Generalized statistical mechanics description of fault and earthquake populations in Corinth rift (Greece) Georgios Michas

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1 Generalized statistical mechanics description of fault and earthquake populations in Corinth rift (Greece) Georgios Michas Ph.D. Thesis University College London 2016

2 Declaration I, Georgios Michas, confirm that the work presented in this thesis is my own. Where information has been derived from other sources, I confirm that this has been indicated in the thesis. ii

3 Abstract The aim of the present thesis is to provide new insights into fault growth processes and the evolution of earthquake activity in one of the most seismically active area in Europe, the Corinth Rift (central Greece). The collective properties of fault and earthquake populations are studied in terms of statistical mechanics and the generalized framework termed as Non-Extensive Statistical Mechanics (NESM). By compiling a comprehensive dataset for the fault network in the Rift, the scaling properties of fault trace-lengths are studied by applying the NESM framework. In the debate of power-law versus exponential scaling in natural fault systems, the analysis indicates the transition from the one end-member to the other as a function of increasing strain in the Rift, providing quantitative evidence for a combination of crustal processes in a single tectonic setting. The results further imply that regional strain, fault interactions and the boundary condition of the brittle layer may control fault growth and fault network evolution in the Rift. The fragment-asperity model, which is derived in the NESM framework, is further used to describe the frequency-magnitude distribution of seismicity and estimate the recurrence times of large earthquakes in the region, supplemented and compared with the empirical Gutenberg-Richter scaling relation. The NESM based analysis of the temporal properties of earthquakes in the Rift indicates that seismicity evolves in temporal clusters, characterized by multifractal structures and both short-term and long-term clustering effects, which indicate highly non-random behavior. Such properties further imply non-linear diffusion phenomena in the evolution of the earthquake activity, a hypothesis that is tested for two case studies of induced seismicity in the Rift. The spatiotemporal properties of the two earthquake sequences are studied in terms of the Continuous Time Random Walk (CTRW) theory and the NESM framework and indicate a non-linear sub-diffusion process in the spatial relaxation of the earthquake activity in the region. Overall, the present thesis, based on the principles of generalized statistical mechanics, provides a physical rationale for the scaling properties of fault and earthquake populations in the Corinth Rift and demonstrates how these properties can provide new insights into the evolution of the earthquake activity and the fault network in the region. iii

4 Acknowledgements I would like to acknowledge a number of people for their help and support over the course of my doctoral studies. I am truly grateful to my supervisors, Prof. Peter Sammonds and Prof. Filippos Vallianatos, for their guidance and support. It was their constructive ideas and sound advice that encouraged me to accomplish much of this work and to publish my research. I would like to extend my gratitude to the members of UCL Institute for Risk and Disaster Reduction for their valuable feedback on my research and for the numerous fruitful discussions on disaster risk reduction. I would like to thank Rosie, Leisa, Barbara, Jen, Celine and Danuta for their administrative and IT support. I am also grateful to all students and staff in the UCL IRDR and the UCL Earth Sciences for making this journey both fun and constructive. Special thanks to Giorgos and Alexis, who shared with me the extensive journey on non-extensive statistical mechanics. I am truly grateful to Alexis for her guidance and assistance in the rock physics laboratory. I would also like to thank all the people at the TEI of Chania for their kindness and hospitality during my stay there. I would like to acknowledge fruitful and constructive discussions with Prof. Gerald Roberts. I am also grateful to Francesco Pacchiani and Vasilios Karakostas for kindly sharing with me their earthquake datasets. I acknowledge financial support from the IRDR, the UCL Graduate School, the UCL Earth Sciences Department, the Thales project and the AGU for attending a number of conferences and training courses. Finally, I would like to express my true gratitude to my family and friends. Without your love and endless support nothing of all these would be possible. The Thesis was examined by Dr. Luciano Telesca and Prof. Ferruccio Renzoni. The PhD project was funded by the State Scholarships Foundation (IKY) through the action Program for scholarships provision I.K.Y. through the procedure of personal evaluation for the academic year from resources of the educational program Education and Life Learning of the European Social Register and NSRF iv

