial guess of the rake λ are taken as 18o, 18o and 104o, respectively. The hypocentre we use is located at 35.9oS, 72.7oW and depth 35 km . In Figure 3.4 (left) is shown some of the traces of both observed and synthetic waveforms. As is often the case in W-phase inversions, they match quite well (Kanamori and Rivera, 2008; Duputel et al., 2012a) giving a percent misfit (L1 norm) of 12.1%, while our WPPS yields 16.3%. As one would expect, the higher number of parameters of the finite fault solution leads to a better fitting compared to WPPS.

The cumulative slip distribution is shown in Figure 3.4 at the right. Our solution exhibits some properties which are present in most published ﬁnite-fault inversions of this earthquake: bilateral rupture with most signiﬁcant slip concentrated northwest of the epicentre and little slip close to the epicentre (Koper et al. (2012). This model is also included in Figure 3.4 at the center, for comparison). As pointed out by (Vigny et al., 2011) a high slip close to the trench in this event §3.4 Models for the Maule and Tohoku Earthquakes 39

Figure 3.5: Results for the Tohoku event. The details are the same as ﬁgure 3.4. However, in this case the reference solution (center) is from Ammon et al. (2011). The source rigidity structure we used for both the W-phase solution (right) and the reference solution is from Lay et al. (2011) . might be supported by the great tsunami generated afterwards and the fact that a number of aftershocks have been located in this region. We ﬁnd a total moment of 22 M0 = 2.0 · 10 Nm. The rake angle (Figure 3.6, top) is mostly dip-slip in the two regions in which the slip is mostly concentrated. Between these two regions, above the hypocentre, there is an area of small slip mostly of strike-slip geometry.

3.4.2 Tohoku earthquake

The Tohoku earthquake was characterized using 65 LHZ channels with an maximum azimuthal gap of 35o. We employ the same rigidity structure described in Lay m et al. (2011). For that event we use th = 15s, vr = 2.5 km/s and Nt = 4. The last value attempts to describe the broader rupture velocity range for this event described in the literature. In fact, Ammon et al. (2011) using teleseismic P wave inversion found that a slow rupture propagation followed by a faster propagation was required for a satisfactory solution. In addition, backprojection analysis (Koper et al., 2011; Wang and Mori, 2011) shows that after a slow rupture initiation (∼ 1.0 km/s) the rupture could reach up to 3 km/s. In this event we locate the hypocentre at 38.3oN, 142.4oE and depth 30 km, based on the USGS solution (http://earthquake.usgs.gov/earthquakes/eqinthenews/2011 /usc0001xgp/#details). The fault plane is deﬁned by strike φ = 196o and δ = 12o, and the initial rake is λ = 85o. The solution predicts the observations well as can be seen in picture Figure 3.5 (left), with a percent misﬁt (L1 norm) of 12.9%, as 40 W-phase Finite Fault Inversion

compared to our WPPS misﬁt of 15.9%.

Figure 3.5 (right) presents the spatial distribution of the slip we obtained. Our result shows a rake of around 90o (see Figure 3.6, bottom) and concentrates the higher slip in the shallowest part of the fault, above the hypocentre. This feature can be found in most of the slip distributions calculated for this event (e.g., Lay et al., 2011; Hayes, 2011). For reference, in Figure 3.5 (center) we show the solution from Ammon et al. (2011) after including the rigidity structure from Lay et al. (2011), that we also use for our model. The maximum slip value is ∼60m in the reference solution (Figure 3.5, center) and ∼50m in our model (Figure 3.5, right). While the exact value of the highest slip is hard to constrain in our approach, this general slip pattern turns out to be quite stable to variations of λ1 and λ2. 22 Moreover, the moment we find with this model is M0 = 4.1 · 10 Nm which is in agreement with most of the values in the literature. Finally, as mentioned earlier, the misfit of this model is well below what we find in the point source solution.

3.5 Conclusion

We have performed a ﬁnite fault inversion of the 2010 Maule (Mw = 8.8) and the 2011 Tohoku (Mw = 9.0) earthquakes, using a dataset consisting of W-phase waveforms, mostly from the Global Seismic Network. In both cases our results for the cumulative coseismic slip show consistency with other models in the literature and can describe ﬁrst-order characteristics of the rupture. We employ a multiple-time-window method (Olson and Apsel, 1982) allowing for rupture velocity variations while keeping the problem linear. The simplicity of the resulting methodology and the accuracy of the results suggest that this method might be suitable to obtain preliminary rupture models in the case of large events. One problem would be, however, to rapidly determine or estimate all the parameters required in the inversion . Nevertheless, some of these critical parameters may be obtained using simple approaches. In Chapter 4 we discuss how to determine some of those parameters so that the inversion can run automatically. §3.5 Conclusion 41

Figure 3.6: Rake distributions for the Maule (top) and the Tohoku (bottom) events. The rake angle is indicated by the arrows and the color scale indicate the slip. 42 W-phase Finite Fault Inversion Chapter 4

Automated Algorithm for Finite Fault Inversion in Megathrust Events

In Chapter 3 we perform FFI using the W-phase for two megathrusts events. Since the W-phase has not been previusly employed to recover slip distribution, it is important to establish the it contains signiﬁcant information about the rupture process. The results of Chapter 3 suggests that the W-phase provides meaningful slip patterns. In this Chapter we develop a method to perform automated FFI using the W-phase. This can result in rapid rupture models useful for disaster management. We apply this approach to the recent Illapel event (Chile, 2015, Mw = 8.3). The content of this Chapter has been adapted from our article Benavente et al. (2016).

4.1 Abstract

We perform rapid W-phase ﬁnite fault inversion for the 2015 Illapel great earthquake (Mw = 8.3). To evaluate the performance of the inversion in a near real time context, we divide seismic stations into 4 groups. The groups consider stations up to epicentral distances of 30o, 50o, 75o and 90o, respectively. The results for the ﬁrst group could have been available within 25 minutes after the origin time and the results for the last group within 1 hour. The 4 results consistently show a peak slip of ∼10 m near the trench with trench perpendicular rake which is consistent with the tsunami genesis of the event. The slip location is similar to that in the preliminary USGS solution. The inversion is automated and provides meaningful results within 25 minutes after the event. This makes the method particularly suited to emergency management and early warning at regional and teletsunami distances.

43 44 Automated Algorithm for Finite Fault Inversion in Megathrust Events

4.2 Introduction

In finite fault inversion (FFI) surface ground motions from earthquakes are used to infer the spatio-temporal evolution of the rupture process. Over the past decade FFI has become routine, so that results produced by different groups are available within several days or even hours after a large event. Traditionally, these models are used to enhance our understanding about the physics of the rupture and the tectonic processes that cause the earthquake. While rapid finite fault inversion is useful to evaluate population exposure to events (Jaiswal and Wald, 2011) and their tsunami potential (Satake and Tanioka, 1999; Hill et al., 2012), automated and rapid inversion has only been studied recently. Important examples include works that employ GPS static offsets (e.g., Minson et al., 2014a; Crowell et al., 2012) and the USGS effort to release rapid finite fault models based on teleseismic body and surface wave data after large events (e.g., Hayes et al., 2011). Continuous GPS data have also been employed to obtain rapid slip models that can in turn be used to carry out tsunami simulations (Melgar and Bock, 2013, 2015; Zhang et al., 2014). In addition, GPS data have been used for slip characterization in a real-time environment for a moderate event (Grapenthin et al., 2014). In comparison to seismic data, GPS data are useful for rapid FFI since they do not saturate even for stations close to the source of a large event. However, dense continuous GPS networks are not available in all earthquake regions and are currently limited to land-based installation. Therefore, global teleseismic data are a valuable alternative and can provide additional information. Most teleseismic studies use body and surface waves, but these can be sensitive to crustal 3D heterogeneities which complicate the FFI. Methods usually involve manual review of data and results but this causes a slower process which is not practical for rapid analysis and automation. Results from Chapter 3 suggests that the long period seismic waves known as the W-phase (Kanamori, 1993) may alleviate some of these difficulties.

The W-phase is a fast, long period seismic wave which is widely used for reliable, rapid point source moment tensor inversions (Kanamori and Rivera, 2008; Duputel et al., 2012a). As a result, the W-phase point source algorithm is applied at several early warning centers. Moreover, the W-phase can also be used to infer more detailed source aspects, such as the inversion for multiple point sources of complex ruptures (Duputel et al., 2012c). Yoshimoto and Yamanaka (2014) carry out FFI for the Sumatra 2004 earthquake with full waveform inversion for several phases including part of the common W-phase frequency band. In chapter 3 we employed vertical component W-phase records and obtained meaningful rupture §4.3 Inversion scheme 45

models for the Maule (2010, Mw = 8.8) and Tohoku (2011,Mw = 9.0) events. We concluded that W-phase data contain sufficient information about the rupture process of megathrust events. Further, Dettmer et al. (2014) applied rigorous trans-dimensional FFI of W-phase records to the Maule event and quantified slip uncertainty. That study concluded that spatial slip patterns can be resolved from W-phase data. In addition, the uncertainties revealed that some slip features are much better constrained than others and that resolution varies significantly across the rupture. The main advantage of W-phase data is the ability to obtain finite fault models for megathrust events via a reliable procedure which is straight forward to automate.

We illustrate our rapid FFI method with W-phase records from the 2015 Illapel great earthquake (Mw=8.3). The Illapel event is the largest earthquake recorded in 2015 to date and one of the largest recorded by modern seismic instrumentation. According to the USGS (2016), the earthquake and subsequent tsunami killed 15 people and destroyed more than a thousand homes. Tidal gauge data show a peak tsunami amplitude of 4.75 m and post tsunami survey data reveal a maximum runup as large as 10.8 m (Aránguizet al., 2016). The W-phase point source solution1 (WPPS) shows a thrust event, which is consistent with the subduction of the Nazca under the South American Plate in this region. The preliminary USGS rapid finite fault inversion (USGS, 2016) estimates that the rupture extends more than 200 km along strike with peak slip of ∼9 m located close to the trench. In the following sections, we discuss our FFI algorithm and how some of the parameters required by our linear FFI approach can be estimated from prior tectonic knowledge about the region. We put particular emphasis on practical aspects that matter for automation and on timing to judge how quickly meaningful finite fault solutions can be obtained.

