Ground Deformation Related to Caldera Collapse and Ring-Fault Activity

Home , Fault scarp, Magma chamber

Thesis by

Yuan-Kai Liu

In Partial Fulﬁllment of the Requirements

For the Degree of

Masters of Science

King Abdullah University of Science and Technology

Thuwal, Kingdom of Saudi Arabia

May, 2018 2

EXAMINATION COMMITTEE PAGE

The thesis of Yuan-Kai Liu is approved by the examination committee.

Committee Chairperson: Sigurj´on J´onsson Committee Members: Juan Carlos Santamarina, Sigurdur T. Thoroddsen 3

©May, 2018

Yuan-Kai Liu

ABSTRACT

Volcanic subsidence, caused by partial emptying of magma in the subsurface reservoir has long been observed by spaceborne radar interferometry. Monitoring long-term crustal deformation at the most notable type of volcanic subsidence, caldera, gives us insights of the spatial and hazard-related information of subsurface reservoir. Several subsiding calderas, such as volcanoes on the Galápagos islands have shown a complex ground deformation pattern, which is often composed of a broad deflation signal affecting the entire edifice and a localized subsidence signal focused within the caldera floor. Although numerical or analytical models with multiple reservoirs are proposed as the interpretation, geologically and geophysically evidenced ring structures in the subsurface are often ignored. Therefore, it is still debatable how deep mechanisms relate to the observed deformation patterns near the surface. We aim to understand what kind of activities can lead to the complex deformation. Using two complementary approaches, we study the three-dimensional geometry and kinematics of deflation processes evolving from initial subsidence to later collapse of calderas. Firstly, the analog experiments analyzed by structure-from-motion photogrammetry (SfM) and particle image velocimetry (PIV) helps us to relate the surface deformation to the in-depth structures. Secondly, the numerical modeling using boundary element method (BEM) simulates the characteristic deformation patterns caused by a sill-like source and a ring-fault. Our results show that the volcano-wide broad deflation is primarily caused by the emptying of the deep magma reservoir, whereas the localized deformation on the caldera floor is related to ring-faulting at a shallower depth. The architecture of the ring-fault to a large extent determines the deformation localization on the surface. Since series evidence for ring-faulting at several volcanoes are provided, we highlight 5 that it is vital to include ring-fault activity in numerical or analytical deformation source formulation. Ignoring the process of ring-faulting in models by using multiple point sources for various magma reservoirs will result in erroneous, thus meaningless estimates of depth and volume change of the magmatic reservoir(s). 6

ACKNOWLEDGEMENTS

My sincere thanks must go to my committee chair, Professor Jónsson, and my committee members, Professor Santamarina and Professor Thoroddsen for their eval- uation of this thesis. This research would not have been possible without my adviser, Prof. Jónsson, for his instruction and mentoring throughout the journey of my graduate study. His scientific and editorial suggestions were essential to the completion of this research. His inspiring advice has taught me innumerable lessons and insights on my future academic pursuits in general. Particular thanks to Prof. Thoroddsen and his students for providing the ex- ceptional laboratory instruments; Prof. Santamarina for the helpful guidance of experimental methods in research; Dr. Ruch for kindly assisting the designing and conducting of laboratory works; Dr. Vasyura-Bathke for his patient guidance of numerical methods and modeling techniques. I would also like to express my gratitude to my friends and colleagues at KAUST. People in the Crustal Deformation and InSAR (CDI) group, including Rishabh, Re- nier, Rémi, Daniele, Shaozhou, Xinying, Ayrat, and members from the CES group (Laura, Luigi, Olaf, Luca, Eva, Nicolas, Mohammad, Jayalakshmi, Subhadra, James, Claudia, and Maria), they have made my time at KAUST a great and memorable experience. My appreciation goes to my parents for encouraging me in all of my pursuits. Finally, my heartfelt gratitude is extended to my wife, Bella, for her warm patience and constant understanding during the past two years in Saudi Arabia. It is my pleasure to have her accompany and support. 7

TABLE OF CONTENTS

Examination Committee Page 2

Abstract 4

Acknowledgements 6

List of Figures 9

List of Tables 10

1 Introduction 11

2 Method: Analog Sandbox Experiments 21 2.1 LaboratoryApparatus ...... 21 2.2 DimensionalAnalysis...... 22 2.3 ImageProcessing ...... 26

3 Results: Analog Sandbox Experiments 30 3.1 EvolutionofSubsidence ...... 30 3.2 Comparison between Near and Far Field Subsidence ...... 37

4 Method: Boundary Element Numerical Modeling 41 4.1 BoundaryElementMethod...... 41 4.2 GeometryofRing-FaultsandSills ...... 47 4.3 BoundaryConditionsandAssumptions ...... 49

5 Results: Boundary Element Numerical Modeling 51 5.1 Focused Subsidence Depending on Ring-Fault Architectures ..... 51 5.2 WrappedSurfaceDisplacements ...... 54

6 Discussion 57 6.1 SummaryofResults ...... 57 8 6.2 Laboratory Limitations and Numerical Assumptions ...... 59 6.3 CausesforOverlappingDeformation ...... 62 6.4 Estimating Source Parameters: Sill Depths and Volume Changes. . . 64

7 Implications 69

8 Conclusions and Outlook 70 8.1 ConcludingRemarks ...... 70 8.2 Outlook ...... 71

References 72 9

LIST OF FIGURES

1.1 A Schematic illustration of caldera collapse processes ...... 13 1.2 Ring-fault structures related to caldera collapses ...... 14 1.3 Calderas in the Gal´apagos islands, Ecuador...... 15 1.4 A multiple-chamber model explaining the complex overlapping deformation observed using interferometric synthetic-aperture radar (InSAR) 16

2.1 Photosofanalogexperimentalsetup ...... 22 2.2 Schematic graphs of analog experimental apparatus ...... 23 2.3 Application of image processing on retrieving kinematic information in theexperiments...... 27

3.1 Surface structural evolution of experimental caldera collapse . . . . . 32 3.2 Thedownsagstageoftheexperiment ...... 34 3.3 Thecollapsestageoftheexperiment ...... 36 3.4 Analysis of diﬀerential surface displacements ...... 38 3.5 Topography changes and related ring-fault structures ...... 40

4.1 NumericalBEMsources ...... 42 4.2 NumericalBEMsourcemodelingsetup ...... 48

5.1 BEMsurfacesubsidencesimulations...... 52 5.2 BEM simulated surface displacements wrapped by fringes ...... 55

6.1 Comparing diﬀerent scenarios of BEM modeling of caldera subsidence 63 6.2 Estimation of deformation source parameters ...... 65 6.3 Schematic summary of models explaining complex overlapping surface deformation ...... 67 10

LIST OF TABLES

1.1 Comparison of volcanoes with ring-fault structures ...... 18 1.2 Explanations of complex deformation patterns ...... 18

2.1 Scalingfactorsusedintheexperiments...... 25

3.1 Analog experiment details and parameters...... 31

4.1 Deformation source parameters of the BEM numerical models..... 47 4.2 Elastic half-space parameters in the BEM numerical modeling. . . . . 50

6.1 Estimation of source parameters using diﬀerent BEM deformation sources 66 11

Chapter 1

Introduction

Volcano collapses are commonly associated with the underlying magmatic activities [Branney, 1995] [Cole et al., 2005] [Lipman, 1997]. Magma in the crust is stored and transported mostly in plumbing systems. A plumbing system is a network composed of reservoirs where magma accumulates, and of planar conduits (e.g., vertical dikes, horizontal sills) along which magma moves [Galland et al., 2015] [Kennedy et al., 2018]. Resurgence and accumulation of magma in the subsurface reservoir of the plumbing system is a precursor of eruptive events [Amelung et al., 2000]. It increases the pressure in the shallow crust, often resulting in intrusions of magma near the surface or leading to eruptions. Since the magma activity is a critical factor in assessing volcanic hazards, it is important to study where the majority of magma rests in the subsurface, how large the volume is, and how rapidly it moves. To resolve the problems, information retrieved from places of interest is required. Various techniques such as petrological analysis [Geist et al., 2014][Michon et al., 2009][Nau- mann et al., 2002], surface or subsurface geological surveys [Ekström, 1994] [Nettles and Ekström, 1998] and geological field work [Geshi, 2009] [Michon et al., 2009] can be applied to gain information. Among all the techniques, satellite-based remote sensing has been the primary method to get information from the most remote and inaccessible volcanic areas on the globe. Remote sensing techniques such as interferometric synthetic-aperture radar (InSAR) allow us to obtain surface displacement data at active volcanoes caused by changes of the underlying magma bodies [Biggs et al., 2014]. The acquired surface displacement data is then typically, compared with 12 analytical or numerical models of a possible deformation source. Such deformation sources are used to represent the subsurface reservoir(s), dike intrusions, or related phenomena. Parameters of the deformation source are then interpreted as properties of the underlying magma bodies in nature [Dzurisin, 2003], providing information to the understanding of the subsurface plumbing system and the potential hazard. Volcanic crustal deformation occurs during episodes of activity, e.g., before, during and after eruptions or during other periods of unrest. Inflation (Fig. 1.1a) occurs mostly in pre-eruptive stages as magma accumulates within reservoirs and the volu- metric increase pushes the overlying host-rock upward, causing uplift at the surface [Chadwick et al., 2006] [Jónsson et al., 2005]. In contrast, deflation (Fig. 1.1b∼c) occurs during co-eruptive and sometimes, during post-eruptive stages as reservoir volume decreases due to significant magma drainage during eruptions, which leads to subsidence at the surface [Peltier et al., 2009] [Walter and Motagh, 2014]. In some cases, such subsidence can later develop into a vertical collapse at the summit of the volcano (Fig. 1.1d) and result in dramatic topographic changes and caldera formation [Geshi et al., 2002] [Geyer et al., 2006]. Calderas are by far the most significant type of vertical collapse induced by the emptying of magma chambers [Branney, 1995] [Cole et al., 2005]. Caldera collapse deformation has been studied at numerous volcanoes such as at Miyakejima in Japan [Geshi et al., 2002] [Geshi, 2009], Piton de la Fournaise volcano on Réunion Island [Michon et al., 2009], and at Wolf volcano, Galápagos [Xu et al., 2016] (Fig. 1.3). The structural evolution of collapses has been widely studied with analogue models [Acocella, 2007] [Roche et al., 2000] [Ruch et al., 2012] [Walter and Troll, 2001]. The primary in-depth structure related to caldera collapses is a composite, double ring fault system (Fig. 1.2). This composite fault system consists of an outward-dipping reverse ring-fault connected by an inward-dipping normal ring-fault near the surface [Geshi et al., 2002][Michon et al., 2009]. These ring-fault systems are also commonly 13

a b

Uplift Subsidence

Conduit Host-rock Magma depleting Chamber

Inflation Deflation Pre-eruptive inflation Co-eruptive deflation c d

Caldera floor Subsidence Collapse

NF RF Partially emptied Sinking chamber Ring-faults roof Deflation Deflation Post-eruptive deflation Post-eruptive collapse

Figure 1.1: An illustration of caldera collapse processes from inflation, eruption, post- eruptive subsidence to collapse. (a) Inflation stage. The supply of magma increases the pressure in the reservoir, causing dike intrusions and crustal expansion in the host- rock above the reservoir (the roof). (Inflation sometimes also induces faulting [Cole et al., 2005]; then these structures will be reactivated during subsidence [Acocella et al., 2000].) (b) Deflation stage. The magma chamber is depleted through an eruption and the reducing magmatic pressure leads to downsag of the roof and slight surface subsidence. (c) Continuous magma depletion partially empties the reservoir, leading to larger subsidence on the surface. (d) Losing magma support may lead to sinking of the roof initiation of ring-faulting (RF: reverse fault; NF: normal fault), and eventual vertical collapse at the surface with a caldera.

observed in nature, e.g., inferred from active or ancient volcanoes (Table 1.1), and are usually related to the stress field change caused by inflation or deflation. [Branney, 1995] [Burchardt and Walter, 2010]. 14

a b Normal faulting Normal faulting

Reverse faulting Reverse faulting

Deflation Deflation

c Normal faults

Reverse faults

Sand

Silicon

Deflation Figure 1.2: Ring-fault structures related to caldera collapses. (a) Interpretative cross sections of Miyakejima, Japan (after [Ruch et al., 2012][Geshi, 2009]).(b) Interpreta- tive cross sections of the Dolomieu collapse (after [Michon et al., 2009]). (c) Evidence from analog models studying caldera structures showing obvious double ring-fault structures in the sand-stack (after [Roche et al., 2000]). All drawings are not to the same scale.

Considering the direct information provided by recent caldera collapses, in this research, we study the surface subsidence deformation at calderas and their related circumferential ring-faults in the subsurface. Such buried ring structures are initiated above partially depleted reservoirs, at which the roof loses support and downsags by gravity into the emptying reservoir (Fig. 1.1d). 15 a b Wolf

Ecuador Darwin

Fernandina

Alcedo Wolf

Figure 1.3: Calderas in the Galápagos islands, Ecuador. (a) A space shuttle photo from 1988 shows five major calderas at the Galápagos islands. The diameter of the calderas range up to 8 km (after [NASA, 1988]). (b) A photo from the north of Wolf caldera. The summit caldera is around 6 km × 7 km in size within a 17 km × 25 km volcanic edifice (after [Ramon, 2005]).

Monitored by InSAR images, several subsiding calderas such as Wolf volcano [Xu et al., 2016], Fernandina volcano [Bagnardi and Amelung, 2012] [Chadwick et al., 2011] on the Galápagos islands and Piton de la Fournaise on Réunion island [Peltier et al., 2009] [Holohan et al., 2017], have shown complex but similar surface deformation patterns. These patterns include an overlapping, complex deformation composed of a broad deflation signal affecting the entire volcano edifice, and of a localized subsidence signal focused within the caldera (Fig. 1.4a) of these volcanoes. Explanations were proposed for the 2015 eruption at Wolf volcano [Xu et al., 2016] about the superimposed surface deformation pattern. They showed that the overlapping displacements with both long-wavelength and short-wavelength signals are possibly, caused by the emptying of two active magmatic reservoirs at two different depths beneath the caldera (Fig. 1.4b). It was suggested that the confined subsidence inside the caldera was likely induced by the depletion of the shallower magma chamber, while the broader deformation around the volcano edifice was due to the depletion of the deeper chamber (Fig. 1.4c). In addition, [Xu et al., 2016] 16

a b Collapse 0º N Subsidence Subsidence

Galápagos 91º W Shallower reservoir

Pathway Magma depletion Deeper reservoir

Shallower source Deeper source Disp.