5 Table of Contents Declaration... ii Abstract... iii Acknowledgements... iv List of Figures... ix List of Tables... xvii Chapter 1 Introduction Introduction Thesis Outline... 7 Chapter 2 Earthquake Statistics and Generalized Statistical Mechanics Introduction The concept of Fractals Scaling in Fault and Earthquake Populations Scaling properties of fault attributes Frequency-size distribution of earthquakes Spatiotemporal scaling properties of seismicity Stochastic models of seismicity Critical phenomena and Self-Organized Criticality Statistical Mechanics and Information Theory Foundations of Statistical Mechanics Information Theory Statistical Mechanics and Earthquakes Generalized Statistical Mechanics The non-additive entropy S q Optimizing S q Applications to Seismicity Spatiotemporal description Earthquake magnitudes Fault populations v

6 2.7 Summary Chapter 3 Statistical Mechanics of the Fault Network Introduction Tectonic setting Regional Geodynamics Physiography and Geological setting Deformation patterns in the Rift Short-term deformation Long-term deformation Fault dataset Fault map and fault trace-lengths Possible errors in the dataset Generalized Statistical Mechanics Formalism Analysis of the fault dataset The cumulative distribution function Possible biases in the fault-length distribution Scaling properties of the fault population Scaling properties for the short-term strain rates Scaling properties for long-term deformation Robustness of the results Discussion Discussion of the results Implications for fault growth Implications for the fault network evolution in the Rift Chapter 4 Earthquake-Size Distribution Introduction Earthquake activity in the Corinth Rift Seismic Networks and Earthquake Catalogues Implementing the G-R and F-A models to earthquake data Estimating the a and b values in the G-R scaling relation Estimating the a E and q E values in the fragment-asperity model vi

7 4.5 Analysis of the earthquake-size distribution Dataset Seismicity Declustering Earthquake-size distribution in the Corinth Rift Expected seismicity rates Chapter 5 Temporal Properties of Seismicity and the Inter-event Time Distribution Introduction Datasets The HUSN catalogue The CRLN catalogue The Inter-event Times Distribution Inter-event Times Histograms A dynamic mechanism for the inter-event time distribution Multifractality in the earthquake time series Multifractal Detrended Fluctuation Analysis Multifractal Analysis of the inter-event time series The effect of the threshold magnitude Temporal variations Discussion Chapter 6 Earthquake Diffusion Introduction Continuous Time Random Walk approach to earthquake diffusion The 2010 Efpalion earthquakes Spatiotemporal scaling properties Aftershock diffusion Discussion The 2001 Agios Ioannis earthquake swarm sequence Spatiotemporal scaling properties Diffusion of the swarm sequence Discussion vii

8 Chapter 7 Discussion Summary of the Results Wider Implications Implications for fault growth and fault network evolution Implications for earthquake physics and the evolution of seismicity Implications for earthquake hazard in the Rift Directions for Future Work Chapter 8 Conclusions Appendices Appendix A: Table of Faults Appendix B: Results for the Declustered Datasets Appendix C: Matlab Codes List of References viii

9 List of Figures 2.1 The classic techniques used to estimate the fractal dimension in a fractured medium (Bonnett et al., 2001) Traces of fault populations in a range of scales (Main, 1996) The various functions that are most frequently used to describe the distribution n(w) of a fault property w (Bonnett et al., 2001) Fault displacement (D) vs. length (L) for various published fault datasets (Schlische et al., 1996) Number of earthquakes per year N with magnitudes greater than m as a function of m on global scale (Turcotte, 1997) Cumulative number of earthquakes with moment magnitudes (m) equal or greater than m as a function of m, for the shallow earthquakes in the Harvard Centroid Moment Tensor Catalog, in the Japan-Kurile-Kamchatka zone (Kagan and Jackson, 2013) The probability distribution PS,L(T) of inter-event times T for various threshold magnitudes and spatial scales of California seismicity (Bak et al., 2002) a) Probability distributions Dxyt(τ) of inter-event times τ in California, for several stationary periods and various spatial scales, b) Probability distributions D(τ) of inter-event times τ for various spatial scales L and threshold magnitudes mc (Corral, 2003) Graphic representation of the slider-block model (Rundle et al., 2003) The q-exponential function for various values of q on a) log-linear axes and b) log-log axes The q-gaussian distribution for various q-values, in a) linear axes and b) loglinear axes The cumulative distribution of inter-event times P(>τ) for the Aigion aftershock sequence (Vallianatos et al., 2012b)...51 ix