4.3 Inversion scheme

We apply the multiple-time-window (MTW) method (e.g., Olson and Apsel, 1982; Ide, 2007) to parametrize the rupture process for the inversion. In MTW, the source time function for each subfault is given by a sequence of simple functions (typically triangles), effectively accounting for spatial differences from a prescribed maximum rupture velocity. This time discretization allows for the problem to be approximated as a linear inverse problem which is advantageous since efficient algorithms can be 1http://earthquake.usgs.gov/earthquakes/eventpage/us20003k7a#moment-tensor 46 Automated Algorithm for Finite Fault Inversion in Megathrust Events

used to ﬁnd the optimal solution. Since we cannot know the fault discretization independently, we apply an over-parametrized regularized approach (e.g., Aster et al., 2005). This means that the discretization is ﬁxed both in time and space and needs to be chosen to be below the resolution of the data and regularization must be applied to stabilize the inversion, typically including both smoothness and positivity regularization. The former means that models with smooth slip distribution across the fault plane are favored. The latter limits the rake angle to a 90o region around an initial guess. No temporal smoothing regularization is utilized in this study. We consider the inversion for this parametrization as an optimization problem that seeks a model m which minimizes the objective function

T −1 2 T T ψ(m) = (Am − d) Cd (Am − d) + λ m L Lm, m ≥ 0, (4.1) where A is a N × M matrix for N data and M parameters. Columns of A contain the Green’s functions convolved with the moment rate function for each parameter. Column vector d contains the concatenated W-phase traces for all stations, Cd is the data covariance matrix, and λ is the smoothing parameter. The matrix L is the smoothness operator given by the finite difference approximation of the Laplacian of the scalar moment distribution across the fault. The inequality m ≥ 0 specifies positivity. To ensure positivity, we employ the non-negative least squares (NNLS) algorithm (Lawson and Hanson, 1974) and in particular the efficient implementation by Luo and Duraiswami (2011) which exploits multi-core architecture. In contrast to equation 3.1, here we do not use a damping term. We favour this approach since it is computationally demanding to independently estimate both smoothing and damping parameter.

The estimation of Cd is itself a challenging problem since it contains both theory and measurement errors (Dettmer et al., 2007). Recently, some studies in seismic source inversion have addressed this issue in a probabilistic framework. Yagi and Fukahata (2011) developed an inversion scheme which jointly solves for the data errors and the optimal model, while Duputel et al. (2012d) models some aspects of the covariance matrix associated with theory error. Unfortunately, these methods are computationally demanding and do not treat all theory and measurement errors. In addition, the resulting inverse problem is non-linear and unlikely to be solved in near real time. Dettmer et al. (2014) approximate the data covariance matrix iteratively in a Bayesian framework and include hierarchical scaling to reduce dependence on the iterative method. This is also computationally expensive and not suitable for rapid FFI. §4.3 Inversion scheme 47

Figure 4.1: Slip distributions obtained using diﬀerent bandpass ﬁlters. For all cases we use the largest dataset (stations up to 90o epicentral distance) and a maximum rupture velocity of 1.5 km/s. Periods considered are: 100-250 s (left), 150 - 500 s (center) and 200 - 1000 s (right).

Here, we adopt a simple but practical approach to estimate a diagonal data covariance matrix. A stationary Cd whose diagonal is formed by the standard deviation of the residuals at each station is estimated and then iteratively updated. Since we do not have independent knowledge about the noise at each station (including measurement and theory error) prior to the inversion, we perform a preliminary inversion without smoothing (the second term on the right hand side of equation 4.1 is neglected). Then, we compute the standard deviation of the p residuals of each station and construct a preliminary data covariance matrix Cd. Next, we apply this estimate in equation (4.1) to obtain an optimal value for the smoothing parameter λ objectively, following the discrepancy principle (Constable et al., 1987). This procedure is discussed in the next section. Once an appropriate p λopt is found, we perform another inversion assuming Cd = Cd to produce a new set of data residuals. We iterate this process 5 times (Cd do not change significantly with additional iterations) to find a new value for Cd and subsequently for λ. This approach accounts for theory and data errors while keeping the inversion procedure straightforward. However, it involves the assumption that the residuals of the unregularized solution contain sufficient information about data errors. In practice, we have found this to work well for this problem and it leads to meaningful results.

We consider only W-phase data to constrain the slip distribution. This early- arriving seismic phase constrains moment tensor solutions for large events well and provides reliable point source solutions for early warning purposes. The FFI utilizes three component W-phase data with 4 s sampling rate. The data are bandpass 48 Automated Algorithm for Finite Fault Inversion in Megathrust Events

filtered from 120 to 500 s but the results seem to be stable even at longer periods such as 200-1000 s (see Figure 4.1). Since the estimation of the optimal smoothing parameter is based on the residuals of the unregularized solution (see Section 4.5) we exclude stations with unusually high noise levels from the analysis. In particular, we carry out this data selection process by performing a W-phase point source inversion prior to the FFI (Kanamori and Rivera, 2008; Duputel et al., 2012a). The final data set is comprised by channels that pass the screening processes including a misfit criterion that rejects traces with poor fit for the point source model (Duputel et al., 2012a).

4.4 Preliminary Rapid W-phase point source solution

Specification of some parameters for the FFI requires a rapid WPPS inversion to be carried out first. Our procedure is similar to Duputel et al. (2012b), but differs in the grid search for the optimal centroid location. (the most computationally expensive part of the algorithm). Our implementation applies an abbreviated grid search for efficiency and since our final results do not depend strongly on the centroid location. The abbreviated grid search first estimates optimal centroid latitude and longitude while fixing centroid depth at the preliminary hypocenter depth. Then, for fixed centroid latitude and longitude a 1D search for the optimal centroid depth is carried out. The solutions we obtain with this approximate grid search agree well with results from the original implementation. In order to use a realistic preliminary hypocenter location (which is required by the WPPS algorithm) we employ the hypocenter and origin time from the first preliminary USGS earthquake report:

Region: OFFSHORE COQUIMBO, CHILE Geographic coordinates: 31.577S, 71.652W Magnitude: 7.9 Depth: 8 km Universal Time (UTC): 16 Sep 2015 22:54:30 Time near the Epicenter: 16 Sep 2015 19:54:31

The results of our rapid WPPS for each dataset considered in this work are summarized in Table 4.4. Figure 4.2 shows the station distribution and source mechanism for each of the four data sets. §4.4 Preliminary Rapid W-phase point source solution 49

Figure 4.2: Rapid WPPS for dataset with stations up to 30o (A), 50o (B), 75o (C) and 90o (D) of epicentral distance. Station locations (blue dots) considered in the inversion and the source mechanism (beachball, displayed at the centroid location) are shown for each case.

Table 4.1: Summary of the WPPS for each dataset. ∆ refers to the maximum epicentral distance considered for the dataset, ‘WPPS comp time’ indicates the computational time required by the W-phase rapid moment tensor inversion (after the data acquisition, see Table 4.2 for the time required to acquire data). The strike, dip and rake angles are denoted φ, δ and λ, respectively. ‘Time delay’ is the centroid time shift with respect to the considered origin time. ∆ WPPS comp φ δ λ Centroid location Mw Time time [s] delay [s] 30o 15 349o 15o 71o (30.58oS, 72.25oW , 11.5 km) 8.37 53 50o 32 356o 16o 83o (30.58oS, 72.25oW , 11.5 km) 8.32 50 75o 62 347o 20o 68o (30.58oS, 72.25oW , 13.5 km) 8.28 51 90o 104 353o 17o 81o (30.58oS, 71.45oW , 13.5 km) 8.30 52 50 Automated Algorithm for Finite Fault Inversion in Megathrust Events

Fault geometry from CMT-W & Slab1.0

Acquire data (FDSN), process W-phase & construct A (eq. (1)) v1 v n parallel rup rup Data sets after execution ... 14, 22, 32, 38 min

p Estimate Cd for λ=0

opt p Estimate λ for Cd

opt Iterative update for Cd and λ

opt Final inversion with Cd and λ

Figure 4.3: Diagram of the algorithm employed in our FFI method. FDSN refers to the international Federation of Digital Seismograph Networks.

4.5 Estimation of required parameters

A challenging aspect of the MTW method is to objectively and efficiently prescribe the fault geometry as well as the spatio-temporal discretization of slip. Furthermore, automated and objective smoothing must be adopted to avoid manual review and subjective choices which are known to substantially affect slip models. In the absence of aftershock information, we need to rely on either rapid moment tensor solutions or prior knowledge to constrain the fault geometry. The rapid W-phase point source solutions we perform (Section 4.4) show that, while all solutions are consistent with a megathrust event, the inferred fault geometries do not always align well with prior knowledge about the trench and subducting slab. Moreover, slab curvature is not accounted for by the planar fault derived from the point source approximation. Therefore, we apply prior slab geometry from the SLAB1.0 model (Hayes et al., 2012) in our inversion. The use of independent slab information was previously considered(Hayes, 2011) as a means to improve early finite fault models. Following that work, we prescribe a fault length L based on an estimate of rupture duration tr and a reasonably fast maximum rupture velocity vr. However, instead of an empirical relation for tr we apply tr = 2td, where td is the centroid delay from a rapid WPPS solution. It has been shown (Duputel et al., 2013) that td can provide robust estimates for the rupture duration in contrast to empirical relations which may not work well for anomalous events. We set subfault size to 25 km by §4.5 Estimation of required parameters 51

25 km (625 km2) which is empirically below the resolution power of W-phase data (Chapter 3; Dettmer et al., 2014) but suﬃciently large to reduce computational cost. The faulting surface extents from 10 to 80 km depth, spanning the main seismogenic zone for megathrust events.

In order to generate a set of subfaults that matches these a priori conditions and properly tessellates on the SLAB1.0 geometry, we employ routines from the OpenQuake hazard library (Silva et al., 2014). In this process, the area and shape of the subfaults is slightly modiﬁed so they can approximately follow the slab geometry. The actual area of each subfault is provided by OpenQuake so we can properly map scalar moment into slip. The minimum and maximum actual subfault areas are 554.3 km2 and 683.2 km2, respectively. While this approach produces a set of subfaults within a curved surface, we neglect curvature in rupture propagation. Because of the small dip angle in the shallow part of the slab this approximation has little eﬀect on shallow slip patterns which are the most relevant to evaluate tsunamigenecity. The procedure we used to obtain the fault’s discretization is summarized as follows:

1. From the rapid WPPS, extract centroid latitude and longitude.

2. Project centroid latitude and longitude along the radial direction onto the slab surface (given by SLAB1.0) to obtain the slab depth at that point. This point is the center of the fault.