Combined Distance Vertical

Figure 1.4: A multiple-chamber model explaining the complex overlapping deformation observed using interferometric synthetic-aperture radar (InSAR). (a) Co-eruptive InSAR data of the 2015 Wolf volcano eruption on the Galápagos islands. The InSAR image spans the time period from late May to early July, with each fringe corresponding to 10 cm line-of-sight (LOS) displacement. Overlapping surface displacements can be observed: short-wavelength subsidence at the center of the caldera plus long- wavelength deflation around the whole volcanic edifice. (b) Schematic model showing the explanation of the overlapping deformation. The model is composed of a shallow and a deep source, generating both broad, smooth deflation and narrow subsidence focused at the center of the volcano (after [Xu et al., 2016]).

suggested that the two magma chambers might be hydraulically connected to each other through a vertical plumbing system, since contemporaneous pressure changes in the two chambers during eruptions were observed. Other examples at Fernandina and Cerro Azul volcanoes reveal similar deformation patterns. In the May 2005 eruption at Fernandina volcano, global positioning system (GPS) and InSAR observations showed overlapping surface deformation signals and were interpreted as two hydraulically connected magma reservoirs [Chadwick et al., 2011]. Another InSAR investigation of the same region [Bagnardi and Amelung, 2012] covering the surface displacements from January 2003 to September 2010, re- ported that the magma was stored in at least two crustal storages. Their modeling 17 results with analytical elastic deformation sources, suggest that there may be a shallow reservoir at ∼ 1 km and a deep reservoir at ∼ 5 km below sea level. Similarly, at Cerro Azul volcano, the wide mineralogical composition range in petrological data [Naumann et al., 2002] was also explained as the variable degrees of fractionation and crystallization of magma at different depths, indicating that the magma was accumulated in multiple chambers before eruptions. Although previous papers on several subsiding calderas have proposed the multi- chamber source model as the cause of overlapping surface deformation, it is still uncertain how other deep processes at those subsiding calderas may relate to the observed deformation pattern (Table 1.2). In addition to the activity of the magmatic chamber, fracture systems in the host-rock of the volcanic area could also affect the local deformation pattern [Acocella, 2007] [Geshi, 2009]. As mentioned above, ring-fault systems are commonly observed or inferred from active or ancient volcanoes. The tangled interplay between the depressurization in conjectured reservoirs at different locations and possible ring-fault activity brings more difficulties in ascertaining the source of the surface deformation. For simplicity, models based on continuum mechanics that assumes elastic deformation, are commonly used for inverting source parameters without taking possible ring-fault activity into account [Riel et al., 2015] [Xu et al., 2016]. Nevertheless, the important role of ring-faults at subsiding calderas has been em- phasized in some studies (Table 1.2). The sharply bounded deformation confined within the calderas of Tendürek volcano [Bathke et al., 2015] and Bárdabunga volcano [Gudmundsson et al., 2016], was explained as the result of the buried ring-fault activities. In Galápagos, as the main co-eruptive subsidence during the 2015 Wolf eruption limited on the caldera floor, they hypothesized that the overlapping surface deformation during the eruption could alternatively be interpreted as a single depleting magma chamber combined with the activity of a buried ring-fault within the 18 ] ]; ] ] ]; ]; ]; ] ] ]; ] ]; ] Saunders, 2001 ]; [ Konstantinou et al., 2003 ńsn 2009 Jónsson, ]; [ [ Geshi et al., 2002 Nairn et al., 1995 Bathke et al., 2013 Michon et al., 2007 Yılmaz et al., 1998 [ Scarpati et al., 1993 Chadwick et al., 2006 De Natale et al., 2006 [ Simkin and Howard, 1970 ihnradTaˇic 2010 Fichtner and Tkalˇcić, [ ]) Jones and Stewart, 1997 kt¨m 1994 Ekström, [ [ Bathke et al., 2013 2,100 [ ] − .6 [ ] 300 [ 330 [ ∼ ∼ ] plitude of ] ]; ] ]; ]; 1,600 ] res (after [ ]; eDisplacement(m) References ] ∼ ńsn 2009 Jónsson, [ Xu et al., 2016 Riel et al., 2015 Bathke et al., 2015 Peltier et al., 2009 Bathke et al., 2013 Holohan et al., 2017 ńsnetJónsson al., 2005 Chadwick et al., 2011 Gudmundsson et al., 2016 nclined uptoseveralmeters [ Bagnardi and Amelung, 2012 inclined normal faulting normal and reverse inclined, reverse and faulting, trapdoor like faulting, trapdoor like inclined, normal faulting ring-faulting? plus ring-faulting plusring-faulting [ plus host-rock fracturing Table 1.1: Comparison of volcanoes with ring-fault structu Observationsof Explanations Table 1.2: Explanations of complex deformation patterns DirectionofSurface MethodofRing- FaultOrientation(s) Am Wolf InSAR two-sourcedepletion [ Bádarbunga subsidence seismic partlyoutwardandinward ? (Galápagos) New Guinea) Miyakejima(Réunion) subsidenceFernandina seismic, subsidence seismic,(Turkey) inwardinclined photographic partlyoutwardandinward normalfaulting Volcano Movement FaultsDetection andSlipSense(s) Surfac Fernandina(Iceland) InSAR,GPS two-sourcedepletion [ InSAR onesourcedepletion [ Tendürek subsidence geodetic eventuallyinward 0.08 [ Rabaul(PapuaCampi Flegrei(Italy) uplift uplift/subsidence seismic, geodetic seismic,geodetic inward i (Galápagos) inwardandoutward(Iceland) gravimetric 1 photographic inclined, (Galápagos) Bádarbunga(Galápagos) PitondelaFournaise InSAR(Réunion) GPS multi-sourcedepletion multi-sourcedepletion [ [ onesourcedepletion [ SierraNegra uplift geodetic outwardinclined,reverse 1-3SierraNegra [ InSAR trapdoorfaulting [ Volcano GroundMovement (Modeling) References (Italy)Dolomieu subsidence seismic,geodetic inwardinclined photographic normalfaulting (Galápagos)Tendürek(Turkey) InSAR InSAR two-sourcedepletion [ onesourcedepletion [ 19 host-rock [Xu et al., 2016]. The effect of ring-like fracturing on surface displacements has also been discussed for Piton de la Fournaise volcano, e.g., in association with the March-April eruptions of 2007. The temporal geodetic data of the 2007 eruption was divided into three sub- phases: pre-eruptive uplift, co-eruptive, and post-eruptive subsidence. The inverted optimum magma chamber for each sub-phase varied highly in both depth and geometry. Overall, the location of the optimized chamber moved toward the surface from the pre-eruptive to the post-eruptive phase [Peltier et al., 2009]. The possible explanation to this enigmatic phenomenon has been proposed in another study [Holohan et al., 2017] in which a forward modeling based on the discontinuous element method was applied to simulate host-rock fracturing processes in the subsurface. As they invert for simple analytical sources using the synthetic displacements produced by the discontinuous element method, the similar upward migrating trend of the deformation sources was discovered. The results have been drawn into a conclusion that the presence of the host-rock fracturing can cause a superimposed deformation pattern and can progressively change estimated shape and location of simple analytical deformation sources during magmatic drainage [Holohan et al., 2017]. At deforming volcanoes, this effect may lead to erroneous inferences of magma source depths and incorrect interpretations such as multi-chamber or chamber migration models. In general, there are several other natural or anthropogenic sinking processes related to ring-structure activities that are comparable to caldera collapse. Such phenomena include sinkhole or pit crater formations [Poppe et al., 2015] [Nof et al., 2013] and underground nuclear-test sinks [Howard, 2010] [Vincent et al., 2003]. A simulation of a mine collapse based on InSAR data led to similar conclusions that a component of faulting is required to explain the observations [Plattner et al., 2010]. Nevertheless, we still lack knowledge about the connection between the overlapping surface deformation observed by InSAR and the integrating effect of subsurface defor- 20 mation sources, including depleting reservoir(s) and particularly, the buried ring-fault system. In this research, we will analyze the role of ring-fault activity in controlling localized deformation at subsiding calderas. The analysis includes both scaled laboratory analog experiments and boundary element method (BEM) numerical modeling. In these two complementary approaches, we will test if the scenario combining a single depleting magma chamber with ring-fault activity can reproduce a surface deformation pattern comprising of a longer wavelength, volcano-wide deflation and a shorter wavelength localized subsidence confined to the caldera floor. We then further discuss differences of source parameters (e.g., source depths and volume changes) between our proposed model and the scenario that includes multiple depleting chambers. Estimat- ing parameters of deformation sources is important, as they help us to better constrain the architecture of the plumbing system, the related geologic structures, and they can also be linked to volcanic hazard assessments [Kennedy et al., 2018]. Therefore, a better insight of how deep processes beneath subsiding calderas relate to observed surface deformation is essential for improving both the knowledge of the underlying volcanic structures and evaluations of potential hazards at active volcanoes. 21

Chapter 2

Method: Analog Sandbox Experiments

2.1 Laboratory Apparatus

Our experimental setup (Fig. 2.1) was designed as an analogy of nature and was constructed according to various caldera subsidence analog models [Acocella et al., 2000] [Ruch et al., 2012]. The key elements of the setup include a glass sandbox and a movable piston controlled by a motor. The transparent sandbox has a dimension of 30 × 20 × 10 cm3 and it is loaded with horizontal layers of sand. The granular material used to simulate the deformation of the shallow earth crust was well-sorted natural quartz with high purity, composed of 96% crushed silica sand. The quartz grains were dry and had the diameter ranging from 40 to 200 µm [Holohan et al., 2008]. A semi-circular opening with a 10 cm diameter at the bottom of the glass box was connected to a piston which was driven by a motor below, and could ascend and descend vertically with adjustable moving velocity (Fig. 2.2). With this moving piston, we simulated the movements of a magma chamber ceiling during a source depressurization process at depth. To better imitate the behavior of a viscous magma body, a viscous silicone putty was placed between the sand-stack and the piston (Fig. 2.2c, orange silicone putty) [Acocella et al., 2000]. The experimental setup allowed us to model a roof collapse above a magma chamber induced by reservoir depressurization, which simultaneously produces surface subsidence and generates ring-fault structures in the subsurface. The entire subsiding processes were observed by several cameras from diﬀerent angles at the same time. 22

Figure 2.1: Analog experimental setup. (a) The key elements of the setup include a sandbox, a movable piston and a motor controlling the piston. (b) The section view of caldera collapse in the our laboratory. (c) Several cameras capture photographs simultaneously during subsiding processes.

2.2 Dimensional Analysis

To ensure experiments are comparable to nature, parameters of analog apparatus should obey the principle of physical similarity and should be scaled to nature [Hub- bert, 1937] [Ramberg, 1981]. Therefore, geometrical scaling (length and shape in the 23

a Perspective view b Map view Top cameras Steel base

Glass box Semi-circular opening Zone of interest r = 5 cm

D c Cross-section view Sand Sand T D Steel base Steel base Silicone putty Silicone putty Side camera Piston Piston Motor

Figure 2.2: The configuration of the analog apparatus: (a) Perspective view of the sand-filled glass box. We monitored the deformation with five cameras from the side and the top. The grey areas are showing the zone of interest, where the main deformation occurred. (b) Map view of the basal steel plate of the setup. Note the deformation source was a semi-circular opening, allowing to see the processes in the cross-section view. (c) Cross-section view of the setup. The silicone putty (shown in orange) was placed between the sand-stack and the piston, imitating the behavior of viscous magma.

model), dynamic scaling (forces that are applied on the model) and kinematic scaling (velocity and strain rate of processes) [Ruch et al., 2012] were taken into account in our research. To achieve physical similarities between the model and nature, the following physical quantities were our primary concerns: geometric linear dimension of the setup l (m), density of the material ρ (kg/m3), stress within the setup σ (kg/ms2) and gravity g (m/s2). According to Buckingham’s Π theorem [Buckingham, 1914], dimensionless ratios that are composed of concerned physical quantities must be similar between the laboratory and nature to properly scale the experiments. If p is the number of fundamental basis dimensions required in a system for measuring the n 24 kinds of physical quantity, the number of dimensionless ratios, i, is

i = n − p. (2.1)

In our case, the physical quantities of our interest include linear dimension of the apparatus (m), density of the granular material (kg/m3), gravity (m/s2), and stress (kg/m · s2) in the system. The number of kinds of physical quantity n = 4, and the number of fundamental units p = 3, therefore, imposing the number of dimensionless ratios to be taken into account i = 1. Through further operation of dimensional matrices, this particular dimensionless ratio derived from our concerned physical quantities is

ρgl Π= , (2.2) σ where ρ is the rock density, g is the gravity, l is the linear dimension, and σ is the stress in the system. For correct scaling, this ratio Π must be similar in the model and nature. The physical meaning behind this derived ratio is the balance between the stress acting on/in the system and the weight per unit area of the system. For proper scaling, we deﬁned that a scaling ratio for any given physical quantity

∗ can be represented as X = Xmodel/Xnature (scaling results are listed in Table. 2.1). Given that Π is roughly constant from nature to the laboratory, the stress ratio between the model and nature σ∗ can be represented as the multiplication of the density ratio ρ∗, the gravity ratio g∗, and the length ratio l∗ through standard operation:

σ∗ ≈ ρ∗g∗l∗. (2.3)

Calderas in nature vary widely in size, from one to more than 50 km [Mouginis- Mark et al., 1996][Walker, 1984]. To model a 10 km wide caldera in a 10 cm setup in

∗ the experiment, the length ratio between the model and nature, l = lmodel/lnature ≈ 25 Table 2.1: Scaling factors used in the experiments. Parameter Symbol Model/Nature ratio Brittle crust Length l∗ 10−5 Density ρ∗ 0.5 Gravity g∗ 1.0 Stress σ∗ 5 × 10−6 Cohesion c∗ 5 × 10−6 Viscous magma Viscosity η∗ 10−6 Time T ∗ 0.2 Deformation rate V ∗ 2 × 10−6

10−5. The shallow crustal material has a natural density around 2600 − 2800 kg/m3 and the quartz sand in our setup is 1.38 g/cm3, resulting in a density ratio ρ∗ of 0.5. The gravity ratio g∗ is 1.0 since our experiments are run under the same gravitational condition as in nature. Because the cohesion has the same dimension as stress, the resulting stress ratio σ∗ and cohesion ratio c∗ by equation 2.3 is then 5 × 10−6. We assumed a linear Mohr-Coulomb failure criterion with an internal friction angle φ = 40◦ and a natural cohesion c = 107 P a [Norini and Acocella, 2011][Ruch et al., 2012] for the rock in the brittle crust near the volcanic area. Considering our cohesion ratio above, the cohesion of granular material in the experiments should be around 50 P a, which agrees to our choice of sand with a cohesion of 12 − 123 P a [Galland et al., 2015]. While the above scalings for the brittle behavior of the shallow crustal material are time-independent, we also need to consider the time-dependent ductile behavior when scaling the viscous magma body. The viscosity η (P a·s) can be derived by multiplying stress σ (P a) and time T (s). Therefore, the time ratio between the model and nature is the viscosity ratio over the stress ratio, representing as 26

T ∗ = η∗/σ∗. (2.4)

The natural viscosity of high viscous magma is about 1010 (P a · s)[Galland et al., 2015] [Holohan et al., 2008] [Plattner et al., 2013] while the silicone putty we used to simulate the magma body has a viscosity of about 104 [Galland et al., 2015], imposing the viscosity ratio between model and nature η∗ ≈ 10−6. Combining with the stress ratio we inferred before, we get a time ratio T ∗ ≈ 0.2. The duration of our analog experiments needs to be about 10 minutes to create a collapse with mature structures and prominent deformation features to be observed. Therefore, by using this time scaling ratio, the time duration in our experiments corresponds to few hours in reality. The ratio of deformation rate (V ∗) of the subsidence is the length ratio (l∗) divided by the time ratio (T ∗), which is 2×10−6. The motor speed that induces both subsidence or uplift in the experiments is 0.04 (mm/s), corresponding to a constant subsidence or uplift rate in reality of 20(m/s).