10 2.13 Illustration of the relative motion of two irregular fault planes (a, b) and the fragments of size r that fill the space between them (Sotolongo-Costa and Posadas, 2004) a) Normalized cumulative distribution of earthquake magnitudes in Greece. b) Normalized cumulative magnitude distribution for the de-clustered dataset (Antonopoulos et al., 2014) a) Probability distribution function (PDF) of incremental avalanche sizes x, normalized to the standard deviation σ, for the OFC model on a small world topology and on a regular lattice. b) PDF of incremental earthquake energies for the Northern California earthquake catalogue (Caruso et al., 2007) Normalized cumulative distribution function P(>L) of fault lengths in Mars and the corresponding fit according to the q-exponential distribution, for a) compressional faults and b) extensional faults (Vallianatos, 2013a) Fraction of active faults as a function of increasing strain and the main stages of fault population evolution (Spyropoulos et al., 2002) The broader area of Greece and the main tectonic setting Topography of the Corinth Rift region a) Simplified geology and the tectonic framework of the Corinth Rift. b) Geodetic horizontal extension rates across the Rift. c) Late Quaternary uplift rates. d) Summation of upper-crust extension (black) and whole-crust extension along three transects in the rift The total sediment thickness in the Gulf of Corinth estimated from seismic reflection profiles by Taylor et al. (2011) Evolution of the Corinth Rift since onset of distributed extension in the Pliocene (Leeder et al., 2012) Spatial coverage of the published geologic maps that were used to compile the fault dataset in the Corinth Rift Mapping fault traces using DEM Map showing the seismic lines of geophysical surveys in the Corinth Rift...80 x

11 3.10 a) Uninterpreted and b) Interpreted seismic profile from the west Gulf of Corinth (Bell et al., 2008) Simplified geology and the fault network in the Corinth rift Histogram of frequency versus fault trace-lengths of the fault dataset Cumulative number of fault lengths for the entire fault population in the Corinth Rift Simplified geology and the fault network in the Corinth Rift, indicating the current high and low strain rate settings Cumulative number of fault trace-lengths for the high and low-strain settings, according to the short-term deformation pattern in the Rift Cumulative number of fault trace-lengths a) for the current high strain rate setting and b) for the low strain rate setting The western, central and eastern zones of the Rift considered as the long-term strain regimes Cumulative number of fault trace-lengths for the eastern, central and western zones of the Rift respectively Cumulative number of fault trace-lengths, a) in the western zone, b) in the central zone and c) in the eastern zone Analysis on synthetic faults datasets that scale according to the q-exponential distribution, with parameter values of q=1.15 and L0= Analysis on synthetic faults datasets that scale according to the q-exponential distribution, with parameter values of q=1.25 and L0= Summary of the observed properties in the different strain zones of the Rift The stress field around a normal fault (Gupta and Scholz, 2000b) The main stages of fault network evolution according to the numerical model of Cowie (1998b) Cross-section indicating the geometry and interactions of faults at depth, at the western zone of the Corinth Rift (Bell et al., 2008) xi

12 4.1 The earthquake activity in Greece for Mw 4, according to the NOA bulletins Historic earthquakes of magnitude greater than 6 in the Corinth Rift since 1694 AD time series of a) the number of seismic stations of the NOA network, b) the annual rate of events, c) a proxy to the completeness magnitude Mc (Mignan and Chouliaras, 2014) Seismic stations of HUSN operating in the Corinth Rift Spatial variations of Mc for four periods of the regional earthquake catalogue produced by NOA and HUSN (Mignan and Chouliaras, 2014) The CRL network Seismicity map for the shallow earthquakes (depth 30 km) in the Corinth Rift Cumulative number of events for , a) for the entire and the declustered dataset, b) for Mw Rates of earthquake magnitudes (Mw) over time for Cumulative seismic moment (in dyn cm) over time Cumulative frequency magnitude distribution (squares) and the non-cumulative (incremental) frequency magnitude distribution (triangles) on a log-linear scale. The solid line represents the G-R scaling relation The estimated b-values and their uncertainties as a function of the minimum magnitude M The cumulative frequency magnitude distribution and the corresponding fits according to the F-A model for various values of the minimum magnitude M The estimated qe values and their uncertainties (95% confidence intervals) as a function of the minimum magnitude M The misfits between the observed distribution and the G-R scaling relation and the fragment asperity (F-A) model, for various values of M xii