3. From the fault center, compute the average strike angle (from SLAB1.0) within a region of ±3o surrounding the center.

4. Assuming a reasonably high rupture velocity, employing the strike from the previous step, and the event duration from the rapid WPPS, compute the along strike bounds of the fault (see section 3, main document). We use 2.5 km/s as the rupture velocity in this step following Hayes (2011).

5. Apply bounds on the rupture depth. We use 10 to 80 km, with the shallow limit given by our Green Function’s database. Beyond 80-km depth, megathrust rupture is unlikely.

6. Construct the subfaults using Openquake (Silva et al., 2014) (requires an area for each subfault). We choose a subfault area of 25 km by 25 km.

The resulting set of point sources, which represent the subfaults, is shown in Figure 4.4. Note that the area, dip, strike and center location for each subfault can be obtained directly from an Openquake complex fault surface object which we use 52 Automated Algorithm for Finite Fault Inversion in Megathrust Events

Figure 4.4: The fault model used in the inversions. Each dot shows the location of a point source in our ﬁnite fault model. The color scale represents the depth of each subfault, the yellow star the hypocentre. The discretization is 25 km.

to model the subfaults. Importantly, for real time applications, this discretazation process can be carried out independent of the event and does not result in computational cost during the real time inversion. Given a centroid location and a estimated fault length, an appropriate set of subfaults can be selected to carry out the inversion by an automated algorithm.

In our approach, objective smoothing is crucial and we select the smoothing parameter via the discrepancy principle (e.g., Constable et al., 1987; Aster et al., 2005). Under the discrepancy principle we seek a solution which ﬁts the data to the expected value of the noise level which is the number of data N for noise of unit standard deviation (the residuals after being standardized by the standard deviation estimate). This means that the parameter is adjusted until the condition is satisﬁed by the residuals. In order to obtain standard deviation estimates to standardize the residuals, we perform a preliminary inversion assuming λ = 0 (see Section 5.2 and Figure 4.3). The main advantages of this criterion for this problem §4.5 Estimation of required parameters 53

Table 4.2: Summary of the inversion results for each data set. Rows show the values for the preferred model of each data set (Labels refer to Figure 4.5). ∆max corresponds to the maximum epicentral, ‘Cha’ is the number of channels used in the inversion,‘Gap’ m is the azimuthal gap, vr is the preferred maximum rupture velocity, “Time to WPPS” is the time span to obtain the WPPS from the origin time. “Comp. time” is the time spent on computation of the finite fault model and the total time to obtain a result is ‘Time to FF’ (total time from origin time to solution). ‘Peak slip’ and Mw are the maximum slip and the moment magnitude computed for the model, respectively. m Label ∆max Cha Gap vr Time to Comp. time Time to Peak Mw [km/s] WPPS [min] [min] FF [min] slip [m] a) 30o 20 158o 1.5 13.90 10.16 24.06 10.47 8.51 b) 50o 31 81o 1.5 21.66 9.31 30.97 11.47 8.45 c) 75o 71 70o 2.0 31.48 11.05 42.53 8.24 8.33 d) 90o 110 48o 1.5 37.23 21.95 59.18 10.52 8.42 are that it can be automated and applied with positivity constraints. Automation is achieved by minimizing the absolute value of the difference between the misfit and its expected value for a variable smoothing parameter λ. Some other methods such as ABIC can be implemented in a minimization scheme as well, however these are not compatible with positivity constraints (e.g., Fukuda and Johnson, 2008). In practice, we impose reasonably large limits on λ, set through experience with previous inversions, and carry out a bounded search which requires ∼10-15 inversions. A disadvantage of the discrepancy principle is that it relies on previous knowledge of the noise level (standard deviation). We address this with the data covariance matrix estimate Cd (Section 5.2). The residuals are standardized by 2 Cd and the expected value for the χ misfit (first term on the right hand side of equation 4.1) is given by the number of data N.

The time dependence of rupture is parametrized with the MTW method (e.g., m Olson and Apsel, 1982; Ide, 2007), where a maximum rupture velocity vr is speci-

ﬁed and subfaults are allowed to rupture a prescribed number of times (Ntw) after the arrival of the ﬁrst hypothetical rupture front. We follow the classic approach of modeling the source time function of each subfault by a triangle sequence of half sf duration th which is also the spacing of successive triangles for a given subfault. m sf Importantly, vr , Ntw and th need to be determined prior to the inversion. Since

Ntw is proportional to the total number of parameters there is an intrinsic trade-oﬀ between the inversion speed and the maximum allowed rise time per subfault. In this sf study we ﬁxed th to 12s and Ntw to 5, allowing for a rise time of 72s per subfault. sf Smaller values of th lead to similar results but the higher number of parameters required to keep the same rise time makes the inversion extremely slow. Given that early point source solutions estimate a total duration of around 100s (see Table 4.4) 54 Automated Algorithm for Finite Fault Inversion in Megathrust Events

this rise time is sufficient to capture the vast majority of the moment release. For a different event, the subfault’s rise time can be incremented by increasing Ntw if required. A simple strategy for an automated procedure would be to increase the number of time windows until the subfault’s rise time is approximately equal to the m half duration of the event obtained from WPPS. To address vr , we consider several values concurrently on multiple computer cores (1.5km/s, 2.0km/s, 2.5km/s and 3.0km/s) and select the model with the best fit to the data. The full inversion procedure is summarized in Figure 4.3. Note that additional efficiency improvements are possible (additional parallelisation and optimization, more sophisticated computer hardware). The time to obtain a solution is intrinsically limited by the propagation time of the W-phase which is currently about 1/2 of the time required to produce the first solution. A detailed description of the times involved are given in Table 4.2. We note that the column labelled ‘Time to WPPS’ includes the maximum W-phase travel time defined by the maximum travel distance and the computational time of the WPPS. The latter ranges from tens of seconds (smallest data set) to around a hundred seconds (largest data set).

4.6 Inversion Results

We carried out inversions for four data sets based on the algorithm in Section 4.5 and Figure 4.3. The data sets are based on 4 choices of maximum epicentral distance (30, 50, 75, and 90◦) and are available at 14, 22, 32, and 38 minutes after the origin time. The timings of data availability at these distances is dictated by the W-phase propagation times. With these inversions, we study how quick a reliable ﬁnite fault solution can be obtained in a real-time scenario. With increasing epicentral distance, the data sets contain more stations and should produce better results. First, we compute a rapid WPPS following closely (Duputel et al., 2012b), for each data set. Next, the automated FFI algorithm (Figure 4.3) is applied to m each data set for several choices of maximum rupture velocity vr . This process objectively produces one ﬁnite fault model for each data set.

Figure 4.5 shows the main results of this paper in terms of the slip magnitude and rake for each of the 4 data sets. Table 4.2 summarizes relevant information about each inversion. We highlight that the earliest model (Figure 4.5 a) is obtained only 25 minutes past the origin time and the ﬁnal model (Figure 4.5 d) after ∼1 hour. Therefore, these solutions are potentially valuable for both early warning and disaster management. All results were obtained using a desktop computer with a recent 4-core processor (Intel R CoreTM i7-2600 CPU @ 3.40GHz). §4.6 Inversion Results 55

Figure 4.5: Preferred slip distributions for each data set, see Table 4.2 for additional information about each model. ∆max indicates the maximum epicentral distance considered for a given model. In each solution the yellow box encloses the considered fault region, the star shows the hypocentre (i.e., nucleation point) and the red line indicates the trench. The contours enclose diﬀerent slip levels and the white arrows are the slip vectors. 56 Automated Algorithm for Finite Fault Inversion in Megathrust Events

Figure 4.6: Observed (solid green) and predicted (dashed blue) W-phase waveforms for the data sets and models considered in Figure 4.5. The name and component of each station is shown in the label at the top left corner of each plot. ∆ corresponds to the epicentral distance of each stations. The x-axis shows the time in seconds from the theoretical P-wave arrival.

The results in Figure 4.5 are for increasing epicentral distance, such that the number of channels considered in the inversion increases from (a) 20 to (d) 110 (see Table 4.2). The four models exhibit a similar peak slip of around 10m located at the trench just north of the epicenter. Model (c) presents a slightly lower peak slip compared to the other preferred solutions (Table 4.2). This seems to be related to the higher (2.0 km/s) rupture velocity selected in these data set since all solutions computed for such a rupture velocity exhibit a similar peak slip (Figure 4.8). In m general, the solutions obtained for vr = 1.5 km/s and 2.0 km/s produce a comparable misfit suggesting that, within this range, the W-phase cannot resolve variability of rupture velocity. In practice this discrepancy results in a difference of ∼2 m in peak slip but does not significantly affect the distribution of slip (Figures 4.7 and m 4.8). In addition, since data misfit does not significantly change with changes in vr , data information appears to be limited with regard to slip magnitude variability of this order. These discrepancies suggest that uncertainty estimation is important for real-time FFI but not addressed in this work. Our results are in close agreement with the USGS rapid finite fault solution (USGS, 2016). The rake angle indicates a clear thrust mechanism in all cases although Figure 4.5 d presents a slight strike-slip component. These rake values indicate high tsunami potential, which is consistent with tsunami records in the region following the event. Figure 4.6 shows the fit of predicted and observed data for some selected stations for each data set. A complete account of data fits can be found at the end of this Chapter (Figures 4.9 to 4.16). The fit to the data is generally excellent but some phase differences are visible for the nearest stations. The quality of data fit is similar for the 4 data sets. §4.6 Inversion Results 57

Figure 4.7: Slip distributions obtained for a maximum rupture velocity of 1.5 km/s for the four data sets (see Table 4.2). Legends are the same as Figure 4.5.