2.3 Image Processing

To retrieve quantiﬁed displacement information from the laboratory experiments, image processing tools were employed for kinematics analysis after capturing optical images from various perspectives [Galland et al., 2016]. Here, we present two image processing methods for dealing with the observations from the top view and from the section view correspondingly (Fig. 2.3). For the top view observations, we applied the structure-from-motion (SfM) technique (Fig. 2.3b). We used a stand-alone spatial photogrammetry software package, Agisoft Photoscan® to construct a three-dimensional (3-D) topographic surface model from multiple two-dimensional (2-D) photographs of the experiments. By combining temporally synchronized photographs focused on the same region from diﬀerent 27 a b

PIV technique: SfM technique: Cross correlation between Extraction of 3-D topography the successive photos from multiple, overlapping photos

Particle movements Time-series DEM DEM

3 c d 0 -2 cm 5 cm

Rupturing processes Topographic changes

Figure 2.3: Reconstructing quantiﬁed information from analog experiments. Op- tical images were captured through time and later used for image processing. (a) Correlating successive photographs captured from section-view gives us in-depth particle movements. (b) Photographs captured from four synchronized top-view cameras were used to build a DEM. (c) A reconstructed set of data illustrating both the active structures and surface topography at the instant t = 520′ (seconds) during one of the experiment. (d) Time-series DEMs enables studying topographic changes.

perspectives and camera positions, the software applies photogrammetric computa- tion of digital images to estimate the 3-D coordinates of points of the object. It automatically reconstructed the detailed point clouds and polygonal meshes of our analog model. Printed reference scales were placed at the surface of the sand-stack and away from the deformed region to serve as fixed reference points (Fig. 2.3b, the L-shape scale bars in the photos). Therefore, we can both scale the model and 28 reference the images by defining the ground control points (GPCs) at the specific points on the scales. Once the GCPs are defined and their relative locations are given, a high horizontal resolution (0.18 mm/pixel) digital elevation model (DEM) can then be calculated from the referenced point clouds of each set of synchronized images. Using four images from different angles with a small baseline (Fig. 2.2a), the maximum theoretical error of the DEMs in the vertical direction can be minimized to ∼ 0.4 mm. We then extracted multi-temporal DEMs (∆t =1s) of the sand-stack that monitored the overall deformation processes on the surface. Through this approach, we had both the spatial and temporal information of the surface topography from our experiments. Therefore, time-series of surface deformation maps were produced from the experiments (Fig. 2.3d). In-depth processes (the section view) including ring-fault rupturing and subsidence deformation was investigated with another method, the particle image velocimetry (PIV) technique (Fig. 2.3a). PIV is a practical and effective application of two- dimensional digital image correlation. It is commonly used in the field of experimental mechanics in order to retrieve the in-plane displacement measurement on planar objects [Pan et al., 2009][Raffel et al., 2013]. Utilizing the open-source graphical user interface software, PIVlab [Thielicke and Stamhuis, 2014] which is based on Matlab®, we can easily apply digital image correlation. The software stacks the selected image pair (usually two successive images, Fig. 2.3a) and calculates offsets of mismatching pixels by cross-correlation. Printed reference scales were put on the section-view glass pane. Therefore, we can apply calibration by selecting reference distance on the scale which is in the image and specifying a real distance to it. With the referenced scaling between the image and the real object, offsets of mismatching pixels can be firstly converted to the particle displacement field and then to the velocity field once the time interval of two images is given. The resolution of the observation is 0.08 mm/pixel. This time-resolved PIV tool helped us to extract a series of (1) velocity fields and 29 (2) vortex fields (rotational strain, which is valuable information for identifying shear structures in depth) [Burchardt and Walter, 2010] [Ruch et al., 2012] of the granular particles in the experiment through time from the acquired photographs (Fig. 2.3c). Therefore, we can visualize the sudden motion and the cumulative motion of the particles. With these two image processing methods, we can track the processes acting both at depth and at the surface simultaneously. We can then have a better insight of the relationship between the activities of the buried structures and the surface deformation as the subsidence evolves. 30

Chapter 3

Results: Analog Sandbox Experiments

In total, 14 sets of the experiment with different parameters were conducted, and we list the details of 10 sets (the earliest four sets were initial trial tests) in Table 3.1. Here, we present results from one of the representative caldera collapse tests (CAL 13) with the sand thickness of T = 10 cm. As described in the previous chapter, the diameter of the sinking source was D = 10 cm in all the tests. To simplify the operation of the experiments, we did not include dome topography representing a volcano edifice in our analog models of the caldera collapse. Instead, we simulated direct magma withdrawal under a flat topography by withdrawing the piston with a constant speed of 0.04 mm/s. The sand-stack in depth was marked with colored layers every one centimeter with the same granular material.

3.1 Evolution of Subsidence

We show the selected map-view images from different timings during the experiment (Fig. 3.1). The subsiding processes can be divided into an earlier downsag stage and a later collapse stage by the occurrence of surface rupturing of the ring-fault system. Our experiments showed that the structural characteristics of the surface motion are distinctive between the two stages. The downsag stage (Fig. 3.1a∼c), which we defined as the period before the ring-fault system reached the surface, began with the central depression and the forming of compressive ridges (P) around the central depressing region (Fig. 3.1a). Then, the reverse ring-faulting (RR) appeared 31 Table 3.1: Analog experiment details and parameters. Cohesion Friction angle T/D Cameras Motor rate Featureda Experiment (Pa) (deg) - (side + top) (mm/s) settings CAL 5 < 30 ∼ 25 0.5 1+1 0.02 CAL 6 12 ∼ 123 ∼ 40 1 1+1 0.02 RF; RA CAL 7 12 ∼ 123 ∼ 40 1 1+4 0.02 RF; RA CAL 8 12 ∼ 123 ∼ 40 1 1+4 0.04 RF; RA CAL 9 12 ∼ 123 ∼ 40 0.6 ∼ 1 1+4 0.04 TP CAL 10 12 ∼ 123 ∼ 40 1 1+4 0.04 RF; CY CAL 11 12 ∼ 123 ∼ 40 1 1+4 0.04 CAL 12 12 ∼ 123 ∼ 40 1 1+4 0.04 CY CAL 13 12 ∼ 123 ∼ 40 1 1+4 0.04 CY CAL 14 12 ∼ 123 ∼ 40 1 1+4 0.04 DM a Refilling of sand: RF; Reactivation of sinking: RA; Topographic effect included: TP; Cycles of uplift and sinking reactivation: CY; Doming caused by inflation before deflation phase: DM. at the surface, creating a larger reverse fault scarp at the location of the previous compressive ridges (Fig. 3.1b). At the end of the downsag stage, the peripheral depression occurred on the outskirts of the central depression, and it also produced some extensional fractures (T) at the locations with maximum tensile stress (Fig. 3.1c). Entering the collapse stage (Fig. 3.1d∼f), these extensional fractures developed into normal faults (NR). The asymmetry of the system let the normal ring-fault develop firstly on the left side, while the right side was still under peripheral depression (Fig. 3.1d). As the subsidence increased, individual extensional fractures well connected to one another, developed into the circumferential normal ring-fault enclosing the inner reverse ring-fault (Fig. 3.1e). At this step, the normal ring-fault was far more active than the reverse ones, creating major displacements at the surface. As the ground continued sinking downward through circumferential normal faulting, the increasing amount of subsidence eventually resulted in slope instability at the surface, producing collapse and landslides that covered some of the original structures (Fig. 3.1f). Note that this effect has also been observed in nature, e.g., where marginal landslides with 32 Downsag stage Collapse stage a Central depression d Normal ring-faulting + peripheral downsag T NR P P RR T 80’ 140’ b Reverse ring-faulting e Normal ring-faulting

T RR RR 100’ 200’ c Peripheral depression f Collapse

T Landslides RR 110’ 250’ 5 cm P : Compressive ridges RR: Reverse ring-fault T : Extensional fractures NR: Normal ring-fault

Figure 3.1: Surface structural evolution from the downsag (a∼c) to the collapse stage (d∼f) of a caldera in the laboratory. This ﬁgure shows photographs from the top view with explanations of the distinctive structural features at diﬀerent steps of the caldera subsidence.

continuing subsidence increased the diameter of the Miyakejima caldera from 800 m to 1.6 km and buried the collapsed structures [Geshi et al., 2002]. Image processed results show both DEM information and PIV particle motions. Map-view images (Fig. 3.2a∼c, Fig. 3.3a∼c) show the surface displacements retrieved from time-series DEMs. For the convenience of comparing to ground deformation observed by satellite interferograms, the surface displacements in our experiments 33 are wrapped with fringes, each of which corresponds to 0.4 mm of displacements in the line-of-sight (LOS) direction. We prescribed the LOS vector as an ascending track satellite with a heading of N345oE and a look angle of 36o. Section-view images (Fig. 3.2d∼f, Fig. 3.3d∼f) show optical photographs overlaid with the rotational strain of the sand-stack calculated using the PIV technique. Highly torqued pixels are marked in blue and red, indicating the ring-fault movement where the sand was sheared. We observe at t = 40′ (seconds; Fig. 3.2d) that the central part of the sand-stack was warped downward, forming gentle flexures in the layers. Since the strain in the sand was subtle, the existence of the rupture could barely be observed by the naked eye. With the assistance of the rotational field calculated by the PIV technique, we can easily map out the fault location at depth. The circular ring-faults had not well- developed at this time, but we can see some diffusive form of the initial structures. The map-view image at t = 40′ (Fig. 3.2a) shows fringes with even spacing, appearing at both inside and outside of the source border (the projected piston source boundary). For simplicity of description, we define the near field as indicating the area inside the piston border and the far field as the area outside of it (Fig. 3.2g). At t = 60′, the increasing subsidence further developed the flexure, evolving into a depression in the middle of the sand-stack enclosed by the outward dipping reverse ring-faults (Fig. 3.2e). The ring-faults slightly extended toward the surface and had larger shear movements. They also developed an asymmetric configuration depending on the uncontrolled heterogeneity in the sand. The map-view image at t = 60′ (Fig. 3.2b) shows the surface subsidence increased faster in the near field and an off-centric subsiding center (offset center to the left) can also be observed at the surface as a result of the asymmetry of the ring-faults at depth. At t = 90′, the section-image (Fig. 3.2f) shows both sides of the outward dipping ring-faults nearly reaching the surface and displacements across the subsurface ruptures were enlarged, causing the prominent depression in between. Another reverse fault with a higher dip angle on the left side 34

a 40’ b 60’ c 90’ 5 cm LW

SW Distance, Y Distance, Y [cm] d e f 0 Source boundary Depth, Z [cm]

10 5 cm Distance, X [cm] - + FF FF Rotation 0.4mm LOS displacement NF g 3 2 3 Source boundary 345o

RR Major active structure Y RR: Reverse ring-fault heading 1 2 NF: Near field 36o FF: Far field Z look angle Piston Line of sight (LOS) X