13 4.16 The annual probability of an earthquake of magnitude M occurring in the Corinth Rift, according to the G-R scaling relation and the F-A model The recurrence time of an earthquake of magnitude M in the Corinth Rift, according to the G-R scaling relation and the F-A model The earthquake activity in the Corinth rift a) Magnitude (ML) rate per day and b) seismicity rate per day and the cumulative number of events, for the earthquake activity in the Rift The inter-event time series τi versus their index number i on semi-log axes, for the earthquake activity in the Rift Spatial distribution of seismicity in the West Corinth Rift according to the CRLN catalogue a) Magnitude rate per day and b) seismicity rate per day and the cumulative number of events, for the earthquake activity in the West Corinth Rift, according to the CRLN catalogue The inter-event time series τi versus their index number i on semi-log axes, for the earthquake activity in the western part of the Rift The histogram of inter-event times τ for the HUSN dataset, in a) linear bin widths and linear axes, b) linear bin widths and log-log axes, c) logarithmic bin widths on log-log axes, d) logarithmic bin widths on log-log axes, with each frequency normalized by the bin width and for total sum of frequencies equal to one The histogram of inter-event times τ for the CRLN dataset, in a) logarithmic bin widths on log-log axes, b) logarithmic bin widths on log-log axes, with each frequency normalized by the bin width and for total sum of frequencies equal to one Probability density p(t) of the rescaled inter-event times T for the HUSN and CRLN datasets and the corresponding fits according to the q-generalized gamma distribution and the gamma distribution Probability density p(t) of the rescaled inter-event times T for the HUSN, CRLN and NOA datasets and for various time periods and threshold magnitudes and the xiii

14 corresponding fits according to the q-generalized gamma distribution and the gamma distribution Probability density p(t) of the rescaled inter-event times T for various time periods of stationary seismicity in the Rift The logarithm of the fluctuation function Fq(n) versus the logarithm of the segment sizes n for a) the HUSN dataset and b) the CRLN dataset The range of generalized Hurst exponents h(q) and their respective errorbars for various values of q, for a) the HUSN dataset and b) the CRLN dataset. The mean h(q) that resulted from 10 randomly shuffled copies of the original inter-event time series is also plotted The mass exponent τ(q) for the various orders of q, for the HUSN dataset and b) the CRLN dataset. The mean τ(q) that resulted from 10 randomly shuffled copies of the original inter-event time series is also shown The singularity spectrum f(a) as a function of the Hölder exponent a, for a) the HUSN dataset and b) the CRLN dataset. The mean singularity spectrum f(a), which resulted from ten randomly shuffled copies of the original inter-event time series, is also plotted for the two datasets respectively Singularity spectrum s width W as a function of the threshold magnitude Mth, for the CRLN and the HUSN datasets Singularity spectrum s width W, as a measure of the degree of multifractality over time, for a) the HUSN dataset and b) the CRLN dataset Spatial distribution of the 2010 Efpalion earthquake sequence Coulomb stress changes due to the coseismic slip a) of the first main shock and b) of the second strong event The cumulative distribution of inter-event times P(>τ) for the Efpalion earthquake sequence The cumulative distribution of inter-event distances P(>r) for the Efpalion earthquake sequence xiv

15 2 6.5 The mean squared displacement x (in km) from the first strong event (triangles) and the second strong event (circles) of their respective subsequent seismicity over the course of time t (in days) on log-log axes The cumulative number of aftershocks as a function of time after the first strong event and the corresponding fit according to the modified Omori formula Map of the Western Corinth Rift showing the 2001 seismicity (from Pacchiani and Lyon-Caen, 2010). The south cluster of events corresponds to the 2001 Agios Ioannis swarm sequence The 3-D spatial distribution of the Agios Ioannis earthquake swarm sequence Cumulative distribution function P(>τ) of the inter-event times τ for the Agios Ioannis earthquake swarm sequence Cumulative distribution function P(>r) of the inter-event distances r for the Agios Ioannis earthquake swarm sequence The mean squared displacement x (in km) of the earthquake swarm sequence over the course of time t (in days) on log-log axes, with the first event of the sequence at the origin Summary of the observed properties in the different strain zones of the Corinth Rift (reproduced from Fig.3.22) Probability density p(t) of the rescaled inter-event times T for the HUSN, CRL and NOA datasets, for various time periods and threshold magnitudes and the corresponding fits according to the q-generalized gamma distribution and the gamma distribution (reproduced from Fig.5.10) B.1 Aftershock identification intervals in a) space and b) time, as a function of the mainshock magnitude B.2 Cumulative and non-cumulative (incremental) frequency magnitude distribution for the declustered NOA dataset B.3 Cumulative frequency magnitude distribution for the declustered NOA dataset and the corresponding fits according to the F-A model for various values of the minimum magnitude M xv