Figure 4.8: Slip distributions obtained for a maximum rupture velocity of 2.0 km/s for the four data sets (see Table 4.2). Legends are the same as Figure 4.5. 58 Automated Algorithm for Finite Fault Inversion in Megathrust Events

4.7 Conclusion and discussion

We have implemented and applied, after the event, a fully automated and rapid FFI for the 2015 Illapel great earthquake. The inversion procedure employs W-phase three component records only and applies both prior slab information and rapid W-phase point source results to constrain fault extent and geometry. For megathrust events this information is normally available and thus desirable to employ. However, the method is flexible enough to allow for a different fault geometry in case it is necessary (e.g., a planar fault inferred from a point source solution when no other information is available). The inversion is parametrized as a linear inverse problem with positivity constraints and smoothing which is efficiently solved with non-negative least squares. Objective smoothing is obtained by iterative estimation of the noise level at each station and subsequent application of the discrepancy principle.

The slip distributions for each data set consistently show a large slip patch (∼ 10,000 km2 of slip > 2 m) at the shallowest portion of the fault, with a rake consist with a thrust event. Similar peak slip is found by Ye et al. (2015) although in that solution it is placed slightly deeper. The estimated magnitude from our ﬁnite fault models ranges from 8.3 to 8.5 depending on the data set considered (Table 4.2). While the typical frequency band of the W-phase seems too low to allow the recovery of these spatio-temporal details of the source, we think that the high signal-to-noise ratio in such a frequency band allows us to utilize the phase information contained in the W-phase waveforms. Since a well distributed network of stations are employed to perform the inversion, data can be very sensitive to time shifts of the various arrivals included in the W-phase time window. The potential impact of the event in terms of population exposure and tsunami potential can be better estimated having this information available. This illustrates the importance of obtaining early yet reliable slip distribution models. While the method proposed here can be applied to any megathrust earthquake, more case studies are required for a full validation. In particular, events with complex rupture patterns (e.g., Samoa-Tonga 2009, Mw = 8.0), that cannot be address by the a priori fault geometry, or anomalous source time functions (e.g., tsunami earthquakes) may need further analysis.

The method produces meaningful results as early as 25 minutes after the origin time without practitioner interaction. The ﬁnal solution for a full data set to 90◦ distance is available within 1 hour past origin time. The objective approach to §4.7 Conclusion and discussion 59

o Figure 4.9: Waveform fits for data up to ∆max = 30 . Legends are the same as in Figure 4.6. estimate station noise/weights and apply these in the quantitative determination of the smoothing parameter is critical in that it raises the confidence in the results. One of the most common criticisms of FFI methods is that subjective and ad-hoc choices can profoundly affect the results (Beresnev, 2003). We have addressed several such shortcomings resulting in a stable algorithm that produces reliable results. In addition, the method relies on simple hardware (an off-the-shelf 4-core computer) and freely available software so that it can be utilized in most seismic observatories.

The quality, rigour, and level of automation make our method highly relevant for early warning and disaster management in regions where no real-time GPS networks are available. In addition, even in regions with extensive GPS networks, the W-phase FFI approach can add valuable additional constraints by improving azimuthal coverage and slip resolution oﬀshore where typically no GPS stations are available. 60 Automated Algorithm for Finite Fault Inversion in Megathrust Events

o Figure 4.10: Waveform ﬁts for data up to ∆max = 50 . Legends are the same as in Figure 4.6. §4.7 Conclusion and discussion 61

o Figure 4.11: Waveform ﬁts for data up to ∆max = 75 . Legends are the same as in Figure 4.6. 62 Automated Algorithm for Finite Fault Inversion in Megathrust Events

Figure 4.12: Continuation of Figure 4.11. §4.7 Conclusion and discussion 63

o Figure 4.13: Waveform ﬁts for data up to ∆max = 90 . Legends are the same as in Figure 4.6. 64 Automated Algorithm for Finite Fault Inversion in Megathrust Events

Figure 4.14: Continuation of Figure 4.13. §4.7 Conclusion and discussion 65

Figure 4.15: Continuation of Figure 4.14.

Figure 4.16: Continuation of Figure 4.15. 66 Automated Algorithm for Finite Fault Inversion in Megathrust Events Chapter 5

Estimating Uncertainty from Rapid Finite Fault Models

In previous Chapters ﬁnite fault inversion has been treated as a classic optimization problem. In this context, the solution is given by a model which produces accurate predictions to the data and satisﬁes, to some extent, additional conditions such as smoothness. In this Chapter we introduce a probabilistic Bayesian approach to solve the inverse problem. The main purpose of this is to be able to obtain expressions for uncertainty in the solution which are naturally available in the Bayesian frame work. We employ analytic expressions for the posterior distribution to carry out rapid inversions which provide an uncertainty estimate. The approach is applied to simulated and measured data showing that reasonable and meaningful solutions accompanied by appraisal of uncertainty can be obtained.

5.1 Introduction

Seismic source studies typically rely on incomplete and noisy measurements of the displacement ﬁeld at the Earth’s surface. Eq. 2.11 shows how slip on a fault surface can be related to displacements of the elastic medium, which can be used as data in an inverse problem. By using these data, we can study the spatio-temporal evolution of rupture (Olson and Apsel, 1982). Similar to many other geophysical inverse problems, the incomplete and noisy observations lead to uncertain knowledge about the inferred model. This non-uniqueness of the solution means that many models can ﬁt the observed data reasonably well, although some of those models may not necessarily be plausible. To avoid implausible models, a common remedy is to include additional constraints on the solution. Typical examples in Finite Fault Inversion (FFI) are Tikhonov regularization (e.g., Tikhonov et al., 2013) and positivity constraints. Once these conditions have been adjusted to the particular problem of interest, a single optimal solution can often be selected as the most plausible which explains the data to a reasonable

67 68 Estimating Uncertainty from Rapid Finite Fault Models

extent. However, to better understand the information observed data provide about rupture, it is desirable to consider not only optimal models but all models that suﬃciently explain the data, a concept referred to as uncertainty estimation.

An approach which quantiﬁes uncertainty is given by Bayesian probability: We update prior knowledge about the model with information from observations to gain insight into the range of solutions. Importantly, this approach explicitly acknowledges that model inferences are limited by the data information. Bayesian methods interpret the data and parameters as random variables and probabilities express degrees of belief. The information is in terms of such probabilities and the solution to the problem is given by a probability density function (PDF) known as a posterior PDF: The state of information about the parameters given prior and data information. From the posterior, we can extract various parameter estimates such as the most probable parameter values and parameter uncertainties.

This probabilistic framework has been employed previously in FFI, addressing diﬀerent levels of complexity in the inverse problem. Early applications (e.g Yoshida, 1989; Yabuki and Matsu’ura, 1992; Ide et al., 1996) posed the inverse problem probabilistically and solved for the maximum a posteriori model using conventional least squares techniques. These works applied Akaike’s Bayesian Information Criterion (ABIC; Akaike 1980) to objectively determine optimal values for the rupture complexity (e.g., regularization by smoothness) and the noise level on the data (i.e., the noise standard deviation). However, even after several decades of application, the procedures to select regularization and noise parameters are still poorly discussed in many other FFI studies.

When the FFI problem is formulated as linear (no positivity constraints and linear source time function), it is possible to write an analytic expression for the posterior PDF which in this special case is a multivariate Gaussian (Fukuda and Johnson, 2008). In addition, the maximum a posteriori model can be found directly from linear least squares inversion. These analytic solutions can be exploited to eﬃciently explore, within a probabilistic framework, dependence of the solution on nonlinear model assumptions such as relative weights of diﬀerent datasets and geometry of the faulting plane (e.g., Fukuda and Johnson, 2010; Minson et al., 2014a). In these cases, the use of analytic expressions for the linearized part of the problem results in a more computationally tractable problem since nonlinear Bayesian sampling is only employed for some parameters. §5.1 Introduction 69

Nonlinear use of Bayesian techniques involves numerical sampling, that is, the posterior PDF is evaluated directly with large numbers of random (Monte Carlo) samples of the parameters. We refer to these methods as nonlinear Bayesian. Note that nonlinear methods provide a natural way to estimate regularization parameters by including them as unknowns in the problem (Fukuda and Johnson, 2008). With increasing computer power, nonlinear Bayesian methods are increasingly utilized in FFI (e.g Fukuda and Johnson, 2008; Monelli and Mai, 2008; Minson et al., 2014b; Dettmer et al., 2014; Duputel et al., 2015; Kubo et al., 2016). While Bayesian sampling is general and does not rely on linearization to estimate the posterior PDF, it is computationally demanding due to the requirement to evaluate the forward problem for extremely large numbers of candidate models.

Therefore, a common way of parametrizing FFI of waveform data is to linearize the problem (e.g., Olson and Apsel, 1982; Hartzell and Heaton, 1983; Ide, 2007). The linearization is achieved by ﬁxing fault geometry and discretizing the fault surface as subfaults whose source time functions are approximated by linear combinations of simple functions (e.g., triangles). Each function represents a time window in which the subfault may exhibit slip. The approach is referred to as the multiple time window method (MTWM), is linear and can account for some variability in rupture velocity and shape of the source time function. A disadvantage is the requirement for large numbers of parameters, since each subfault requires slip parameters for each time window and many time windows are often applied.

Assuming the linear approximation is reasonable, the posterior PDF is given analytically by a Gaussian PDF (e.g., Tarantola, 2005; Fukuda and Johnson, 2008) and can be estimated eﬃciently. However, uncertainties are rarely considered in linearized inversion since positivity constraints are often employed as additional regularization to obtain plausible inferences. Positivity constraints restrict the solution to only allow slip in certain directions and avoid unlikely, opposing slip directions across the fault. Under positivity constraints the posterior is no longer Gaussian (e.g., Fukuda and Johnson, 2008).

Therefore, the posterior cannot be found analytically which means that no analytic expression exists for the posterior covariance matrix. Such analytic expressions are required, for example, to apply the ABIC and to infer slip uncertainties. In the absence of analytic expressions for the ABIC, Bayesian sampling can be applied to objectively constrain noise and smoothing parameters when positivity constraints are applied (Fukuda and Johnson, 2008; Kubo et al., 2016). In that 70 Estimating Uncertainty from Rapid Finite Fault Models

nonlinear approach, positivity constraints are implemented by bounded priors which restrict the parameter space of the inversion. Moreover, integrals of the posterior PDF are evaluated numerically and do not require analytic solutions, account for uncertainty of the smoothing parameters, and are less likely to exhibit linearization error. However, computational cost is much higher and such methods are therefore unsuitable for time-sensitive application.

This work considers a linearized Bayesian approach with positivity constraints which does not require Bayesian sampling. Our work is aimed at applications where rapid solutions are required (e.g., emergency response after large earthquakes) or where computational infrastructure is not suﬃcient to carry out Bayesian sampling.