Figure 3.2: The downsag stage of the experiment showing processes before surface rupturing at t = 95′ (seconds). (a)∼(c) Wrapped surface displacements (0.4 mm per fringe) in line-of-sight (LOS) direction (cumulative displacements since t = 0′). Results came from top-view images. (d)∼(f) Section-view photos overlaid with vortex (rotation) field images, revealing the location of ring-faults in depth (cumulative motion since t = 0′). The ring-fault started to form at t = 40′, and it propagated upward as the subsidence increased. As the outward dipping reverse ring-fault developed closer to the surface at t = 60′, localized surface subsidence started to form in the near field, and the amount increased through time. At t = 90′, we observed the short-wavelength displacements (SW) were well enclosed by the source boundary, separating itself from the long-wavelength displacements (LW) in the far field. (g) Overlay of the structures showing the reverse ring-fault tended to become more vertical from step 1 to 3. A branch of higher angle reverse ring-fault later developed on the initial one. All reverse faults located within the near field. 35 was branched from the pre-existing reverse fault from its bottom end and propagated toward the surface. At the surface, at t = 90′ (Fig. 3.2c), the subsidence continued to progress more in the near field than in the far field. Comparing between the pattern of the near-field fringes and the far-field fringes, we observed that the subsiding rate is very distinctive between them. The short-wavelength subsidence was confined in the near field while a long-wavelength sagging affected the far field. The location of the sudden change from short-wavelength to long-wavelength displacement corresponded to the projected boundary of the deformation source underneath. After the ring-faults penetrated through the surface, the central subsidence developed into a circular caldera collapse bordered by the normal ring-fault trace (Fig. 3.1d∼f). At t = 110′ (Fig. 3.3d), the reverse faults on the left side developed branches with even higher angles. The numerous branches of subvertical fault grew in succes- sion, enabling secondary depressions around the central and major depression near the surface. In the map view (Fig. 3.3a), we observed the sharp boundary separated the short-wavelength displacements from the long-wavelength ones. This boundary corresponded well to the projected source border. At t = 130′ (Fig. 3.3e), an inward dipping normal fault had formed as a result of gravitational instability, and it controlled the major displacements happening in the sand-stack. At the surface (Fig. 3.3b) the localized displacements seemed to extend into the far field, asymmetrically in the right flank of the region. At t = 160′ (Fig. 3.3f), branching of the ring-fault development produced not only the internal depression enclosed by the outward dipping reverse faults (RFs) but also induced a secondary external depression enclosed by the newly formed inward dipping normal faults (NFs) (also Fig. 3.3g). The normal faults structure had just appeared since approximately t = 110′ − 130′ on the left side (Fig. 3.3d∼e), later on, they became dominant and controlled the major displacements in the subsurface. Notice that due to asymmetry, the right side of the ring-faults had not developed an inward dipping normal fault. In the map view (Fig. 36

a 110’ b 130’ c 160’ 5 cm LW ESW

SW Distance, Y Distance, Y [cm] d e 0 f Source boundary Depth, Z [cm]

10 5 cm Distance, X [cm] - + FF FF Rotation 0.4mm LOS displacement NF g 2 2 1 1 Source boundary 345o RR Major active structure Y NR RR: Reverse ring-fault heading NR: Normal ring-fault 36o NF: Near field Z FF: Far field look angle Piston Line of sight (LOS) X

Figure 3.3: The collapse stage of the experiment showing processes after surface rupturing at t = 95′ (seconds). (a)∼(c) Wrapped surface displacements (0.4 mm per fringe) in line-of-sight (LOS) direction (cumulative displacements since t = 0′). Re- sults came from top-view images. (d)∼(f) Section-view photos overlaid with vortex (rotation) field images, revealing the location of ring-faults in depth (cumulative motion since t = 0′). The ring-fault reached the surface, and the near-field subsidence intensified during t = 110′ −160′. The short-wavelength subsidence (SW) was well enclosed by the source boundary and was distinct from the pattern of long-wavelength sagging (LW). Branches of higher angle faults later developed on the initial ones, followed by the forming of inward dipping normal faults at t = 160′. The outward growing normal fault system induced the external depression and also created the extended short-wavelength displacements (ESW) beyond the boundary to the far field. (g) Overlay of the structures showing the normal faults developed after the reverse fault from step 1 to 2. Reverse faults located near the central part of the near field while normal faults extended to the far field. 37 3.3c), the region of short-wavelength displacements extended and covered part of the far field. The extending localized displacements are due to the outreaching of the normal ring-faults (Fig. 3.3g). More explanation will be provided in the next section.

3.2 Comparison between Near and Far Field Subsidence

The wrapped surface displacements allowed us to better investigate the distinctive pattern of deformation from the near field to the far field. Figure 3.4 shows the number of fringes through time. At t = 40′, the near-field and far-field subsidence were similar, with both corresponding to one fringe of displacement. During t = 60′ − 90′, the far- field subsidence increased with a constant and slow rate, and was exceeded by the rapid increase of the near-field subsidence. At t = 60′, the displacement difference between the near field and the far field was 0.4 mm, and it increased to 2 mm at t = 90′. By the end of the downsag stage, we observed a rapid increase of near- field fringes while the number of far-field fringes rose at a stable rate. During the collapse stage, at t = 110′, the displacement difference increased to 3.2 mm and then it remained stable until t = 160′. During all the experiments, we observed the prominent rising of differential subsidence at the end of the downsag stage. In contrast, in the collapse stage, we found that the differential subsidence remained stable sometime after the surface rupturing of the ring-faults. This transition of differential subsidence, which evolved from an increasing trend at the later downsag stage to a stable trend at the collapse stage, was mainly determined by the structures of surface rupturing (Fig. 3.1). The ring-fault system initially appeared at the outer rim of the near field during the later downsag stage (Fig. 3.5a and Fig. 3.2g), and the reverse faulting increased the subsiding rate in the near field, raising the differential displacement. During the collapse stage, the active normal ring-faults reached out to the far field (Fig. 3.5b and Fig. 3.3g), allowing both the near and far fields to subside with a similar rate and thus, the differential 38

16 Near-field (within source diameter) 13 Far-field (out of source diameter) 11 3.2 mm * One fringe = 0.4 mm

(LOS displacement) 3.2 mm 8 8 3.2 mm

2 mm 5 Number of fringes

3 0.4 mm 3 3

0 mm 2 1 1 4

Downsag stage Collapse stage 2

Differential Surface rupturing

displacements [mm] 0 40 60 80 100 120 140 160 Time [seconds]

Figure 3.4: Analysis of differential surface displacements. Histograms show number of fringes in the near field and in the far field at time t = 40′, 60′, 90′, 110′, 130′, 160′ (seconds). Each fringe corresponds to 0.4 mm of displacement along line-of-sight (LOS) direction. Numbers labeled by the histograms are differential displacements between the near field and the far field. The blue-curve plot shows how differential displacement changed through time. Notice the rapid rise in the downsag stage just before the surface rupture and a plateau in the later collapse stage.

subsidence remained stable. The differential subsidence can also be observed on topographic profiles retrieved from DEM time series. By comparing the elevation changes through time during the experiment of top-view photos, we have more information about the relationship between the ring-faults and the induced surface displacements (Fig. 3.5). As the amount of subsidence increased since t = 40′, the localized displacements rapidly enlarged within the near field (Fig. 3.5c,d, the grey shaded region), producing short- 39 wavelength displacements. During t = 0′ ∼ 90′, the subsidence in the near field was mainly caused by the central depression as a result of the outward dipping reverse faulting (Fig. 3.5a, Fig. 3.2g). This localized subsiding area was widened later on as the normal ring-faults developed (Fig. 3.5b, Fig. 3.3g) and led to the gravitational instability and collapse of the edifice at t = 160′ (also Fig. 3.1f). As a result, as the area of collapse enlarged, and the normal ring-faults migrated outward. Short- wavelength displacements can also extend to part of the far field (Fig. 3.5b, Fig. 3.3c). We observe broad and smooth subsidence in the downsag stage at t = 40′ − 90′ which affected both the near and the far field (Fig. 3.5c,d). This broad sagging was a long-wavelength deformation and can be better identified on the north-south direction profiles because the west-east direction profiles were closer to the boundary of the sandbox and the displacements were reduced due to friction on the glass pane. Although this far-field long-wavelength sagging had been evident in the downsag stage, it was influenced by the extending collapse due to the normal ring-faulting at collapse stage around t = 130′ − 160′. In summary, long-wavelength displacements affecting the entire region appeared since the beginning the downsag stage of the subsidence (from 40’∼90’, Fig. 3.5c and d), corresponding to the steady and constant increase of the fringe number in the entire area (Fig. 3.2a∼c Fig. 3.4). On the other hand, short-wavelength displacements were initially observed in the near field at the end of the downsag stage (after 90’, Fig. 3.5c and d). After that, the short-wavelength pattern dramatically changed the topography in the near field. At the end of the collapse stage, this short-wavelength pattern then rapidly intensified along with the increased activity of the normal ring- fault, outreaching to the far field and affecting a larger area (at 160’, Fig. 3.5b, c, and d). 40

a t = 90’ b t = 160’ Top-view photo NR 5 Source boundary RR RR

A A’ B B’ 0 Distance, Y Distance, Y [cm] DEM profile 0 Source boundary A A’ B B’

-6

Depth, Z [mm] -12

-5 0 5 -5 0 5 Distance, X [cm]

c d WE NS 0 t = 40’ t = 40’ NR 60’ 60’ NR NR 90’ 90’ -6 110’ 110’ 130’ 130’ RR

Depth, Z [mm] RR 160’ RR 160’ -12 -5 0 5 15 10 5 0 Distance, X [cm] Distance, Y [cm] RR: Reverse ring-fault e N NR: Normal ring-fault Short-wavelength displacements Long-wavelength displacements W S E

Figure 3.5: Topography changes and related ring-fault structures. (a) The downsag stage of top-view photos and elevation profiles. Subsurface faults are drawn according to the sectional observation. At t = 90′, the beginning of surface rupturing created reverse fault scarps. (b) The collapse stage. At t = 160′, a large amount of subsidence was mainly induced and enclosed by the normal ring-faults. (c) W-E profiles of time- series elevation profiles show the localized subsidence was confined within the near field, bordered by the ring-fault system. In the far field, a smoother displacement happened since t = 40′ and remained observable until t = 90′ when the normal faulting took over. (d) N-S profiles. (e) A schematic graph showing short-wavelength displacements are mainly in the near field and long-wavelength in the far field. 41

Chapter 4

Method: Boundary Element Numerical Modeling

In the previous chapter, we showed the observations from our analog experiments quantified by structure from motion photogrammetry (SfM) and particle image velocimetry (PIV). The experimental method helps us to relate the surface deformation and structural evolution to the in-depth faulting and source depletion. However, the analog experiments do not reveal their individual contribution to the surface kinematics. To understand the relative contribution of the reservoir depletion and ring-faulting, numerical simulations are needed. Therefore, here we simulate reservoir depletion and ring-fault activity using the 3-D boundary element method (BEM) modeling and inspect the evolving surface deformation from initial downsag to later collapse. By using this complementary numerical method, we are able to better char- acterize the distinctive subsidence patterns caused by different deformation sources and fault structures and can therefore, understand their relative influences on localized ground deformation.

4.1 Boundary Element Method

The boundary element method (BEM) is a computational approach to solve linear partial diﬀerential equations which have been formulated as integral governing equations [Segall, 2010]. This method is eﬃcient because only the boundaries of the source in the modeling need to be discretized and match the given boundary conditions [Bathke et al., 2015]. The deformation source in our modeling comprises a 42

Active: Passive: Source depleting Uniform pressure change +Freely sliding = + Ring-faulting

Surface displacement

Sill-like Ring-fault d1 contracting source

Figure 4.1: Numerical BEM sources contain one sill-like contracting source plus one cone-shape ring-fault. The geometries of sources are discretized into triangular elements. The kinematics of each element is controlled by the analytical solutions for triangular dislocations with given stress condition or vice versa.

circular, sill-like source, interacting with a tapered cylindrical ring-fault lying above it. The sill-like source simulates the depletion mechanism of the reservoir [Bathke et al., 2013] [Bathke et al., 2015]. The complex geometry of the deformation sources can be constructed by a mesh composed of several triangular elements (Fig. 4.1). The kinematics of each element can then be described by slip on the analytical solutions for triangular dislocations [Nikkhoo and Walter, 2015]. Given appropriate boundary conditions on each triangular dislocation, we can simultaneously calculate the strain and stress ﬁelds on all boundaries of the deformation sources (i.e., the ring-fault and the sill-like source). Therefore, we will explain the principle of BEM and its application to our volcanic deformation problem in the following paragraphs. In the upper crust, strain and stress accumulate on some geologic structures (e.g., dikes and faults) due to tectonic loading. As rocks have limited strength, the strain and stress will ultimately, if not completely, at least partially release, resulting in events such as volcano eruptions and earthquakes. To understand the physics of the 43 underlying geologic processes, the elementary physical conditions of interest are the in situ displacement, strain, and stress. For linear elastic material, the stress is related to the strain according to a generalized Hooke’s Law [Segall, 2010]:

σij = C ijklǫkl, (4.1)

where C ijkl is the symmetric fourth-rank stiﬀness tensor due to the symmetry of the strain and the stress tensor. By considering the symmetry of the stiﬀness tensor and assuming the isotropy of the crustal medium [Segall, 2010], Equation 4.1 can be written as:

σij =2µǫǫij + λǫǫkkδij, (4.2) where µ and λ are the Lame constants, among which µ is the shear modulus that relates shear stress to shear strain. In our numerical modeling, we use the sill-like source and the ring-fault to model the sources of volcanic deformation and ring-faulting respectively; the ring-fault and the sill-like source are delineated by M and N triangular elements respectively. The strain and stress at the centroids arise from displacements on all the elements (fault slip on the ring fault and opening/closing of the sill-like source), that can be calculated using triangular dislocations [Nikkhoo and Walter, 2015]. Then, we relate the strain tensors to the stress tensors using Hooke’s Law for isotropic and homogeneous elastic medium. The stress tensors at each triangle can be further resolved into the stress tractions both parallel and normal to the elements that are necessary for our modeling. All these can be mathematically described as:

σ = G · s, (4.3)

where σ is a vector composed of tractions on both the sill and the ring-fault; s 44 is a vector composed of displacements on the sill and on the ring-fault; G is the stiﬀness matrix relating s to σ (i.e., G is composed of stress at each element due to unit dislocation displacement on each triangular element). Note that there are three components (i.e., strike, dip and tensile components) of both the traction and the displacement on every element depending on the orientation of the plane of each triangular element. According to our formulation, a linear system in Equation 4.3 can be explicitly expressed as:

Rs Rs Rs R g g g T s   Rs Rd St  S gSs gSs gSs  T s     Rs Rd St  R     xs   R gRd gRd gRd  T d    σ G · s  Rs Rd St  × R σ =   = G s =   xd  , (4.4)  S   Sd Sd Sd    T d  g g g       Rs Rd St  xS   R  t  T   Rt Rt Rt   t  g g g     Rs Rd St   S    T t  gSt gSt gSt   Rs Rd St  where: 45

R T s : the strike-component traction acting on the ring-fault; S T s : the strike-component traction acting on the sill; R T d : the dip-component traction acting on the ring-fault; S T d : the dip-component traction acting on the sill; R T t : the tensile-component traction acting on the ring-fault; S T t : the tensile-component traction acting on the sill;

R xs : the strike-slip on the ring-fault;

R xd : the dip-slip on the ring-fault;

S xt : the tensile-slip on the sill (opening or closing);

Rs R R g : a matrix of resulting T s calculated from a unit xs ; Rs

Rs R R g : a matrix of resulting T s calculated from a unit xd ; Rd

Rs R S g : a matrix of resulting T s calculated from a unit xt ; St

Ss S R g : a matrix of resulting T s calculated from a unit xs ; Rs

Ss S R g : a matrix of resulting T s calculated from a unit xd ; Rd

Ss S S g : a matrix of resulting T s calculated from a unit xt ; St ......