16 B.4 Probability density p(t) of the rescaled inter-event times T for the declustered HUSN and CRLN datasets and the corresponding fits according to the q- generalized gamma distribution and the gamma distribution B.5 Multifractal analysis of the inter-event time series for the declustered HUSN dataset B.6 Multifractal analysis of the inter-event time series for the declustered CRLN dataset xvi

17 List of Tables 2.1 Spatiotemporal scaling properties of seismicity according to the q-exponential distribution for various earthquake catalogues and tectonic environments Analyses of the frequency-magnitude distribution of earthquakes according to the q-exponential distribution and the fragment-asperity model, for various earthquake catalogues and tectonic environments Summary of the marine geophysical surveys and the corresponding studies, which were used to compile the fault dataset offshore...81 A.1 Table of Faults xvii

18 Chapter 1 Introduction 1.1 Introduction Earthquakes originate from the deformation and sudden rupture of parts of the lithosphere due to the external forces that arise from plate tectonic motions. The abruptness of the earthquake phenomenon and its devastating consequences in the anthropogenic environment have always attracted peoples fear and wonder. In an evergrowing world, the societal and economic effects of large earthquakes can be substantial. For instance, the 2004 Sumatra-Andaman earthquake (Mw = 9.0) and tsunami caused more than 230,000 deaths, while the 2011 Tohoku (Japan) earthquake (Mw = 9.1) and tsunami killed more than 22,000 people, caused the nuclear meltdown of Fukushima power plant and economic losses of about US$235 billion, as those were estimated by the World Bank (e.g., Sachs et al., 2012). Despite the great scientific effort, understanding the physics that govern the earthquake generation process and the subsequent prediction of future earthquakes still represent an outstanding challenge for scientists (Nature debates, 1999; Scholz, 2002; Kanamori, 2008).

19 Chapter 1 Introduction The complexity 1 of earthquakes is manifested in the wide range of spatial and temporal scales and the nonlinear dynamics that are incorporated in the evolution of seismicity (Keilis-Borok, 1990; Kagan, 1994). Earthquake ruptures occur on a complex network of fractures and faults that vary from microcracks, to major fault zones and plate boundaries, whereas the temporal scales that are incorporated in the process vary from seconds (during dynamic rupture), to the hundreds, thousands and millions of years that characterize the repeat times of characteristic earthquakes and the evolution of fault zones and tectonic plate boundaries, respectively (Scholz, 2002; Rundle et al., 2003). While the history of a seismic rupture can be reconstructed a posteriori through the analysis of the seismic waves recorded in seismographic stations, knowledge of the dynamics that lead to the initiation and propagation of a rupture through a fault system, giving rise to an earthquake, is really limited. The microscopic laws that govern friction, the rupture evolution through a highly heterogeneous medium and the coupling of these processes with chemical alterations and rock-fluid interactions still remain unclear (Main et al., 1992; Sammonds et al., 1992; Sammonds, 2005; Dieterich, 2007; Sornette and Werner, 2009). Despite the complexity of the earthquake generation process, the collective properties of fault and earthquake populations exhibit some universal scaling properties that are valid and consistent. The most prominent are scale-invariance and fractality that are manifested in the frequency-size distribution of earthquakes, the statistics of fault systems and the spatiotemporal properties of seismicity (e.g., Main, 1996; Turcotte et al., 2007b). The frequency-size distribution of earthquakes generally follows the empirical Gutenberg-Richter (G-R) scaling relation (Gutenberg and Richter, 1944), which resembles power-law scaling and fractality in the distribution of the dissipated seismic energy and the fault rupture areas, limited in each case by the size of the seismogenic system (Turcotte, 1997; Scholz, 2002). Power-law scaling and fractal geometries also characterize a variety of fault attributes, such as the distribution of fault trace-lengths or fault displacements (e.g., Sholz and Cowie, 1990; Bonnett et al., 2001). These empirical observations of seismicity and faulting have led to the consideration of the earthquake generation process as a critical point phenomenon undergoing a 1 Although an exact definition of complexity does not exist, a well-accepted one for a complex system is that of a system containing many interdependent constituents that interact nonlinearly (Latora and Marchiori, 2004). 2