We apply a parameter transformation to the scalar moment of each subfault and invert for the natural logarithm of the parameters. The transformation ensures positivity and the posterior can be approximated by a Gaussian distribution in logarithmic space. The solution in linear space around the maximum a posteriori (MAP) solution is given by a log-normal PDF from which we can obtain analytic expressions for marginal distributions to quantify uncertainties. In this article, we ﬁrst develop the appropriate Bayesian framework to be used in the inversion scheme. Particular attention is given to the computation of uncertainties from the posterior PDF. Next, we discuss the problem of estimating the MAP solution, which is a nonlinear optimization problem. Then, we apply our method to simulated W-phase data (Kanamori, 1993) which are modelled to mimic the Iquique event (Mw=8.2, 2014, Chile). The simulation results are compared to nonlinear Bayesian sampling estimates. This systematic comparison provides some insight into the linearization errors due to the posterior approximation. Finally, the method is applied to W- phase observations from the recent Illapel great earthquake (Mw = 8.3, Chile). In particular, we study the parameter uncertainties for two data sets which consider an increasing number of stations as epicentral distances increase.

5.2 Bayesian Inversion Framework

This section presents a Bayesian framework for MTWM FFI. We begin with the formulation of Fukuda and Johnson (2008); Kubo et al. (2016) and then introduce a variable transformation to obtain an approximated posterior which does not require Bayesian sampling. In Bayesian inference, we seek the posterior PDF from Bayes’s theorem (e.g., Tarantola, 2005; Sivia and Skilling, 2006). For N data d and M §5.2 Bayesian Inversion Framework 71

parameters m, the posterior PDF P (m|d) is given by

P (m|d) = CP (d|m)P (m), (5.1) where C is a normalization constant and P (m) is the prior, expressing knowledge about the parameters which is independent from the data. For observed data, the data error distribution P (d|m) is interpreted as a likelihood function (a function of only m) that quantiﬁes the likelihood that the parameters gave rise to the data.

5.2.1 Parametrization and Positivity Constraints

In FFI we seek to infer the spatial and temporal evolution of fault rupture from ground motions detected at the surface. The problem is parametrized by spatial and temporal discretization: The fault surface is assumed to be known and discretized with subfaults. It is common to represent slip by two orthogonal vectors of prescribed rake angles with unknown magnitude to incorporate unknown rake in the inversion. The temporal evolution can be parametrized as nonlinear or linearized. A common linearized method is MTWM which can account for some variability in the temporal evolution of rupture while allowing linear methods to be applied (e.g., Ide, 2007). Here, we apply MTWM discretization to achieve computational eﬃciency.

In MTWM, Ntw rupture fronts propagate with a prescribed rupture velocity m vr , separated by time interval td. A particular subfault can slip each time a rupture front reaches it and the temporal dependence of each slip event follows the prescribed source time function, typically a triangle function of half duration td. The parameter vector m contains the scalar moment of each spatio-temporal unit. To obtain slip from the scalar moment, a rigidity structure must also be assumed.

Assuming Gaussian distributed data noise, the likelihood function P (d|m) in eq. (5.1), can be written

−N/2 −1/2 P (d|m) =(2π) |Cd| 1 (5.2) exp − (Gm − d)T C −1(Gm − d) , 2 d

T where Cd is the data covariance matrix, |(·)| denotes the determinant, (·) the matrix transpose and the data vector d includes the concatenated seismograms. Matrix G is of size N ×M and contains Green’s functions for each parameter. 72 Estimating Uncertainty from Rapid Finite Fault Models

The prior P (m) quantiﬁes information about the parameters that is independent of the data information (e.g., physical or geological constraints). A common approach for the prior is to assume a smooth slip distribution (e.g., Tarantola, 2005; Yabuki and Matsu’ura, 1992; Kubo et al., 2016; Fukuda and Johnson, 2008). A smoothing prior can be expressed as a Gaussian probability distribution where the covariance matrix includes oﬀ-diagonal terms and is given by LT L of size M × M. L is a linear smoothing (Laplacian) operator. In addition, positivity constraints can be included by requiring all components of m to be non-negative

 (πα2/2)−M/2|LT L|−1/2 exp − 1 (Lm)T (Lm),  2α2 P (m|α) = for mi ≥ 0, i = 1,...,M (5.3)  0 otherwise, where α is an unknown scaling parameter controlling the degree of slip smoothness. Other prior conditions can also be implemented by replacing L with alternative operators. For instance, L = I results in a minimum norm prior equivalent to minimum model regularization. Importantly, LT L must be a positive deﬁnite matrix so that P (m|α) is a proper PDF.

The posterior PDF can then be computed from eq. (5.2) and (5.3) using Bayes’ theorem (eq. (5.1))

 ψ(m)  C1 exp − ,  2 P (m|d, α) = for mi ≥ 0, i = 1,...,M (5.4)   0 otherwise, where C1 is the normalization constant given by

Z ψ(m)−1 C1 = dm exp − , (5.5) m≥0 2 and ψ(m) is the objective function

T −1 −2 T ψ(m) = (Gm − d) Cd (Gm − d) + α (Lm) (Lm). (5.6)

The integral in eq. (5.5) is known as the marginal likehood PDF or evidence. Traditional inversion approaches estimate an optimal model, subject to positivity constraints, that minimizes ψ for an optimal value of α. Since the dependence on α causes nonlinearity, a grid search for α is applied. Since the optimal model §5.2 Bayesian Inversion Framework 73

maximizes the posterior PDF given by eq. (5.4), it corresponds to the MAP solution of a Bayesian approach. However, unless the full posterior PDF is estimated, other inferences cannot be obtained. Examples of quantities that require the full posterior PDF are: the mean solution, parameter correlations and parameter uncertainties.

A remaining issue is the dependence on the hyperparameter α. Most commonly, optimal values are determined by grid search and the application of some criterion which addresses the intrinsic trade-off between fitting the data and model smoothness. These criteria include the L-curve test (e.g., Hansen, 1992), the discrepancy principle (e.g., Constable et al., 1987) and generalized cross validation (e.g., Craven and Wahba, 1978). From the Bayesian point of view, α can be treated as an unknown random variable to be estimated from the data since Bayes’ theorem intrinsically and quantitatively addresses the smoothness and data fit trade-off (MacKay, 2003). The ABIC method achieves this by estimating an optimal α value based on an analytic expression for the evidence.

Fukuda and Johnson (2008) note that the ABIC should not be applied in FFI with positivity constraints, and that it does not give optimal results for smoothing since positivity introduces nonlinearity. The main issue is that the ABIC requires computation of the marginal likelihood PDF (see eq. (5.5)), which is not available analytically when positivity constraints are applied. Similarly, the computation of the posterior covariance matrix Z T Cm = dm (m − ˆm) (m − ˆm)P (m|d), (5.7) where ˆm is the posterior mean, cannot be obtained analytically because P (m|d) is not analytic and model uncertainties cannot be inferred. To overcome this diﬃculty, Fukuda and Johnson (2008); Kubo et al. (2016) implement a nonlinear Bayesian method to estimate the posterior PDF by Markov chain Monte Carlo sampling. This sampling has orders of magnitude higher computational cost since ψ needs to be computed for large numbers of parameter vectors.

5.2.2 Positivity by variable transformation

The approach described in section 5.2.1 provides robust estimates of the posterior PDF but the only way to compute it is by Bayesian sampling. However, under some circumstances the computational infrastructure for Bayesian sampling may not be available and some applications, such as disaster response, require rapid ﬁnite 74 Estimating Uncertainty from Rapid Finite Fault Models

fault solutions. In these situations, it is also important to understand parameter uncertainties which can reduce the risk of basing decisions on misleading solutions (e.g., misplacing shallow slip or under/overestimating peak slip). Therefore, it is desirable to apply Bayesian methods that make reasonable assumptions and eﬃciently provide uncertainties.

To avoid Bayesian sampling, we apply two changes in the approach previously described (section 5.2.1): First, positivity is implemented by a variable transformation which considers the natural logarithm of each parameter instead of the parameters themselves. Second, the posterior in logarithmic space is approximated as a Gaussian around the MAP solution. The ﬁrst change is required since it provides positivity for which an analytic estimate of the posterior PDF exists. This is particularly important for subfaults with near-zero slip, for which a Gaussian PDF approximation would result in erroneous results. In contrast, by working in logarithmic space, the new parameters are deﬁned for all real numbers and truncation is not a problem. This idea has been previously proposed by Tarantola (2005) and, to our knowledge, has not been applied to FFI.

To formally state our algorithm, we deﬁne the transformation from scalar moment to logarithmic scalar moment as

T M(m) = [ln m1, ln m2,..., ln mM ] (5.8) and T E(M) = [exp M1, exp M2,..., exp MM ] . (5.9)

Solving for M intrinsically leads to non-negative parameter values in m and we formulate the inversion in terms of parameters M. The likelihood function is given by

−N/2 −1/2 P (d|M) =(2π) |Cd| 1 (5.10) exp − [GE(M) − d]T C −1[GE(M) − d] 2 d and the smoothing prior is

1 P (M|α˜) = (2πα˜2)−M/2|LT L|−1/2 exp − (LM)T (LM) . (5.11) 2˜α2

Note that this smoothing prior is applied in logarithmic space and not equivalent to that in eq. (5.3). However, smoothing in logarithmic space appears to be a §5.2 Bayesian Inversion Framework 75

reasonable choice and in section 5.3, we carry out a quantitative comparison of both smoothing approaches.

Using Bayes’ theorem (eq. (5.1)) and eqns. (5.10) and (5.11), we obtain

! ψ˜(M) P (M|d, α˜) = C exp − , (5.12) 2 2 where " !#−1 Z ψ˜(M) C = dM exp − (5.13) 2 2 and

˜ T −1 −2 T ψ(M) = (GE(M) − d) Cd (GE(M) − d) +α ˜ (LM) (LM). (5.14)

The integral in eq. (5.13) is taken over all real numbers since M is an unbounded quantity, in contrast to m. Note that the new objective function is now non-linear and requires iterative linearization for eﬃcient solution. The main assumption in our method is that the posterior in eq. (5.12) can be approximated by a Gaussian as discussed below.