In our case, the σ is used to represent the stress boundary condition in which only traction in the normal direction occurs on the sill-like source. Once the geometry of the sill-like source and ring-fault are deﬁned (in the next section), G can be constructed, and then estimating s in the Equations 4.3∼4.4 becomes a standard linear inverse problem:

−1 s = G · σ. (4.5)

In our modeling, we deﬁne the sill as being the only active source in the system 46 by assigning speciﬁc traction on it. On the contrary, the ring-fault is considered as a passive source with a traction free boundary condition and thus, the stresses on it are all zero. Here we generalize the boundary conditions as

R T s  OM×1 S T  O ×   s   N 1       R   T d  OM×1 σ =   =   · p, (4.6)  S    T  O ×1   d   N   R   T  O ×   t   M 1      S    T t  U N×1  where:

OM×1: a M-by-1 zero vector.

ON×1: a N-by-1 zero vector.

U M×1: a N-by-1 unit vector. p: a tensile traction acting on the sill-like source determining the magnitude and the sign convention of the deformation; (positive p is expansion while negative p is contraction)

Note that all tractions of the ring-faults are zero. By making the ring-fault free of traction, it slides passively with the deformation of the sill-like source. To resolve the linear system of equations in Equation 4.5, we then applied the least square method [Menke, 2012]. The mathematical expression of a least square method is

s = [(GT G)−1GT ]σ. (4.7)

Finally, we have the displacement vector s on all elements. We further want to compute the deformation ﬁeld in the elastic half-space and at the surface caused by the displacements on the ring-fault and the sill. This can simply be accomplished 47 Table 4.1: Deformation source parameters of the BEM numerical models. Sill Sourceparameters Values Unit Radius 0.5km Centerposition,x-axis(W-E) 0.0 km Centerposition,y-axis(N-S) 0.0 km Center depth, z-axis (below surface) 4.08 km Tensile traction (boundary condition) 33 MPa Shear traction (boundary condition) 0 MPa

Ring-fault Sourceparameters Values Unit Upperellipseradius 0.45 km Centerposition,x-axis(W-E) 0.0 km Centerposition,y-axis(N-S) 0.0 km Center depth, z-axis (below surface) varied km Lowerellipseradius 0.5 km Centerposition,x-axis(W-E) 0.0 km Centerposition,y-axis(N-S) 0.0 km Center depth, z-axis (below surface) 4.00 km Tensile traction (boundary condition) 0 MPa Shear traction (boundary condition) 0 MPa

by calculating the resulting deformation from each triangular element on the given grid points. In a linear system, we can then add up a series of solutions from every element, applying the principle of superposition to get the actual deformation ﬁelds in the elastic half-space contributed by the entire ring-fault and the sill-like source [Segall, 2010].

4.2 Geometry of Ring-Faults and Sills

In our numerical modeling of caldera subsidence, the surface displacement is caused by two deformation sources: (1) a contracting sill-like source and (2) a ring-fault sliding in the dip-direction. We implemented several models with the same setup of the sill but diﬀerent conﬁgurations of the ring-fault to imitate the ring-faulting 48

Surface

d1 Ring fault A B C D E Depth d2 Sill-like contracting source

Figure 4.2: Numerical BEM source modeling setup. The schematic illustration shows the triangular mesh of the deformation sources. Diﬀerent models with varying fault depths were implemented to study the processes evolving from initial to mature phase of caldera subsidence.

processes evolving from the initial downsag stage to the mature collapse stage (Fig. 4.2). Modeling was run under a flat surface setting without including any topographic effect. We use the observed rupturing structures from our analog experiments as an input of the ring-fault configuration in the numerical modeling. We also refer to previous studies including numerical modeling [Bathke et al., 2015] [Holohan et al., 2011] and analog experiments [Acocella, 2007] [Roche et al., 2000] [Ruch et al., 2012] at subsiding calderas for more information about the architecture of the ring-faults. The complicated geometry of the ring-fault and the sill is approximated by the mesh consisting of several triangular elements. The sill-like source model was designed as an ellipse disk with a diameter of 1.0 km and was placed at around 4.0 km at depth. Based on the analysis of calderas geometry related to source dimension [Holohan et al., 2011], our collapse model has a thickness to diameter ratio T/D ≥ 1 ∼ 4, representing a thick-roof setting. The effect of the thick-roof setting brings about a relatively 49 high dip-angle ring-fault, characterizing mainly by an outward-dipping reverse fault, rather than an inward-dipping normal fault [Holohan et al., 2011]. Therefore, the ring-fault model is designed as an upward-tapered cylinder. We defined the geometry of the ring-fault based on two ellipses centered at different depths. The ellipses represent the upper and lower edge of the ring-fault, and the fault plane was then linearly interpolated between these two ellipses [Bathke et al., 2015]. The ring-fault in different models has its lower ellipse right above the sill at 4.0 km in depth with a diameter of 1.0 km, while the upper ellipse of the ring-fault has a diameter of 0.9 km and depth varied from 3.0 km below the surface to the surface (Fig. 4.2). Under our meshing setup, a small space between the lower ellipse of the ring-fault model and the sill-like source needs to be preserved to prevent computational instability [Nikkhoo and Walter, 2015]. The detailed source parameter setup can also be found in Table 4.1. Therefore, by defining a smaller upper ellipse and a larger lower ellipse, we generated an outward-dipping ring-fault that corresponded to experimental observations [Acocella, 2007] [Roche et al., 2000] [Ruch et al., 2012] and ring-structure geometry revealed by seismicity in nature [Nettles and Ekström, 1998] [Ekström, 1994].

4.3 Boundary Conditions and Assumptions

To achieve simplicity in our numerical modeling, assumptions of the ring-fault and the sill-like source had to be made. We assumed the fault plane is frictionless and free of traction, i.e., the stress boundary condition on the ring-fault is zero. The sill on the other hand, as an active deformation source, was assumed to have only tensile stress as a boundary condition while shear stresses on the sill are zero. By using the above stress boundary conditions in the triangular dislocation, we simulate a displacement scenario in which the sill closes with a speciﬁc stress drop, and the ring-fault responds with slip in dip-direction accordingly. Because including detailed information of the earth’s heterogeneous medium into 50 Table 4.2: Elastic half-space parameters in the BEM numerical modeling. Parameter Value Units Lam´e1stconstant 33 GPa Shearmodulus 33 GPa Poisson’sratio 0.25 - Tractiononthesill 33 MPa

our modeling is both computationally complex and time-consuming, we took the subsurface medium as isotropic, linearly elastic and homogeneous. The physical co- eﬃcients involved in our numerical modeling include Poisson’s ratio and Lame’s two constants. For Poison’s ratio we used γ =0.25 and for the Lame’s constants we used λ = µ =33 GPa [Bathke et al., 2015]. Finally, we speciﬁed a uniform tensile traction of −33 MPa on the sill as a stress boundary condition to generate about 1 to 2 m of surface displacement under our elastic half-space settings (Table 4.2). 51

Chapter 5

Results: Boundary Element Numerical Modeling

In this chapter, we show the numerical simulation results imitating the ring-faulting processes, evolving from the initial downsag stage to the mature collapse stage, during increasing caldera subsidence (Figure 5.1). Note that the surface displacement is an integrating consequence of (1) a contracting sill-like source and (2) a ring-fault slipping in the dip-direction.

5.1 Focused Subsidence Depending on Ring-Fault Architec- tures

We assigned the traction on the sill p = −33 MP a as the change of tensile stresses (negative as contraction of the sill, Equation. 4.6). Using this assuming uniform stress change at Model A (Figure 5.1a), the derived maximum contraction of the sill is about 1.0 m (maximum closing ≃ 1.0 m) at the sill center. Applying the same sill traction on different models from Model A to Model E (Fig. 5.1a∼e) results in increasing maximum sill closure. In other words, longer faults result in larger sill closure under the same stress change on the sill. To assess the key differences of the surface displacements between the different ring-fault architectures in a linear system, we can linearly scale the maximum contraction of the sill to 1.0 m among different models (Fig. 5.1). This way, we can better compare different models under the same sill contraction. We discovered models with a shorter ring-fault and a deeper ring-fault tip (the upper edge of the ring-fault) inclined to have broader and smaller 52 Surface profiles Model configuration a 0 0 Ring-fault Sill

2 -25 3 km bs Model A Faulting Sill 4 Fault = 33% 1 1 Total 0 0 -50 -1 1 km 0 b 0

2 2 km bs -25 Model B Faulting Sill Fault 4 = 56% Total 1 1 -50 0 0 c -1 0 0 0 1 km bs

2 -25 Model C Sill 4 Faulting Fault [cm] = 78% Total 1 1 -50 0 0 Sill closing -1

Vertical displacement [cm] Vertical d 0 0 300 m bs 100

2 -25 Model D Sill

Faulting 0 Fault 4 = 93% Total 1 1 -50 0 0 -1 e 0 0 0 km bs [cm] 2 -25 Fault dip-slip Z [km] Model E Faulting Sill Fault 4 = 100% 60 Total 1 1 -50 0 0 -5 0 5 Y [km] -1 Distance [km] X [km]

Figure 5.1: BEM surface subsidence simulations. Each model setup from A to E imitates different stages of caldera subsidence from initial to mature. The surface deformation was caused by two deformation sources: the sill-like source and the ring- fault. As the ring-fault became shallower and longer, the displacements on the surface were localized. 53 surface displacements (Fig. 5.1a∼b). In contrast, models with a longer ring-fault and a shallower ring-fault tip tended to have larger and more focused displacements within the near field (Fig. 5.1c∼e). Unlike analog experiments, numerical simulations enable us to not only observe the overall consequences of surface displacements as a whole but also to differentiate displacements from different deformation sources accordingly. By analyzing displacements contributed by the sill contraction and the ring-faulting in the models, we can understand their relative significance under different settings of fault geometry. We highlighted that the models resembling the downsag stage of caldera subsidence, in which the configuration of the ring-fault has a deeper upper edge location (Fig. 5.1a∼b), produced very distinctive surface profiles from models with shallower ring- fault tip, which resembles the collapse stage (Fig. 5.1d∼e). Our results show that for the same sill location and closure, models with longer ring-faults and shallower fault tips generated more localized subsidence in the near field. Model A and Model B (Fig. 5.1a∼b) represent models with deeper ring-fault tips at 3.0 km and 2.0 km below the surface, respectively. These models produced 10-15 cm of subsidence affecting the area up to 5.0 km away from the epicenter of deformation. Notice that the displacements due to the ring-faulting (red curve), compared to the ones due to the sill closure (blue curve), are small. Model C (Fig. 5.1c) had a depth of the fault tip at 1.0 km below the surface, and it resulted in larger total subsidence (black curve) which slightly corresponded to the displacements resulted from the ring-faulting (red). Model D (Fig. 5.1d) had a depth of the fault tip at 300 m below the surface, corresponding to 93% of faulting ( = fault length/sill depth), creating 35 cm of subsidence mainly confined in the near field. Model E (Fig. 5.1e) displayed a sharp collapse was created and confined within the near field by the surface ring-faulting. This collapse profile was well defined by the fault-induced displacements (red) in the near field. Though the far-field region seemed to have some 54 uplift component caused by the reverse ring-faulting (red), the net surface movements remained negative (black) because the broad deflation caused by sill contraction (blue) overtook this uplift. In summary, the near-field localization of the subsidence grew more intense as the ring-fault developed to a shallower and longer architecture. In the end, we can numerically reproduce a sharp, focused collapse in the near field overlaid by a far-field deflation by assigning the upper edge of the ring-fault at a shallow location in the subsurface (Fig. 5.1e).

5.2 Wrapped Surface Displacements

Wrapped displacements with 2 cm per fringe are shown in this section to see better the deformation pattern and to compare it to that from our analog experiments and satellite observations shown in Figure 1.4 [Xu et al., 2016]. In addition to the previous five models, here we implemented a model with only a single contraction source and without a ring-fault. The contraction of a single sill-like source could induce a broad deflation affecting the entire range up to 10 km radius in the model (Fig. 5.2a). This long-wavelength deflation can be clearly observed from the topographic profile (Fig. 5.2g, the grey curve). With a ring-fault included into the model setup, the subsidence tended to concentrate closer to the sill-like source boundary as we saw from Model A with fault depth at 3.0 km (Fig. 5.2b) and Model B (2.0 km below the surface, Fig. 5.2c). By including a ring-fault with a shallower upper edge into the model, we increased the localization in the near field. Model C with a ring-fault at a depth of 1.0 km (Fig. 5.2d) produced apparent short-wavelength subsidence within the near- field border. Model D, with a ring-fault at a depth of 300 m (Fig. 5.2e) created focused displacements in the near field, distinguishing itself from the long-wavelength deformation in the far field. Model E, with a ring-fault at the surface (Fig. 5.2f) produced such sharp displacements that it caused fringe aliasing. The induced sharp collapse was difficult to observe from the fringe pattern since the displacements at 55

5 a Only sill b - 3 kmc - 2 km

-1

-5 Model A Model B 5 d - 1 kme - 0.3 kmf 0 km Distance [km] 1

+ -1

2 cm -5 Model C Model D Model E

-5-1 1 5 -5-1 1 5 -5-1 1 5 - Distance [km]

g Models: 0 Only sill Model C Model A Model D Model B Model E -25 Deformation: Short-wavelength Long-wavelength Vertical disp. [cm]

-50

-5 0 5 Distance [km]

Figure 5.2: Localisation of subsidence is closely related to the depth of the ring-fault upper edge. (a)∼(f) Wrapped surface displacements from different models with different ring-fault configuration. The surface displacements tend to be focused in the near field when a shallower ring-fault tip is installed. When the ring-fault is not included in the model or a ring-fault with a deeper fault tip is installed, the surface displacements are broader and smoother. (g) The profiles of vertical surface displacement from different models. Models with deeper ring-fault tips are comparable to the downsag stage of our experiments while models with shallower ring-fault tips are comparable to the collapse stage. 56 the border of the collapse were too intense to be differentiated. However, this severe topographic change could easily be seen in the surface profile (Fig. 5.2g, the green curve), revealing both a flat caldera floor and a sudden elevation change at the border enclosed by the reverse ring-fault. Based on our analysis, the surface deformation is mainly controlled by the sill contraction if a ring-fault is not included or a ring-fault with a rather deep upper edge is involved in the process. In this case, the deformation is gentle and shows broad subsidence. In models with a ring-fault tip at shallower depths of the half- space, the synthetic deformation on the surface tends to be dominated by the buried ring-faulting, resulting in concentrated subsidence. Because the closing of the sill-like source remained the same in all these models, the displacement differences observed at the surface are determined only by the ring-fault architecture (Fig. 5.1). Therefore, we conclude that, for the same sill location and closure, models with longer ring-faults and shallower fault tips generate more localized subsidence in the near field. 57

Chapter 6

Discussion

In this work, we aim to explain the deformation feature observed (e.g., InSAR, GPS) at some subsiding calderas (Fig. 1.4) [Holohan et al., 2017][Xu et al., 2016], composed of a volcano-wide smooth deflation and a localized subsidence on the caldera floor. We have so far described our results from both analog subsiding experiments and numerical modeling using the boundary element method. In the following section, we will first summarize our research results and then relate them to cases in nature and propose an explanation of this complex deformation. Finally, we will discuss several limitations in this type of research.