20 Chapter 1 Introduction phase-transition (Rundle et al., 2003; Sornette, 2006). In this context, the Earth s crust has been considered as a self-organized critical (SOC) system that spontaneously organizes in a dynamical, statistically stationary state, which is characterized by powerlaw earthquake size distributions, fractal geometries and long-range interactions (Bak and Tang, 1989; Sornette and Sornette, 1989; Main, 1996; Sornette, 1999). Although the scaling properties of seismicity can be reproduced by statistical models or SOC, the fundamental challenge that still remains elusive in earthquake physics is to understand the transition from the microscopic scale and the laws that govern friction, rupture propagation, plasticity, fluid migration, chemical reactions and so on, to the macroscopic scale of large earthquakes, fault networks and tectonic plate boundaries (Main, 1996; Sornette and Werner, 2009; Kawamura et al., 2012). Clearly, to specify completely this problem, one would have to incorporate a large number of degrees of freedom and knowledge of exact dynamics in the Earth s crust that are generally inaccessible (Rundle et al., 2003). Although the physics of each individual earthquake is not well understood and their definitive prediction is extremely difficult, an ensemble of many events clearly demonstrates organization patterns that can be studied with different approaches than the physics of the individual events. These observations motivate the statistical mechanics approach to seismicity. Statistical mechanics constitutes one of the pillars of contemporary physics that remarkably establishes the link between the mechanical microscopic laws and the macroscopic (large-scale) description of a physical system. By using the mathematical tools of probability theory and statistics, statistical mechanics can be used to estimate the macroscopic properties of large populations from the specification of the relevant microscopic constituents and their interactions (Sornette and Werner, 2009). Originally, statistical mechanics and the associated concept of entropy (S) were used in thermodynamics and the kinetic theory of gases. The concept of entropy was later incorporated into information theory as a measure of uncertainty (Shannon, 1948) and since mid-1950 s it has been extended to other fields beyond classic thermodynamics in order to provide a general principle for inferring the least biased probability distribution from limited information (Jaynes, 1957). According to statistical mechanics principles, the relative probability that the system possesses a given state can be estimated by optimizing (i.e., maximizing) the entropy, subject to our limited 3

21 Chapter 1 Introduction knowledge about the system that is expressed through the constraints. Optimization of the classic definition of entropy due to Boltzmann and Gibbs (SBG), for a large number of independent non-interacting elements, yields the exponential (Boltzmann) distribution as the most probable state for a wide class of equilibrium systems, although the theory has been extended beyond the classic thermal equilibrium state (e.g., Dugdale, 1996; Kondepudi and Prigogine, 1998). Boltzmann-Gibbs (BG) statistical mechanics have been applied to earthquake physics in a series of works, which address the theory as a variational principle for deriving the large-scale properties of earthquake populations (Main and Burton, 1984; Main, 1996) or to investigate the thermodynamic state of the crust (Rundle et al., 1995; 1997; Main and Al-Kindy, 2002; Main and Naylor, 2010), although the theory fails to describe fragmentation processes and the scaling properties observed in the size of the fragments (Englmant et al., 1987). However, there is an important class of complex out-of-equilibrium systems that violate some of the essential properties of BG statistical mechanics (Bouchaud and George, 1990; Gell-Mann, 1994; Zaslavsky, 1999; Sornette, 2006). The macroscopic behavior of such systems, instead of the Boltzmann distribution, typically presents power-law distributions with heavy tails, enhanced by (multi) fractal geometries, long-range memory effects, intermittency or large fluctuations between the various possible states; properties that seem to correspond well to the phenomenology of earthquakes. For such systems, where BG statistical mechanics has limited applicability, a generalized framework has been introduced by Tsallis (1988), termed as non-extensive statistical mechanics (NESM). NESM is a generalization of BG statistical mechanics and its main advantage is that it considers all-length scale correlations among the elements of the system, leading to broad distributions with power-law asymptotic behavior and heavy tails. NESM has been widely applied across various non-linear dynamic systems during the last two decades (Tsallis, 2009a and references therein). Consistent results between the theory and observations have established the generalized framework as a powerful tool for describing the complex behavior of physical systems by specifying a priori the relevant microscopic configurations. Given the non-linear dynamics that control the evolution of fragmentation processes and the phenomenology that fault and earthquake populations exhibit, NESM has been considered as the appropriate statistical mechanics framework to describe their 4