5.2.3 Uncertainty estimation by linearization

Slip uncertainties can be inferred from the model covariance matrix deﬁned by eq. (5.7). In principle, this requires numerical integration to evaluate eq. (5.7) using eq. (5.12). However, we can linearize the problem by approximating the posterior by a Gaussian PDF around the MAP solution of M, a procedure that is commonly applied to weakly non-linear inverse problems (e.g., Sivia and Skilling, 2006). In section 5.3, we evaluate the accuracy of this approximation and in the following we discuss its application to our problem.

For a PDF similar to eq. (5.12) to be reasonably approximated by a Gaussian, a sufficient condition is that the objective function ψ˜ is a quadratic function of M (Tarantola, 2005). For linear forward problems this condition is satisfied exactly. When the forward problem is non-linear, the strategy is to expand the objective function around the MAP and retain terms up to second order. For highly non-linear problems, higher order terms may be significant and a second-order approximation 76 Estimating Uncertainty from Rapid Finite Fault Models

may be poor. A second-order approximation is given by

1 ψ˜(M) = ψ˜(Mˆ ) + (M − Mˆ )T ∇∇ψ˜(Mˆ )(M − Mˆ ), (5.15) 2 where ∇∇ψ˜ is the Hessian matrix of ψ˜ and Mˆ is the MAP solution. The ﬁrst-order term includes the factor ∇ψ˜(Mˆ ), which is the gradient of ψ˜ at the MAP, which must vanish and is omitted in the expansion eq. (5.15).

Substituting eq. (5.15) in eq. (5.12) gives

1 P (M|d, α˜) ∝ exp − (M − Mˆ )T C−1(M − Mˆ ) , (5.16) 2 M where 1 C−1 = ∇∇ψ˜(Mˆ ) (5.17) M 2 is the covariance matrix of the Gaussian posterior PDF. In eq. 5.16 the normalization constant is omitted, hence the proportionality. These expressions can be evaluated analytically by computing the Hessian matrix. From eq. (5.14), the gradient and Hessian of the objective function are

T −1 −2 T ∇ψ(M) = 2E ⊗ G Cd (GE − d) + 2α L LM (5.18) and

T T −1 ∇∇ψ(M) = 2EE ⊗ G Cd G T −1 + 2 diag(E) ⊗ G Cd (GE − d) (5.19) + 2α−2LT L, respectively. For convenience we do not write explicitly the dependence E(M). The symbol ⊗ denotes the element-wise product and the operator diag creates a diagonal matrix whose elements are the elements of the argument vector. Derivations for eqns. (5.18) and (5.19) are given in the Appendix A.

5.2.4 Inversion strategy

The goal of our work is to obtain an analytic approximation of the posterior PDF so that uncertainties can be estimated. The procedure can be summarized as follows. First, we parametrize the rupture process following the MTWM. This involves prescribing the number and geometry of subfaults, the number of time windows and the maximum rupture velocity. In this study, we assume that a data §5.2 Bayesian Inversion Framework 77

covariance matrix Cd and an optimal value for α have been previously estimated and are known. Since directly determining these parameters as part of the inversion can strongly aﬀect our linearity assumption, iterative approaches to estimate them are used instead (e.g., Benavente et al., 2016).

After parametrizing the problem, we proceed to find the MAP solution. The optimization is carried out by minimizing the objective function ψ˜. Because of its non-linearity (see eq. 5.14), an iterative approach is applied. Since analytic expressions exist for the linearized gradient and Hessian (eqns. (5.18) and (5.19), respectively), efficient methods can be applied to find the minimum (e.g., Press et al., 1992; Aster et al., 2005). In this work, we use the Newton conjugate gradient (Newton-CG, Wright and Nocedal, 1999) method as implemented by SciPy (Jones et al., 2001–).

As for any other non-linear inversion, some concern remains regarding the uniqueness of the solution. If ψ˜ is a convex function the solution is guaranteed to be unique. While, we do not provide a general proof of convexity, the objective function appears empirically well behaved near the MAP solution. We reach this conclusion since a sufficient requirement for convexity is a positive definite Hessian. Equation (5.19) shows that the Hessian comprises 3 terms. The first and third terms are positive definite as long as the rows of G and L are linearly independent. However, the second term can contribute negative elements to the diagonal which may cause convexity issues. Nevertheless, for solutions close to the MAP the predictions should be comparable to the data and the contribution of the second term to the Hessian should be small compared to the other two. Further- more, all the Hessian matrices we obtained for MAP solutions were positive definite.

After obtaining the MAP, we estimate the uncertainties from the posterior covariance matrix by calculating the Hessian from eq. (5.19) and then the posterior covariance matrix from eq. (5.17). At this point, we have approximate expressions for the posterior PDF in logarithmic space given by 5.16. In principle, the diagonal elements of the covariance matrix are suﬃcient to obtain marginal distributions for single parameters since the marginals of a Gaussian are also Gaussian (e.g., Tarantola, 2005).

However, to interpret the solution in terms of slip in linear space, a coordinate transformation to m is required. To consider the transformation of uncertainties, we note that normally distributed random variables in logarithmic space have log- 78 Estimating Uncertainty from Rapid Finite Fault Models

normal distribution in linear space. That is to say if x = ln y, is normal distributed with mean µ and standard deviation σ, y has log-normal distribution as

1 (ln y − µ)2 P (y) = √ exp − . (5.20) yσ 2π 2σ2

Note the convention that µ and σ are the parameters of the normal distribution in logarithmic space. The mean Ω and the standard deviation Σ of P (y) are given by

Ω = eµ+σ2/2 (5.21) and q Σ = (eσ2+2µ)(eσ2 − 1), (5.22) respectively. In our applications in sections 5.3 and 5.4 we use eq. (5.22) to compute the standard deviation of each parameter using σ from the square root of the corresponding diagonal element in CM. As a measure of uncertainties in the log-normally distributed parameters we use the width of the 95% conﬁdence interval (CI). Details about the computation of CI are given in B.

5.3 Inversion for simulated data

This section applies our method to simulated noisy data to study whether it can provide meaningful slip uncertainties. All simulations are for W-phase waveforms based on a Green’s function database (Kanamori and Rivera, 2008). The W-phase has been extensively used for source inversions of large events (e.g., Kanamori and Rivera, 2008; Duputel et al., 2012a; Hayes et al., 2009; Nealy and Hayes, 2015). W-phase FFI was first considered by Benavente and Cummins (2013) and subsequently by Dettmer et al. (2014); Benavente et al. (2016). These studies concluded that W-phase waveforms contain significant information about the rupture process of megathrust earthquakes. We begin by assessing the effects of our smoothing scheme in logartihmic space. Then, we consider uncertainties for simulated data with different noise levels. Finally, we compare the linearized uncertainty estimates from our method to those obtained by non-linear Bayesian sampling. §5.3 Inversion for simulated data 79

Figure 5.1: Simulation geometry and true model for the simulations. Stations (blue), hypocenter (star) and fault geometry (left) are based on the 2014 Iquique earthquake. The true slip distribution (right) is chosen to be representative for an event of this size.

5.3.1 Eﬀects of smoothing in logarithmic space

An important consequence of the variable transform we apply is that smoothing is applied in logarithmic space. Typical FFIs apply smoothing directly to slip or scalar moment. While it is reasonable to assume some form of smoothness for slip, the precise implementation is subjective. In this simulation, we study to which extent log-space smoothing is diﬀerent from non-logarithmic smoothing which is widely applied.

The simulation geometry is based the 2014 Iquique event (Chile, Mw=8.2) and vertical component W-phase seismograms are computed using the GF database for a simple geometry consisting of 15 subfaults along strike (355o) and 7 along dip (11o). Only one time window was employed resulting in constant rupture velocity. The model includes only positive slip values which are decomposed into two rake components. This is a very simple parametrization intended only to produce a workable example for simulation purposes. Importantly, the model satisﬁes a traditional (not in log-space) smoothing scheme expressed by the prior of eq. 5.3. Figure 5.1 shows the target slip distribution. The model is clearly smooth and exhibits a major slip patch at the shallow-most part of the fault and a very small slip feature in the bottom-right corner.

We compute synthetic W-phase data based on the target model and stations shown in ﬁgure 5.1. Then, we add Gaussian noise and invert the resulting simulated data using our algorithm. We consider two noise levels which are 2% and 5% with respect to the absolute value of the mean of each trace. The data covariance matrix employed is a diagonal block matrix where the diagonal elements of each block are 80 Estimating Uncertainty from Rapid Finite Fault Models

Figure 5.2: Simulation results for the log-space smoothing approach in terms of the MAP model. Results for 2% (left) and 5% (right) noise level are shown. Both results agree well with the true model (Fig. 5.1). equal to the noise variance for that trace. The smoothness was adjusted following the discrepancy principle. The results are given in ﬁgure 5.2. Both results agree well with the true model. However, both results are less smooth and display more abrupt changes in the slip distribution than the true model. In addition, the small slip patch is somewhat misplaced in the result for the data with 5% noise. However, these diﬀerences are likely small compared to the slip uncertainties of the solution. We conclude that log-space smoothing achieves results which are similar to non- logarithmic-space smoothing.

5.3.2 Uncertainty estimation for diﬀerent noise levels

The estimation of slip uncertainties is straightforward once a MAP solution is obtained. Equation (5.17) is evaluated at the MAP solution to approximate the posterior by a Gaussian PDF. The single-parameter marginals of this multivariate Gaussian are also Gaussian distributions with variance given by the appropriate diagonal elements of the covariance matrix. Then, the marginals of each parameter in linear space are given by log-norm distributions with mean and standard deviation given by eqns. (5.21) and (5.22), respectively. We follow this procedure to estimate uncertainties for the slip results in ﬁgure 5.2

However, to obtain uncertainties for the cumulative slip distribution it is necessary to perform an additional step. This is because in the MTWM the parameters given by m correspond to the scalar moment of each spatio-temporal unit which are also decomposed into rake directions. That is, the cumulative scalar moment (cum) Moi of a subfault i is

v 2 2 u Nt ! Nt ! cum u X j1 X j2 Moi = t Moi + Moi , (5.23) j j §5.3 Inversion for simulated data 81

j1 j2 where Moi and Moi are the rake components for the j-th time window and Nt is cum the number of time windows. Thus, to compute the standard deviation of Moi we employ the error propagation rules

q 2 2 σx+y = σx + σy (5.24) and

σx2 = 2xσx. (5.25)

These expression are justiﬁed by assuming that the PDFs of the variables and the results are Gaussian distributions. However, for the square root operation, this procedure may not work well since for very small slip values the Gaussian assumption may be poor (Sivia and Skilling, 2006). Therefore, it is better to approximate the resulting distribution by a Gaussian ﬁrst and then compute the errors of this √ approximated PDF (Sivia and Skilling, 2006). As a result, the mean < x > and √ the standard deviation σ x are obtained from

√ 1 h i < x >2 = x + x2 + σ21/2 and 2 x √ (5.26) 1 2(3 < x >2 −x) σ√−2 = √ + , x 2 2 < x > σx respectively.