6.1 Summary of Results

In our analog experiments, we observed broader subsidence dominated the displacements at the surface in the deflation stage before the ring-fault ruptures were well developed (Fig. 3.2). Then, the near field started to be influenced by localized ground deformation as the ring-fault propagated towards the surface while the far field still showed broader deflation (Fig. 3.3). The distinctive subsiding rates in the near field and the far field led to increasing differential subsidence between the two areas. Shortly before the ring-fault propagated to the surface, the differential subsidence intensified rapidly, making the displacement pattern very different between the long-wavelength deflation in the far field and the short-wavelength subsidence in the near field (Fig. 3.4). Later on in the collapse stage, the localized subsidence widened 58 slightly and reached out beyond the near field as the normal ring-fault system extended further due to gravitational instabilities and collapses on the periphery of the subsiding caldera (Fig. 3.5). In the numerical modeling, under the same contraction of the sill-like source, models with deeper ring-fault tips tend to create broader surface subsidence affecting a larger area (Fig. 5.1a∼b). In contrast, models with shallower ring-fault tips incline to concentrate the deformation and induce strong, focused subsidence in the near field (Fig. 5.1d∼e). The deformation caused by ring-faulting become increasingly dominant on the surface as the upper tip of the ring-fault gets shallower (Fig. 5.1e, Fig. 5.2f). Numerical models with deeper fault tips (Fig. 5.2a∼c) are comparable to the initial downsag stage of the analog experiments (Fig. 3.2), in which ring structures were still rather formless, resulting in broad surface deformation. In contrast, models with shallower fault tips (Fig. 5.2d∼f) are comparable to the mature collapse stage of the experiments (Fig. 3.3) characterized with a well-formed ring-fault system, resulting in complex surface deformation consisting of both long-wavelength and short-wavelength patterns. Our analysis suggests that the occurrence of concentrated subsidence within the near field is closely related to the development of the ring-fault ruptures and their activity. Assuming a depleting source scenario with the ring-fault inactive, or active at a larger depth, we can only see wide-area deflation on the surface caused by the sill closure. The localization of near-field subsidence will only occur once the ring-fault is active at shallow depth. Likewise, as has been observed at subsiding calderas in nature (Fig. 1.4), the overlapping surface deformation composed of short-wavelength focused subsidence and long-wavelength deflation can be interpreted as a consequence of the depressurization within a single reservoir in combination with the activity of a buried ring fault near the surface [Xu et al., 2016]. 59 6.2 Laboratory Limitations and Numerical Assumptions

Our ring-fault and contracting source model provided sufficient evidence to explain thoroughly the complex displacement observed at subsiding calderas. Nevertheless, our results depend on assumptions and limitations of the experimental and numerical methods. In this section, we will briefly review the main limitations of the laboratory apparatus and the primary assumptions behind the BEM numerical source modeling. In the analog experiment, firstly, we demonstrated the setting with a flat surface of the sand-stack rather than a volcano dome. However, calderas in nature are often located within a shield or a cone; topography relief with slopes higher than ∼ 20◦ may have a certain level of contribution to the subsurface stress field and the surface deformation pattern [Cayol and Cornet, 1998] [Corbi et al., 2016]. To verify whether the topographic effects will have a substantial influence on surface displacements, we also carried out an experiment with a gentle dome with a mean slope about 22◦(4 cm high and 20 cm wide. CAL 9, Table 3.1). The results showed consistently that we could reproduce a similarly complex pattern composed of both localized subsidence in the near field and broad deflation in the far field by including topography effects. Therefore, the complex deformation features do not appear to depend on the change of topography. Topography profiles of Galápagos volcanoes show an average slope ranging from 15 to 35◦ [Chadwick and Howard, 1991], with some steep flanks reaching slopes as steep as 37◦, according to results of airborne interferometric radar data [Mouginis-Mark et al., 1996]. Taking these steep slopes into account, topography effects need to be better considered in future work as well. Secondly, the half-circular subsiding design may bring frictional artifacts [Ruch et al., 2012]. Since there was a direct contact between the glass pane and the subsiding granular material, the glass could induce additional frictional forces acting on the material and affect the displacement pattern. To test this effect, we compared the velocity of the in-depth subsiding material to that of the sinking piston and found 60 that they are similar. Nevertheless, the surface displacement profiles in the west-east direction showed that the broad deformation pattern was subtly restrained due to the friction of the glass pane in its vicinity. We conclude that the friction from the glass does not significantly affect the kinematics of the in-depth subsidence while the elevation change on the surface is more sensitive to it. Thirdly, we began the experiment with an initial sinking of the roof above the reservoir. In other words, our subsiding processes started from the initiation of the ring-faulting. This process may be different from the subsiding calderas in nature that contain pre-existing ring-faults. Wolf volcano had an inherited, buried ring-fault in the subsurface before the deformation observed in the 2015 eruption [Geist et al., 2005] [Walker, 1984]. The pre-existing structures are consequences of earlier and possibly repeated caldera collapses. Moreover, this inherited ring-fault was suggested to have been reactivated during the 2015 eruption and to have contributed to the surface deformation [Xu et al., 2016]. To look into this, we also conducted a few experiments with a reactivation of the ring-fault after a one-hour halt of the sinking piston. The results showed that pre-existing faults were reactivated followed by localized surface displacements with the relative prominence of the broad deflation signal depending on the shallowest depth of the reactivated ring-faults. As the emphasis of this research is on the relationship of ring-faulting and subsidence localization rather than fault reactivation, we will not go further into this subject here. It would require significantly more work and laboratory tests [Acocella et al., 2000] [Folch and Mart, 2004]. In the BEM numerical model, firstly, we assumed an isotropic and homogeneous earth model and did not take into account the effects of a heterogeneous medium. While heterogeneities of the crustal material undoubtedly have an impact on the surface deformation, the geological conditions vary from place to place and require information from seismic tomography data, local gravitational surveys or other inves- tigations. Therefore, we here assume the earth structure to be homogeneous. 61 Secondly, in our model, there was a small gap between the contracting sill and the ring-faults, as it is computationally expensive to make a perfect connection between the two sources. The analog experiments show that the lower end of the ring-fault is rooted to the lateral edge of the sinking source (piston). In reality, the root of a developed ring-fault should also connect down to the edge of a reservoir. However, the lower end of the ring-fault in our numerical models cannot extend down to the edge of the sill-like source. This issue comes from the meshing of the triangular dislocations [Nikkhoo and Walter, 2015]. The size of a triangular element determines a critical distance (half of the triangle edge) from the edges of the triangle, such that if any calculating points fall within it, computational instabilities may occur. From several trial tests, a gap of at least 80 m was needed to avoid these computational instabilities under our meshing setup. This gap leads to some discontinuous deformation between the contracting sill and the slipping ring-fault. In the gap between the lower edge of the ring-fault and the sill, the medium behaves completely elastic. This discontinuity results in higher traction on the sill to produce the surface displacement. If the gap was absent, the required traction to produce 1.0 m of sill closure would be less. In other words, this limitation in our numerical setup leads to underestimation of the surface displacements under the given traction (−33 MPa) within the sill. By using finer meshing (much smaller and more triangles), the gap could be reduced, but it would be computationally too intensive. Thirdly, the ring-fault is assumed to be frictionless in our modeling. According to presumed stress boundary conditions, the ring-fault passively undergoes uncon- strained (free-surface) dip-slip along the given fault plane once the sill closes. This simplification violates the reality both in nature and laboratory [Krantz, 1991]. In some cases, large calderas with single and large volume decreases may facilitate piston- like rapid roof collapse bounded by steeply dipping ring faults [Cole et al., 2005][Lip- man, 1997], which are less affected by friction [Spray, 1997]; However, this is not our 62 case. Disregarding friction and shear stresses distribution on the fault plane leads to an inadequate simulation of faulting behavior and the induced surface displacements [Segall, 2010]. Nonetheless, we anticipate the effect of friction does not substantially influence our current results as we limit the extent of the slip by gradually changing the location of the upper fault tip, above which the fault can be regarded too strong to slip. However, we suggest further numerical modeling efforts with detailed frictional conditions will be needed in the future. Lastly, we assumed a single outward-dipping ring-fault in our numerical modeling with a constant dip-angle and without any of the branches seen in the analog experiments. This assumption simplifies what we observe from analog models [Aco- cella, 2007], is seen in discontinuous element modeling [Holohan et al., 2011] and in nature [Geshi et al., 2002][Michon et al., 2009]. Nevertheless, the architecture of the subsurface ring-fault is distinctive from one place to another, depending on the local fracture pattern and geological conditions [Bathke et al., 2013]. Since considering a variety of different fault geometries and implementing numerous fault segments is too complicated and time-consuming [Bathke et al., 2015], we only focused on the general phenomenon with a simple setup of an outward-dipping ring-fault.

6.3 Causes for Overlapping Deformation

Based on the experimental and BEM results, we propose a source model with a contracting sill and a ring-fault to explain the overlapping deformation observed in geodetic data (InSAR) from subsiding calderas. This deformation is complex, composed of (1) a broad deflation signal covering the entire volcanic edifice, and (2) a localized subsidence signal confined within the caldera [Xu et al., 2016]. Without taking physical and geological constraints into account, the surface deformation can also be modeled using different sources at various depths, whether they are realistic or not [Holohan et al., 2017] [Riel et al., 2015]. Therefore, we compare our results to 63

Comparing scenarios: Scenario I Scenario II Scenario III (this research) (similar to Xu et al., 2016 and Bagnardi et al., 2012) BEM numerical Ring-fault Deeper sill Single sill source Sill Shallower sill a b c 0 30’ 30’ 30’ 60’ 60’ 60’

90’ 90’ 90’ 5 Displacement [mm] -15 0 15 -15 0 15 -15 0 15 Distance [cm] Experimental observation BEM numerical expectation

Figure 6.1: Comparing scenarios of BEM modeling of caldera subsidence. Scenario I: the modeling in this research. Scenario II: a model of a contracting sill. Scenario III: a model including two contracting sill sources at diﬀerent depths (similar to [Bagnardi and Amelung, 2012] and [Xu et al., 2016]).

different modeling setups to look for other possible models that may also explain the complex deformation. First, a single sill-like contracting source, and then second, two sill-like contracting sources at different depths. The two-sill scenario is a simplified version of the models used in [Bagnardi and Amelung, 2012] and [Xu et al., 2016]. The comparisons were based on our experiment data at t = 30′, 60′ and 90′ (seconds, Fig. 3.5c). Notice that at around t = 90′, the complex surface deformation which contains long-wavelength broad deflation and short-wavelength localized subsidence already appeared. We then overlaid the synthetic surface displacement from the BEM modeling containing both the ring-fault and sill sources (Scenario I, Fig. 6.1a). For comparison, we linearly rescaled the synthetic displacement curves to match the experiment observations. As expected, the scenario with ring-faulting and single-sill contraction explains nicely the complex deformation feature seen in 64 the analog results. In the second case (Scenario II, Fig. 6.1b), we used a single sill and adjusted the source depth to produce surface displacements that could match the experimental observations. At t = 30′, the single sill-like source can explain the observation, but it fails to explain the concentrated subsidence that becomes obvious at t = 90′. When two sills are used (Scenario III, Fig. 6.1c), vertically stacked with adjusted source depths, they create surface displacements that match the observations. The results therefore suggest that both the two-sill scenario (Scenario III) and a scenario combining a ring-fault and a sill (Scenario I) can be used to model the same observational data consisting of the complex overlapping displacement pattern.