The 95% CIs of the cumulative slip distribution (ﬁgure 5.2) are shown in ﬁgure 5.3. Details about the calculation of the CIs are given in Appendix B. In both cases the uncertainties are larger for slip patches of large magnitude. For the case with 2% noise, the maximum uncertainty value is ∼ 1.2 m and for the case with 5% noise it is ∼ 1.6 m. While the uncertainty distribution is similar for both cases, the magnitude uncertainty is larger for noisier data. While these uncertainty estimates appear reasonable, it is desirable to carry out quantitative comparison with a robust independent method.

5.3.3 Comparison with nonlinear Bayesian sampling

This section applies Bayesian sampling to obtain non-linear uncertainty estimates and compares these to the linearized results of the previous section. The comparison is based on the simulated W-phase data with 5% noise level. The uncertainty estimates are considered separately for each rake component. Scalar moments are transformed to slip to ease comparison with ﬁgures 5.1 and 5.3. The resulting 82 Estimating Uncertainty from Rapid Finite Fault Models

Figure 5.3: 95% CIs of recovered models using log-space smoothing. Target model is shown in ﬁgure 5.1. CIs correspond to the models of ﬁgure 5.2, in the same order.

log-normal marginal distributions are shown as red curves in ﬁgure 5.4 for rake component 1 and ﬁgure 5.5 for rake component 2.

A Bayesian sampling algorithm is applied to obtain independent marginal distributions for the parameters. In this approach, random samples are drawn directly from the non-linear posterior PDF (eq. 5.12). A detailed description of the sampling approach we employ is given by Dettmer et al. (2014). To isolate the effect of the linearization approximation, we fix the smoothing parameter α to the same value we estimated in our linearized method. The Bayesian sampling results in a large ensemble of parameter vectors that numerically approximate the posterior. Marginal distributions are straightforward to obtain as normalized histograms for each parameter. These marginal PDFs are shown for the appropriate rake component in figures 5.4 and 5.5 (blue histograms).

In general, we find very good agreement between the marginals obtained from both methods (figures 5.4 and 5.5). The histograms reveal that most parameters do not exhibit features indicating strong non-linearity (such as multiple modes or other complicated shapes). Rather, the log-norm approximation seems appropriate with similar peak positions and uncertainties throughout. However, we note that fixing α to an optimal value can impact both estimated slip values and uncertainties. In practice, treating α as unknown in Bayesian sampling provides a more general solution than possible in a linearized approach. We conclude that the comparison indicates that our approximation can recover meaningful uncertainty estimates at a fraction of the computational expense required by non-linear methods. For this particular example the non-linear Bayesian method converged to the posterior shown after 12 hours using 12 CPUs while our linearized approach only requires a couple of seconds in a desktop computer. §5.3 Inversion for simulated data 83

Figure 5.4: Marginal PDFs for parameters of rake component 1. Each subplot shows the marginal PDF for the corresponding subfaults. The layout of the subplots is the same as the slip distributions shown in ﬁgures 5.1 and 5.3 so horizontal direction goes along strike and vertical direction along dip. The red curves were obtained using the analytic expression we proposed here and the blue histograms were obtained using Bayesian sampling of the posterior PDF.

Figure 5.5: Marginal PDFs for parameters of rake component 2. Legends are same as ﬁgure 5.4 . 84 Estimating Uncertainty from Rapid Finite Fault Models

5.4 Application: Illapel great earthquake (2015, Mw=8.3), Chile

An interesting aspect of our method is its potential application in rapid inversion after large earthquakes. Since these solutions could be utilized during the first phases of emergency management, uncertainty quantification can play crucial role in their interpretation. Here, we apply our inversion to W-phase waveforms of the recent Illapel great earthquake (2015, Mw = 8.3, Chile). Early finite fault models inferred peak slip of 5 - 10 m (Ye et al., 2015; Tilmann et al., 2016; Benavente et al., 2016; Melgar et al., 2016) and the maximum tsunami run-up was estimated to be 10.8 m (Aránguizet al., 2016). We quantify slip uncertainties for two data sets of increasing epicentral distance and study how uncertainties change as more data are added to the inversion.

In previous work (Benavente et al., 2016), we applied an automated algorithm to obtain rapid ﬁnite fault models from W-phase data for this event. In that work we employed the traditional MTWM with positivity constraints. A block-diagonal data covariance matrix Cd was estimated from the residuals of a previous inversion and optimal smoothness was estimated based on the discrepancy principle. Typically W-phase source inversions make use of data in the range 5o to 90o of epicentral distance (e.g., Kanamori and Rivera, 2008; Duputel et al., 2012a; Benavente and Cummins, 2013). In our previous Illapel study we considered 4 diﬀerent data sets with upper limits on epicentral distance being 30o, 50o, 75o and 90o. We concluded that a meaningful solution can already be obtained by considering data from the smallest dataset.

Here, we carry out estimation of slip uncertainty for the Illapel event based on assuming a single time window. First, we perform an inversion following Benavente et al. (2016). For computational efficiency, we decrease the number of parameters and consider one time window. That is to say, the rupture is parametrized with constant rupture velocity of 1.5 km/s. Then, we apply the approach of sections 5.2 and 5.3 to compute slip uncertainties. In this step we use the data covariance matrix from the previous inversion (first step). The optimal smoothness is chosen according to the discrepancy principle, that is, the χ2 data misfit is approximately equal to the number of data points. In order to assess the impact of including more stations on uncertainty, we compare the results obtained for two datasets with maximum epicentral distances of 30o and 90o. §5.5 Summary and Discussion 85

Figure 5.6: Inversion results for the Illapel event, stations up to 30o of epicentral distance. Legends are the same as ﬁgure 5.1, right. Left: MAP solution of the slip distribution. Right: 95% of CI width for slip on the fault.

Figures 5.6 and 5.7 show the inversion results for the two datasets. As a representative model we plot the MAP model for both figures (left). For convenience we have plotted the subfaults as rectangles but the actual shape may be slightly irregular (see Benavente et al., 2016) In both cases, the peak slip of the MAP model is around 20 m while in the results of Benavente et al. (2016) peak slip was around 12 m. This may be due to smoothing being applied in log space, which will penalize abrupt slip changes less. In addition, parametrization with a single time window may be too simple for this event. However, the location of the peak slip is similar to the results in Benavente et al. (2016). We also compute the width of the 95% CI of slip (see appendix B), for both solutions (figures 5.6 and 5.7). In both cases, uncertainty is largest where slip magnitude is large. The uncertainties for data to 30o also show uncertainty of ∼4 m in the top-right corner (5.6, left) which is likely due to the limited amount of data. The uncertainty in this region is much smaller for the solution of the larger dataset (5.7). The uncertainty magnitudes appear to be reasonable with a maximum 95% CI width of ∼ 8 m for the 30o data and a maximum width of 6 m for the 90o data: The maximum width of the CIs is almost 2 m less for the larger data set. This suggests that significant additional information about the rupture is provided by the additional stations in the largest dataset. Note that this conclusion could not be obtained by only considering optimal parameters.

5.5 Summary and Discussion

We developed an eﬃcient method for slip uncertainty estimation in ﬁnite fault inversion with positivity constraints. A Bayesian framework is employed which incorporates uncertainty estimation naturally as part of the inversion. Positivity is ensured by a parameter transformation to logarithmic space which formulates 86 Estimating Uncertainty from Rapid Finite Fault Models

Figure 5.7: Inversion results for the Illapel event, stations up to 90o of epicentral distance. Legends are the same as ﬁgure 5.6. the inversion in terms of intrinsically positive slip. This procedure results in a non-linear inverse problem for which optimal parameters are estimated by applying a linearised iterative approach. The posterior PDF is approximated by a normal distribution in logarithmic space. Under this linearized approximation, an analytic expression exists for the posterior covariance matrix which provides straightforward uncertainty estimates in logarithmic space. These uncertainties correspond to log-normal distributions for slip in linear space.

Simulation results and comparison with nonlinear Bayesian sampling show that our new method leads to meaningful uncertainties at a fraction of the computational eﬀort required for nonlinear estimation. This suggests that the linearized approximation for the posterior PDF is suﬃcient for robust rapid uncertainty estimation. However, more general results would be obtained if smoothing were treated as an unknown which moderately increases computational cost since it requires additional solutions to the non-linear inverse problem (see Benavente et al., 2016). Slip distributions obtained for the recent Illapel event are consistent with previously published results. The Illapel slip uncertainties we obtained seem reasonable since they do not exceed MAP values (e.g., we are inferring information from the data) and correlate with slip. These results indicate that the method is promising although some limitations still need to be addressed.

An important consideration that we have omitted in our analysis concerns the determination of the smoothness. Strictly speaking α is given by a probability distribution which needs to be inferred (Fukuda and Johnson, 2008; Kubo et al., 2016). We use a point estimate for α, based on the discrepancy principle, but note that this is a simpliﬁcation and that we have currently no information about the linearization error made by using a point estimate. In the nonlinear Bayesian sampling, we employ for comparison (ﬁgures 5.4 and 5.5), we do not treat α as an §5.5 Summary and Discussion 87

unknown and thus cannot quantify the linearization error due to α. This requires further study since strong non-linearities could cause poor linearized uncertainty estimates.

However, we note that many current FFI studies do not address the issue of slip uncertainty and are typically based exclusively on point estimates to infer rupture properties and sometimes they do not indicate how regularization is applied. We assert that this may be one of the reasons that discrepancies of slip estimates exist in the literature for the same event (e.g., Beresnev, 2003; Minson et al., 2013). The Bayesian approach is convenient for FFI since it provides more robust solutions to the ill-posed inverse problem. The eﬃciency of our method is such that it can be carried out on personal computers, making it more broadly applicable in comparison to nonlinear Bayesian sampling which requires high-performance computer infrastructure.