6.4 Estimating Source Parameters: Sill Depths and Volume Changes

As we showed above, though the same deformation pattern can be explained with very different source models, the source parameters are distinctive. Difficulties in constraining the source parameters from the surface deformation appear not only in the laboratory but also in nature [Riel et al., 2015]. These parameters determine the whole picture of the processes in the subsurface. Therefore, applying different models on explaining the surface deformation can lead to very different estimations of the plumbing system including the reservoir depth(s) and the depressurization within the reservoir(s) [Chadwick et al., 2011] [Holohan et al., 2017]. In the following, we will show the difficulties in estimating deformation source parameters in our BEM modeling. The analysis here comprises three steps: (1) An introduction to our ring-fault-sill simulation results from BEM Model A ∼ Model E (Fig. 5.1) that we take as our ”real data” that we want to simulate and (2) modeling with new BEM models consisting of two sill-like sources and (3) comparison of the estimated source parameters. In the ring-fault-sill simulations (Scenario I), from Model A ∼ Model E, the con- 65

Scenario I (Ring-fault Scenario III (2-Sill) 0 a and sill)

-25 3.0 km 3.0 km 4.0 km 4.0 km ΔV diff. = 2.60 km3 -50 = 0.0% = 2.60 km3 ΔV1 ΔV2 0 b 3

2.0 km -25 1.6 km

ΔV diff. 4.0 km = 2.65 km3 -50 = 11.3% ΔV1 = 2.95 km3 ΔV2 0 c

1.0 km 0.9 km -25

4.0 km ΔV diff. 3 = 16.1% = 2.67 km -50 = 3.10 km3 ΔV1 ΔV2 0 d Vertical displacement [cm] Vertical

0.3 km 0.42 km -25

4.0 km ΔV diff. -50 = 20.4% = 3.25 km3 = 2.70 km3 ΔV2 ΔV1 0 e 0.2 km 0.0 km

-25

ΔV diff. 4.0 km = 28.7% = 2.72 km3 -50 = 3.50 km3 ΔV1 ΔV2 -5 0 5 Distance [km] Figure 6.2: Estimation of deformation source parameters. Comparing two BEM modeling scenarios with different deformation sources: (1) A ring-fault plus a contracting sill-like source; (2) Two contracting sill-like sources. Though the same displacement can be simulated by different deformation source models, the estimated source depths and volume changes are distinctive. As the ring-fault tips have been set at shallow depths, the upper sills of the two-sill models are required to migrate upward to fit the surface displacement. The difference of volume change estimation (∆Vdiff ) between the two scenarios increases as the ring-fault gets closer to the surface. 66 Table 6.1: Estimation of source parameters using different BEM deformation sources Scenario 11:1 Scenario 22:2 3 3 3 3 Fig. 6.2 Sill depth Fault depth ∆V Shallow sill Deep sill ∆V ∆Vdiff [km] [km] [km3] [km] [km] [km3] % a 4.0 3.0 2.60 3.0 4.0 2.60 0.0 b 4.0 2.0 2.65 1.6 4.0 2.95 11.3 c 4.0 1.0 2.67 0.9 4.0 3.10 16.1 d 4.0 0.3 2.70 0.42 4.0 3.25 20.4 e 4.0 0 2.72 0.2 4.0 3.50 28.7 1 Deformation source is composed of a ring-fault and a sill-like source. 2 Deformation source is composed of two sill-like sources. 3 Depth below the surface.

tracting sill is ﬁxed at a depth of 4.0 km; the depth of the fault tip varies from 3.0 ∼ 0.0 km, corresponding to an upward propagating ring-fault activity. Results show that the surface subsidence increased as the upward development of the ring-fault while the total volume change of the closing sill slightly increased from 2.60 km3 to 2.72 km3. As we tried to simulate the same surface displacements using the two-sill models (Scenario III), the estimated depth of sills changed. Though the lower sill always remained at 4.0 km depth, the upper sill gradually migrated upward (Table 6.1). When the ring-faulting was at 3.0 km depth, the estimated upper sill was at 3.0 km (Fig. 6.2a). As the ring-fault propagated to 2.0 km, 1.0 km, 0.3 km and 0.0 km depths, the estimated upper sill relocated upward to 1.6 km, 0.9 km, 0.42 km and 0.2 km below the surface, respectively (Fig. 6.2b∼e). Therefore, using a multi-source model, changing of the source depth (i.e., upward migration) is inevitable in order to simulate the surface deformation caused by a deep source depletion combined with upward migrating ring-faulting (Scenario I). In addition to the uncertainty in the estimation of the sill depth, estimation of the volume change in the source is challenging. When the ring-faulting was at 3.0 67 a b c

Two-reservoir depletion Multiple sources? Alternative or source migration?

Figure 6.3: Schematic summary of models explaining complex overlapping surface deformation patterns. (a) The common practice to explain overlapping wavelength of displacements is to use a two-reservoir depletion model. However, it is diﬃcult to constrain the depths and the volume changes of the real reservoirs as the ring-faulting may involve in the deformation processes. (b) At some calderas where subsurface de- veloping ring-fault systems participate deformation processes, using a two-reservoir model may eventually result in multiple sources or source migration phenomenon, which is sophisticated and the reason to it is unclear. (c) A simple alternative model including ring-faulting can better explain the deformation pattern with more realis- tically geological meaning.

km depth, the real volume change and the estimated value were roughly the same, 2.60 km3 (Fig. 6.2a). As the ring-fault propagated to 2.0 km, 1.0 km, 0.3 km and 0.0 km depths, the estimated volume changes for the two-sill model grew from 2.95 km3 to 3.10 km3 and to 3.25 km3, respectively (Fig. 6.2b∼d). When the ring- faulting reached the surface, the real volume change was 2.72 km3 while the estimated value had increased to 3.50 km3 (Fig. 6.2e). In summary, using two-sill simulations (Scenario III) to model deformation caused by a deep source depletion combined with ring-fault activity (Scenario I), the estimated contraction of the source will likely be higher than the true value. This overestimation increases as the ring-fault propagates closer to the surface (Table 6.1). We observed that if we try to use a simple two-sill-source model to simulate the displacements caused by a depressurization chamber with active ring-faulting, the 68 location of the sill-like source migrates upward along with the development of the ring-fault. This phenomenon may correspond to the uncertain multiple-reservoir interpretation of the deformation at Bárdabunga volcano [Riel et al., 2015] and the significant upward migration of the estimated source of Piton de la Fournaise [Peltier et al., 2009]. Based on our results, we suggest that such puzzling changes in the estimated elastic-deformation source are due to the ring-fault activity in the subsurface. This interpretation agrees with the hypothesis at Piton de la Fournaise that the varying parameters (e.g., shape, location, orientation, volume change) of the deformation sources were primarily a consequence of inelastic host-rock deformation that migrated upward from the magma reservoir to the surface [Holohan et al., 2017]. In addition to the difficulties in estimating source depth, we discover that merely implementing two depleting sills into numerical modeling on deformation processes related to the ring- fault activity, may result in overestimation of the volume change in the reservoir(s) by up to 20∼30% depending on the architecture of the ring-fault (Fig. 6.3). 69

Chapter 7

Implications

Constraining source parameters of the observed vertical and horizontal ground surface deformations near active volcanoes, has led to useful estimates of the location and shape of subsurface magma reservoirs and related structures. Therefore, it is one of the most important aspects of volcano geodesy [Dieterich and Decker, 1975] [Dzurisin, 2007]. However, it is a common practice to ignore geological or other geophysical evidence of ring-fault activity at subsiding volcanoes and to model the observed deformation with multiple sill-like sources [Riel et al., 2015]. The geological meaning of such results is often uncertain and can lead to biased estimation of source parameters [Holohan et al., 2017]. Implications of our work emphasize the importance of considering ring-fault systems when constraining the architecture of the plumbing system and related geological structures. We suggest that once substantial evidence of the buried ring-fault structures is provided, the ring-fault must be taken into account in explaining the observed volcanic deformation. Our work can also be linked to associated aspects such as volcanic hazard assessments or anthropogenic sinking processes related to ring-structure activities that are comparable to volcano subsidence. Such cases include sinkhole or pit crater formation [Poppe et al., 2015] [Nof et al., 2013] and underground nuclear- test sinks [Howard, 2010] [Vincent et al., 2003]. Therefore, an improved insight of how deep processes relate to surface deformation is beneﬁcial for both the knowledge of volcano tectonics and the potential hazards at volcanic or non-volcanic subsidence phenomena in general. 70

Chapter 8

Conclusions and Outlook

In this work, we successfully use a combination of experimental models and numerical simulations to reconstruct the 3-D geometry and kinematics of subsiding calderas and related ring structures. We propose an alternative explanation for the complex surface deformation pattern revealed by geodetic data at inﬂating/deﬂating volcanoes. Our results further suggest that ring-fault structures may be critical in explaining the complex surface deformation.

8.1 Concluding Remarks

(1) Our analog experiments revealed the 3-D surface subsiding processes and the cross-section architecture of the ring-fault. Observations showed that the localized subsidence in the near field of the surface became more intense along with the upward incremental development of the buried ring-fault while the far field was constantly under broader downsagging deformation. (2) Numerical BEM modeling allows us to construct a deformation source composed of a ring-fault and an elastic sill-like contracting source. The results show that under the same sill contraction setting, models with shallower ring-faulting produced more localized and sharp surface displacements. Models with deeper ring-faulting, on the contrary, led to broader and smoother displacements. (3) The complex deformation pattern observed by InSAR at some subsiding calderas, which consists of a broad deflation signal affecting the entire volcanic edifice 71 and the focused subsidence confined within the caldera was simulated in our work both experimentally and numerically. (4) We conclude that a more realistic alternative explanation for the observed two- wavelength deformation could be the depressurization of a deep magmatic reservoir along with the activity of a ring-fault at shallower depths of the host-rock. (5) Geodetic [Amelung et al., 2000] [Bathke et al., 2013], geological [Geshi, 2009] [Michon et al., 2009] and geophysical [Ekström, 1994] data have shown that the ring-fault structures are common in nature. Therefore, one should take into account the contribution of the ring-faulting in source interpretations and inversions if the structures of the ring-fault are known. By doing so, one could avoid misestimation of the reservoir depth(s) and the depleted volume.

8.2 Outlook

For prospects, we recommend to simulate/invert data from a real case of deforming calderas using the numerical BEM modeling. Obvious targets include the ones where patterns of complex surface deformation have been revealed, such as the volcanoes on the Gal´apagos islands [Xu et al., 2016]. For these cases, one could demonstrate well how the overlapping wavelengths of deformation may be explained with one magmatic source in combination with ring-fault activity. 72

REFERENCES

[Acocella, 2007] Acocella, V. (2007). Understanding caldera structure and development: An overview of analogue models compared to natural calderas. Earth-Science Reviews, 85(3):125–160.

[Acocella et al., 2000] Acocella, V., Cifelli, F., and Funiciello, R. (2000). Analogue models of collapse calderas and resurgent domes. Journal of Volcanology and Geothermal Research, 104(1):81–96.

[Amelung et al., 2000] Amelung, F., J´onsson, S., Zebker, H., and Segall, P. (2000). Widespread uplift and trapdoor faulting on Gal´apagos volcanoes observed with radar interferometry. Nature, 407:993–996.

[Bagnardi and Amelung, 2012] Bagnardi, M. and Amelung, F. (2012). Space- geodetic evidence for multiple magma reservoirs and subvolcanic lateral intrusions at Fernandina Volcano, Gal´apagos Islands. Journal of Geophysical Research: Solid Earth, 117(B10).

[Bathke et al., 2015] Bathke, H., Nikkhoo, M., Holohan, E., and Walter, T. (2015). Insights into the 3D architecture of an active caldera ring-fault at Tend¨urek volcano through modeling of geodetic data. Earth and Planetary Science Letters, 422:157– 168.

[Bathke et al., 2013] Bathke, H., Sudhaus, H., Holohan, E. P., Walter, T. R., and Shirzaei, M. (2013). An active ring fault detected at Tend¨urek volcano by using InSAR. Journal of Geophysical Research: Solid Earth, 118(8):4488–4502.

[Biggs et al., 2014] Biggs, J., Ebmeier, S. K., Aspinall, W. P., Lu, Z., Pritchard, M. E., Sparks, R. S. J., and Mather, T. A. (2014). Global link between deformation and volcanic eruption quantiﬁed by satellite imagery. Nature Communications, 5:3471.

[Branney, 1995] Branney, M. J. (1995). Downsag and extension at calderas: new perspectives on collapse geometries from ice-melt, mining, and volcanic subsidence. Bulletin of Volcanology, 57(5):303–318. 73 [Buckingham, 1914] Buckingham, E. (1914). On physically similar systems; illustra- tions of the use of dimensional equations. Physical review, 4(4):345.

[Burchardt and Walter, 2010] Burchardt, S. and Walter, T. R. (2010). Propagation, linkage, and interaction of caldera ring-faults: comparison between analogue experiments and caldera collapse at Miyakejima, Japan, in 2000. Bulletin of Volcanology, 72(3):297–308.

[Cayol and Cornet, 1998] Cayol, V. and Cornet, F. H. (1998). Three-dimensional modeling of the 1983-1984 eruption at Piton de la Fournaise Volcano, R´eunion Island. Journal of Geophysical Research: Solid Earth, 103(B8):18025–18037.

[Chadwick et al., 2006] Chadwick, W. W., Geist, D. J., Jónsson, S., Poland, M., Johnson, D. J., and Meertens, C. M. (2006). A volcano bursting at the seams: Inflation, faulting, and eruption at Sierra Negra volcano, Galápagos. Geology, 34(12):1025–1028.

[Chadwick and Howard, 1991] Chadwick, W. W. and Howard, K. A. (1991). The pattern of circumferential and radial eruptive ﬁssures on the volcanoes of Fernandina and Isabela islands, Gal´apagos. Bulletin of Volcanology, 53(4):259–275.

[Chadwick et al., 2011] Chadwick, W. W., Jónsson, S., Geist, D. J., Poland, M., Johnson, D. J., Batt, S., Harpp, K. S., and Ruiz, A. (2011). The May 2005 eruption of Fernandina volcano, Galápagos: The first circumferential dike intrusion observed by GPS and InSAR. Bulletin of Volcanology, 73(6):679–697.

[Cole et al., 2005] Cole, J., Milner, D., and Spinks, K. (2005). Calderas and caldera structures: a review. Earth-Science Reviews, 69(1-2):1–26.

[Corbi et al., 2016] Corbi, F., Rivalta, E., Pinel, V., Maccaferri, F., and Acocella, V. (2016). Understanding the link between circumferential dikes and eruptive ﬁssures around calderas based on numerical and analog models. Geophysical Research Letters, 43(12):6212–6219.

[De Natale et al., 2006] De Natale, G., Troise, C., Pingue, F., Mastrolorenzo, G., Pappalardo, L., Battaglia, M., and Boschi, E. (2006). The Campi Flegrei caldera: unrest mechanisms and hazards. Geological Society, London, Special Publications, 269(1):25–45. 74 [Dieterich and Decker, 1975] Dieterich, J. H. and Decker, R. W. (1975). Finite element modeling of surface deformation associated with volcanism. Journal of Geo- physical Research, 80(29):4094–4102.

[Dzurisin, 2003] Dzurisin, D. (2003). A comprehensive approach to monitoring volcano deformation as a window on the eruption cycle. Reviews of Geophysics, 41(1).

[Dzurisin, 2007] Dzurisin, D. (2007). Volcano Deformation: New Geodetic Monitor- ing Techniques. Geophysical Sciences. Springer-Verlag, Berlin Heidelberg.

[Ekstr¨om, 1994] Ekstr¨om, G. (1994). Anomalous earthquakes on volcano ring-fault structures. Earth and Planetary Science Letters, 128(3):707–712.