The proposed method can also be utilized to determine hyperparameters via the ABIC without lifting positivity restriction. Fukuda and Johnson (2008) point out that ABIC cannot be computed analytically because when positivity constraints are in use no analytic expression for the evidence exists. However, in our formulation it would be possible to make use of the approximate posterior given by eq. 5.16 to evaluate the evidence and therefore the ABIC. This could be done, for instance, in log-space and once the hyperparameters have been determined a full inversion with ﬁxed hyperparameters can be performed. Importantly, the use of the ABIC method allows the determination not only of α but also of a scaling hyperparameter for the data covariance matrix. This can assist in improving the initial estimate of this matrix by relaxing assumptions about it.

While the algorithm has been tested employing only W-phase records, the formulation is quite general and applicable for various types of data. Teleseismic, geodetic, continuous GPS and strong motion data can also be used in similar scheme, since the forward problem is also linear (or linearizable via MTWM). In addition, this approach can also be used in joint FFI, providing a quantitative means to evaluate the uncertainties. This is an interesting aspect to be explored since the impact of combining several datasets on model uncertainties has not been well studied. 88 Estimating Uncertainty from Rapid Finite Fault Models Chapter 6

Conclusion and Discussion

Many inversion techniques for imaging the earthquake rupture have been developed. However, the problem of obtaining rapid coseismic slip distributions for large earthquakes remains challenging. From a practical point of view, rapid source models can be employed to model tsunami travel times and predict ground motions at the source region so they can be useful in emergency response context. In this PhD I have developed techniques that can contribute to overcome some of the issues of rapid finite fault inversion for megathrust events. Throughout all this work we have employed only W-phase waveforms to retrieve slip models. There are two main advantages of this: 1) Broadband seismic data is widely available in realtime and 2) The W-phase is particularly well suited to rapid source characterization for large events. With this motivation three major projects were developed during my PhD: 1) Classic finite fault inversion using the W-phase (Chapter 3), 2) Automated finite fault algorithm for megathrust events (Chapter 4) and 3) A new inversion scheme to obtain uncertainty from rapid finite fault inversions (Chapter 5). Specific conclu- sions for all of them are given in the corresponding Chapters. Here I list the main outcomes of this PhD research:

1. For the first time, the W-phase was employed in finite fault inversion. Despite its long-period, we found that it contains significant information about the rupture process in megathrust earthquakes.

2. We have shown that automated ﬁnite fault inversion algorithms are feasible. The short time required to obtain a solution (∼ 25 min) indicates that they may be useful in emergency response contexts.

3. We have proposed a method to iteratively characterize the noise level of seismic stations in ﬁnite fault inversions. In addition this method allows for the objective estimation of the smoothness of the slip model.

4. We have developed a novel method to estimate uncertainty in rapid ﬁnite fault inversion with positivity constraints. The method relies on a robust proba-

89 90 Conclusion and Discussion

bilistic framework, where the solution to the inverse problem is intrinsically given by a probability distribution. Moreover, the method is quite general and can be used with various data types, including teleseismic and geodetic data.

6.1 Discussion and future work

While I believe the outcomes of this PhD research represent a signiﬁcant contribution to the rapid ﬁnite fault inversion problem, there are some interesting aspects that remain to be explored. Firstly, none of the proposed methodologies have been tested for an event in real time. Since they rely purely on W-phase records and the W-phase point source algorithm is widely used, all the existing machinery in seismic observatories for W-phase point source inversion can be utilized. Obtaining a rupture model in real time within 25 min which only requires data from broadband seismic stations would be an extremely relevant test for this study. However, as a previous step, the automated inversion should be tested against a more comprehensive catalog of megathrust events for which results have already been obtained using other methods.

Another aspect that needs to be further explored is the iterative noise characterization approach we have proposed. While the method has worked reasonably well so far, some concern remains as to how strong the inversion results depend on the initial inversion. This is because the initial inversion is performed with no regularization constraints and the resulting prediction will fit noise features. Of course, we do not actually use the solution but the residuals will be employed as a first characterization of the noise. In addition, the noise level of this initial inversion depends on the station screening we perform in previous W-phase point source inversion. For example, the impact of selecting different thresholds at the misfit screening stage of the W-phase point source inversion in the finite fault solution needs to be quantified. Presumably, since the smoothness is selected based on the noise of the data, the results will be similar but the with a different degree of smoothness.

One of the advantages of using W-phase data is that the global seismic network allows for the characterization of all large events. However, in many cases there may be additional types of data that can be incorporated into the inversion. As we have discussed, static and continuous GPS data have proven to be very useful to characterize the seismic source when available. In principle, there are no fundamental issues in performing joint inversions of GPS and W-phase data. §6.1 Discussion and future work 91

However there may be practical inconveniences that need to be addressed to work with the two data types in real-time. Since the use of various data types in the inversion can assist in constraining the rupture process, this an interesting topic to be explored.

The main purpose of our study in Chapter 5 was to be able to estimate uncertainties from the ﬁnite fault model. Nevertheless, that is not the only advantage of utilizing a probabilistic Bayesian approach. Another import result is that we show that the posterior distribution can be approximated by an analytic expression. This result can be used to objectively constrain the smoothing (or other type of regularization) via the ABIC even if positivity constraints are in use. Compared to the discrepancy principle, the ABIC method has the advantage of estimating a scaling parameter for the noise level. In addition, the expression for the posterior can be employed to evaluate the use of diﬀerent priors for an event. For instance, we can compare if a smoothing prior describes an earthquake better than a minimum norm prior. In the Bayesian approach this can be done by comparing the evidence in both cases. The analytic expression we have obtained for the posterior allows the evidence to be computed and thus makes it possible to perform this type of study. 92 Conclusion and Discussion Appendix A

Computation of Hessian and gradient of the non-linear objective function

In this appendix we derive the expressions for the gradient and Hessian of the objective function eq. (5.14) which is expressed as

˜ T −1 −2 T ψ(M) = (GE(M) − d) Cd (GE(M) − d) +α ˜ (LM) (LM). (A.1)

Introducing the Cholesky decomposition of Cd,

−1 T −1 −T −1 Cd = (KK ) = K K , (A.2) where K is a lower triangular matrix, we can rewrite eq. (A.1) as

ψ˜(M) = (GE˜ (M) − d˜)T (GE˜ (M) − d˜) +α ˜−2(LM)T (LM), (A.3) where G˜ = K−1G, d˜ = K−1d. (A.4)

For convenience we name the ﬁrst and second terms of eq. (A.3) ψ˜(M)(I) and ψ˜(M)(II), respectively, so that

ψ˜(M) = ψ˜(I)(M) + ψ˜(II)(M). (A.5)

93 94 Computation of Hessian and gradient of the non-linear objective function

By applying index notation and rearranging, we ﬁnd that

!2 ∂ψ˜(II)(M) ∂ X X = α−2 L M ∂M ∂M ij j l l i j ! ! −2 X X X = 2α LijMj Limδlm i j m (A.6) ! −2 X X = 2α LijMj Lil i j −2 T = 2α L LM l , where δij is the Kronecker delta. From this result, it can be seen that

˜(II) ∂ψ (M) −2 T = 2α L L lk . (A.7) ∂Mk∂Ml

The derivatives of ψ˜(I)(M) can be obtained similarly:

!2 ∂ψ˜(I)(M) ∂ X X = G˜ E − d˜ ∂M ∂M ij j i l l i j ! ! X X ˜ ˜ ∂ X ˜ ˜ = 2 GijEj − di GikEk − dk ∂Ml i j k ! ! (A.8) X X ˜ ˜ X ˜ = 2 GijEj − di GikEkδkl i j k X ˜ X ˜ ˜ = 2El Gil GijEj − di i j T −1 = 2 E ⊗ G Cd (GE − d) l 95

and " # ∂ψ˜(II)(M) ∂ X X = 2E G˜ G˜ E − d˜ ∂M ∂M l ∂M il ij j i k l k i j

∂El X X + 2 G˜ G˜ E − d˜ ∂M il ij j i k i j X ˜ X ˜ = 2El Gil GijEjδjk i j X ˜ X ˜ ˜ + 2Elδkl Gil GijEj − di (A.9) i j X ˜ ˜ = 2ElEk GilGik i X ˜ X ˜ ˜ + 2Elδkl Gil GijEj − di i j T T −1 = 2 EE ⊗ G Cd G T −1 + diag(E) ⊗ G Cd (GE − d) kl .

Taking the derivative of eq. (A.5) and using eqns. (A.8) and (A.6) we obtain eq. (5.18). Likewise, taking the second order derivatives of eq. (A.5) and using eqns. (A.9) and (A.7) yields eq. (5.19). 96 Computation of Hessian and gradient of the non-linear objective function Appendix B

Conﬁdence intervals for log-normal distribution

In this appendix we explain the computation of the width of the conﬁdence intervals displayed in ﬁgures 5.6 and 5.7. Once we have obtained the posterior covariance matrix we proceed by computing the posterior PDF of the cumulative slip distribution by using the error propagation rules of eqns. (5.24), (5.25) and (5.26). The resulting PDF is assumed to be a log-normal distribution with known median M˜ (M˜ = eMˆ ) and standard deviation Σ (from the error propagation analysis). From here we compute the standard deviation σ of the generating normal distribution as ! 1 r Σ2 1 σ2 = ln + + , (B.1) 2 M˜ 2 4 which follows directly from eq. (5.22) and the median M˜ expression given above. Now we can write explicitly the lognormal PDF (see eq. (5.20)) for the cumulative slip on each subfault.

The cumulative distribution function (CDF) for a log-normally distributed parameter x is ! 1 1 ln x − Mˆ CDF(x) = + erf √ , (B.2) 2 2 σ 2 where σ and Mˆ are the standard deviation and mean of the generating normal distribution, respectively, and erf is the error function. Thus, a speciﬁc parameter value x for a given CDF probability is

h√ i x = exp 2σerf−1(2 CDF − 1) + Mˆ , (B.3) where erf−1 is the inverse of the error function. To calculate the width of the 95% CI, we replace CDF in eq. B.3 by 2.5 and 97.5 to obtain the parameter values for

97 98 Conﬁdence intervals for log-normal distribution

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