[Fichtner and Tkalˇcić, 2010] Fichtner, A. and Tkalˇcić, H. (2010). Insights into the kinematics of a volcanic caldera drop: Probabilistic finite-source inversion of the 1996 Bárdarbunga, Iceland, earthquake. Earth and Planetary Science Letters, 297(3):607–615.

[Folch and Mart, 2004] Folch, A. and Mart, J. (2004). Geometrical and mechanical constraints on the formation of ring-fault calderas. Earth and Planetary Science Letters, 221(1-4):215–225.

[Galland et al., 2016] Galland, O., Bertelsen, H. S., Guldstrand, F., Girod, L., Jo- hannessen, R. F., Bjugger, F., Burchardt, S., and Mair, K. (2016). Application of open-source photogrammetric software MicMac for monitoring surface deformation in laboratory models. Journal of Geophysical Research: Solid Earth, 121(4):2852– 2872.

[Galland et al., 2015] Galland, O., Holohan, E., van Wyk de Vries, B., and Bur- chardt, S. (2015). Laboratory Modelling of Volcano Plumbing Systems: A Review. Springer Berlin Heidelberg, Berlin, Heidelberg.

[Geist et al., 2014] Geist, D. J., Bergantz, G., and Chadwick, W. W. (2014). Gal´apagos magma chambers. The Galapagos: A Natural Laboratory for the Earth Sciences, 204:55.

[Geist et al., 2005] Geist, D. J., Naumann, T. R., Standish, J. J., Kurz, M. D., Harpp, K. S., White, W. M., and Fornari, D. J. (2005). Wolf Volcano, Gal´apagos Archipelago: Melting and Magmatic Evolution at the Margins of a Mantle Plume. Journal of Petrology, 46(11):2197–2224. 75 [Geshi, 2009] Geshi, N. (2009). Asymmetric growth of collapsed caldera by oblique subsidence during the 2000 eruption of Miyakejima, Japan. Earth and Planetary Science Letters, 280(1):149–158.

[Geshi et al., 2002] Geshi, N., Shimano, T., Chiba, T., and Nakada, S. (2002). Caldera collapse during the 2000 eruption of Miyakejima Volcano, Japan. Bul- letin of Volcanology, 64(1):55–68.

[Geyer et al., 2006] Geyer, A., Folch, A., and Mart, J. (2006). Relationship between caldera collapse and magma chamber withdrawal: An experimental approach. Jour- nal of Volcanology and Geothermal Research, 157(4):375–386.

[Gudmundsson et al., 2016] Gudmundsson, M. T., Jónsdóttir, K., Hooper, A., Holo- han, E. P., Halldórsson, S. A., Ofeigsson,´ B. G., Cesca, S., Vogfjörd, K. S., Sig- mundsson, F., Högnadóttir, T., Einarsson, P., Sigmarsson, O., Jarosch, A. H., Jónasson, K., Magnússon, E., Hreinsdóttir, S., Bagnardi, M., Parks, M. M., Hjörleifsdóttir, V., Pálsson, F., Walter, T. R., Schöpfer, M. P. J., Heimann, S., Reynolds, H. I., Dumont, S., Bali, E., Gudfinnsson, G. H., Dahm, T., Roberts, M. J., Hensch, M., Belart, J. M. C., Spaans, K., Jakobsson, S., Gudmundsson, G. B., Fridriksdóttir, H. M., Drouin, V., Dürig, T., Aáalgeirsdóttir, G., Riishuus, M. S., Pedersen, G. B. M., Boeckel, T. v., Oddsson, B., Pfeffer, M. A., Barsotti, S., Bergsson, B., Donovan, A., Burton, M. R., and Aiuppa, A. (2016). Grad- ual caldera collapse at Bárdarbunga volcano, Iceland, regulated by lateral magma outflow. Science, 353(6296):aaf8988.

[Holohan et al., 2011] Holohan, E. P., Sch¨opfer, M. P. J., and Walsh, J. J. (2011). Mechanical and geometric controls on the structural evolution of pit crater and caldera subsidence. Journal of Geophysical Research, 116(B7).

[Holohan et al., 2017] Holohan, E. P., Sudhaus, H., Walter, T. R., Schöpfer, M. P. J., and Walsh, J. J. (2017). Effects of Host-rock Fracturing on Elastic-deformation Source Models of Volcano Deflation. Scientific Reports, 7(1).

[Holohan et al., 2008] Holohan, E. P., van Wyk de Vries, B., and Troll, V. R. (2008). Analogue models of caldera collapse in strike-slip tectonic regimes. Bulletin of Volcanology, 70(7):773–796.

[Howard, 2010] Howard, K. A. (2010). Caldera collapse: Perspectives from comparing Gal´apagos volcanoes, nuclear-test sinks, sandbox models, and volcanoes on Mars. GSA Today, 20(10):4–10. 76 [Hubbert, 1937] Hubbert, M. K. (1937). Theory of scale models as applied to the study of geologic structures. GSA Bulletin, 48(10):1459–1520.

[Jones and Stewart, 1997] Jones, R. H. and Stewart, R. C. (1997). A method for determining signiﬁcant structures in a cloud of earthquakes. Journal of Geophysical Research: Solid Earth, 102(B4):8245–8254.

[Jónsson, 2009] Jónsson, S. (2009). Stress interaction between magma accumulation and trapdoor faulting on Sierra Negra volcano, Galápagos. Tectonophysics, 471(1):36–44.

[Jónsson et al., 2005] Jónsson, S., Zebker, H., and Amelung, F. (2005). On trapdoor faulting at Sierra Negra volcano, Galápagos. Journal of Volcanology and Geother- mal Research, 144(1-4):59–71.

[Kennedy et al., 2018] Kennedy, B. M., Holohan, E. P., Stix, J., Gravley, D. M., Davidson, J. R. J., and Cole, J. W. (2018). Magma plumbing beneath collapse caldera volcanic systems. Earth-Science Reviews, 177:404–424.

[Konstantinou et al., 2003] Konstantinou, K. I., Kao, H., Lin, C.-H., and Liang, W.- T. (2003). Analysis of broad-band regional waveforms of the 1996 September 29 earthquake at B´ardarbunga volcano, central Iceland: investigation of the magma injection hypothesis. Geophysical Journal International, 154(1):134–145.

[Krantz, 1991] Krantz, R. W. (1991). Measurements of friction coeﬃcients and cohesion for faulting and fault reactivation in laboratory models using sand and sand mixtures. Tectonophysics, 188(1-2):203–207.

[Lipman, 1997] Lipman, P. W. (1997). Subsidence of ash-ﬂow calderas: relation to caldera size and magma-chamber geometry. Bulletin of Volcanology, 59(3):198–218.

[Menke, 2012] Menke, W. (2012). Geophysical Data Analysis: Discrete Inverse The- ory, Volume 45, Third Edition: MATLAB Edition. Academic Press, Waltham, MA, 3 edition edition.

[Michon et al., 2007] Michon, L., Staudacher, T., Cayol, V., Bachlery, P., and Marti, J. (2007). April 2007 collapse of Piton de la Fournaise: A new example of caldera formation. Geophysical Research Letters, 34(21).

[Michon et al., 2009] Michon, L., Villeneuve, N., Catry, T., and Merle, O. (2009). How summit calderas collapse on basaltic volcanoes: New insights from the April 77 2007 caldera collapse of Piton de la Fournaise volcano. Journal of Volcanology and Geothermal Research, 184(1-2):138–151.

[Mouginis-Mark et al., 1996] Mouginis-Mark, P. J., Rowland, S. K., and Garbeil, H. (1996). Slopes of Western Gal´apagos volcanoes from airborne interferometric radar. Geophysical Research Letters, 23(25):3767–3770.

[Nairn et al., 1995] Nairn, I. A., Mckee, C. O., Talai, B., and Wood, C. P. (1995). Geology and eruptive history of the Rabaul Caldera area, Papua New Guinea. Journal of Volcanology and Geothermal Research, 69(3):255–284.

[Naumann et al., 2002] Naumann, T., Geist, D., and Kurz, M. (2002). Petrology and Geochemistry of Volc´an Cerro Azul: Petrologic Diversity among the Western Gal´apagos Volcanoes. Journal of Petrology, 43(5):859–883.

[Nettles and Ekström, 1998] Nettles, M. and Ekström, G. (1998). Faulting mechanism of anomalous earthquakes near Bárdarbunga Volcano, Iceland. Journal of Geophysical Research: Solid Earth, 103(B8):17973–17983.

[Nikkhoo and Walter, 2015] Nikkhoo, M. and Walter, T. R. (2015). Triangular dislocation: an analytical, artefact-free solution. Geophysical Journal International, 201(2):1119–1141.

[Nof et al., 2013] Nof, R. N., Baer, G., Ziv, A., Raz, E., Atzori, S., and Salvi, S. (2013). Sinkhole precursors along the Dead Sea, Israel, revealed by SAR interferometry. Geology, 41(9):1019–1022.

[Norini and Acocella, 2011] Norini, G. and Acocella, V. (2011). Analogue modeling of ﬂank instability at Mount Etna: Understanding the driving factors. Journal of Geophysical Research, 116(B7).

[Pan et al., 2009] Pan, B., Qian, K., Xie, H., and Asundi, A. (2009). Two-dimensional digital image correlation for in-plane displacement and strain measurement: a review. Measurement Science and Technology, 20(6):062001.

[Peltier et al., 2009] Peltier, A., Staudacher, T., Bachlery, P., and Cayol, V. (2009). Formation of the April 2007 caldera collapse at Piton de La Fournaise volcano: Insights from GPS data. Journal of Volcanology and Geothermal Research, 184(1):152–163. 78 [Plattner et al., 2013] Plattner, C., Amelung, F., Baker, S., Govers, R., and Poland, M. (2013). The role of viscous magma mush spreading in volcanic ﬂank motion at Klauea Volcano, Hawaii. Journal of Geophysical Research: Solid Earth, 118(5):2474–2487.

[Plattner et al., 2010] Plattner, C., Wdowinski, S., Dixon, T. H., and Biggs, J. (2010). Surface subsidence induced by the Crandall Canyon Mine (Utah) collapse: InSAR observations and elasto-plastic modelling. Geophysical Journal Interna- tional, 183(3):1089–1096.

[Poppe et al., 2015] Poppe, S., Holohan, E. P., Pauwels, E., Cnudde, V., and Kervyn, M. (2015). Sinkholes, pit craters, and small calderas: Analog models of depletion- induced collapse analyzed by computed X-ray microtomography. GSA Bulletin, 127(1-2):281–296.

[Raﬀel et al., 2013] Raﬀel, M., Willert, C. E., Wereley, S. T., and Kompenhans, J. (2013). Particle Image Velocimetry: A Practical Guide. Springer. Google-Books- ID: v3r1CAAAQBAJ.

[Ramberg, 1981] Ramberg, H. (1981). Gravity, Deformation and the Earth’s Crust: In Theory, Experiments and Geological Application. Academic Press Inc, London ; New York, 2nd revised edition edition edition.

[Riel et al., 2015] Riel, B., Milillo, P., Simons, M., Lundgren, P., Kanamori, H., and Samsonov, S. (2015). The collapse of B´ararbunga caldera, Iceland. Geophysical Journal International, 202(1):446–453.

[Roche et al., 2000] Roche, O., Druitt, T. H., and Merle, O. (2000). Experimen- tal study of caldera formation. Journal of Geophysical Research: Solid Earth, 105(B1):395–416.

[Ruch et al., 2012] Ruch, J., Acocella, V., Geshi, N., Nobile, A., and Corbi, F. (2012). Kinematic analysis of vertical collapse on volcanoes using experimental models time series. Journal of Geophysical Research: Solid Earth, 117(B7).

[Saunders, 2001] Saunders, S. J. (2001). The shallow plumbing system of Rabaul caldera: a partially intruded ring fault? Bulletin of Volcanology, 63(6):406–420.

[Scarpati et al., 1993] Scarpati, C., Cole, P., and Perrotta, A. (1993). The Neapolitan Yellow Tuﬀ A large volume multiphase eruption from Campi Flegrei, Southern Italy. Bulletin of Volcanology, 55(5):343–356. 79 [Segall, 2010] Segall, P. (2010). Earthquake and Volcano Deformation. Princeton University Press, student edition.

[Simkin and Howard, 1970] Simkin, T. and Howard, K. A. (1970). Caldera Collapse in the Galpagos Islands, 1968: The largest known collapse since 1912 followed a ﬂank eruption and explosive volcanism within the caldera. Science, 169(3944):429– 437.

[Spray, 1997] Spray, J. G. (1997). Superfaults. Geology, 25(7):579–582.

[Thielicke and Stamhuis, 2014] Thielicke, W. and Stamhuis, E. (2014). PIVlab To- wards User-friendly, Aﬀordable and Accurate Digital Particle Image Velocimetry in MATLAB. Journal of Open Research Software, 2(1).

[Vincent et al., 2003] Vincent, P., Larsen, S., Galloway, D., Laczniak, R. J., Walter, W. R., Foxall, W., and Zucca, J. J. (2003). New signatures of underground nuclear tests revealed by satellite radar interferometry. Geophysical Research Letters, 30(22).

[Walker, 1984] Walker, G. P. L. (1984). Downsag calderas, ring faults, caldera sizes, and incremental caldera growth. Journal of Geophysical Research: Solid Earth, 89(B10):8407–8416.

[Walter and Motagh, 2014] Walter, T. R. and Motagh, M. (2014). Deﬂation and inﬂation of a large magma body beneath Uturuncu volcano, Bolivia? Insights from InSAR data, surface lineaments and stress modelling. Geophysical Journal International, 198(1):462–473.

[Walter and Troll, 2001] Walter, T. R. and Troll, V. R. (2001). Formation of caldera periphery faults: an experimental study. Bulletin of Volcanology, 63(2-3):191.

[Xu et al., 2016] Xu, W., Jónsson, S., Ruch, J., and Aoki, Y. (2016). The 2015 Wolf volcano (Galápagos) eruption studied using Sentinel-1 and ALOS-2 data: The 2015 Wolf Eruption in the Galápagos. Geophysical Research Letters, 43(18):9573–9580.

[Yılmaz et al., 1998] Yılmaz, Y., G¨uner, Y., and S¸aroˇglu, F. (1998). Geology of the quaternary volcanic centres of the east Anatolia. Journal of Volcanology and Geothermal Research, 85(1):173–210.