22 Chapter 1 Introduction ensemble. In a series of works during the last decade, NESM has been successfully applied to earthquake-related phenomena and other geophysical problems. Applications concern the size distribution of the tectonic plates areas (Vallianatos and Sammonds, 2010), the distribution of fault lengths in Crete (Vallianatos et al., 2011a) and Mars (Vallianatos and Sammonds, 2011; Vallianatos, 2013a), the spatiotemporal and size distributions of acoustic emissions (AE) in rock experiments under triaxial deformation (Vallianatos et al., 2012a), the spatiotemporal properties of seismicity at regional (Abe and Suzuki, 2003; 2005; Darooneh and Dadashinia, 2008; Vallianatos et al., 2012b, Vallianatos et al., 2013; Papadakis et al., 2013) and global scale (Vallianatos and Sammonds, 2013) and the size distribution of earthquakes (Sotolongo-Costa and Posadas, 2004; Silva et al., 2006; Vilar et al., 2007; Darooneh and Mehri, 2010; Telesca and Chen, 2010; Telesca, 2010a; 2010b; 2010c; 2011; 2012; Vallianatos et al., 2013; Papadakis et al., 2013; Vallianatos and Sammonds, 2013). Other applications in rock physics concern laboratory electromagnetic emissions (Vallianatos et al., 2012c), stress-induced electric current fluctuations (Cartwright-Taylor et al., 2014), pressurestimulated currents (Vallianatos and Triantis, 2012; 2013; Stergiopoulos et al., 2013) and the relaxation of depolarization current emissions (Vallianatos et al., 2011b). SOC models that have been used to model the earthquake phenomenology, like the sliderblock model (Burridge and Knoppoff, 1964; Rundle et al., 2003) and the Olami-Feder- Christensen model (Olami et al., 1992), also exhibit properties consistent with NESM (Caruso et al., 2007; Hasumi, 2007; 2009; Zhang et al., 2011). Complex correlations during the preparatory phase of large earthquakes have also been studied using NESM approaches in the size distribution of earthquakes (Telesca, 2010c; Minadakis et al., 2012; Vallianatos et al., 2014; Papadakis et al., 2015) or in electromagnetic emissions from the crust (Kalimeri et al., 2008; Papadimitriou et al., 2008; Contoyiannis and Eftaxias, 2008; Eftaxias, 2010; Potirakis et al., 2011). Other related geophysical applications concern the size distribution of landslides (Chen et al., 2011), rockfalls (Vallianatos, 2013b), risk assessments (Vallianatos, 2009) and geomagnetic reversals (Vallianatos, 2011). In all these studies, it has been shown that NESM is a consistent framework for studying the geodynamic behavior of the Earth s crust, as it is expressed through a plethora of phenomena, such as earthquakes, fragmentation processes or electric current and electromagnetic emissions. 5

23 Chapter 1 Introduction The aim of the present thesis is to study the fundamental properties of fault and earthquake populations in the Corinth Rift (Central Greece) and gain new insights into the physics that govern fault growth and the evolution of earthquake activity in the area by applying the novel approach of NESM. My study on this particular region is motivated by, i) the fact that the Corinth Rift is one of the most seismically active areas in Europe, imposing an increasing seismic risk for the population and infrastructures, ii) the rich phenomenology that characterizes the earthquake activity in the area, such as fluctuating behavior and intermittency, clustering effects both in space and time, strong earthquakes followed by aftershock sequences, frequent earthquake swarms triggered by fluid diffusion at depth and stress diffusion phenomena among others (Latoussakis et al., 1991; Drakatos and Latoussakis, 1996; Rigo et al., 1996; Bernard et al., 1997; Papadopoulos et al., 2000; Hatzfeld et al., 2000; Telesca et al., 2002; Lyon- Caen et al., 2004; Pacchiani and Lyon-Caen, 2010; Bourouis and Cornet, 2009; Karakostas et al., 2012), iii) the active rifting processes that have resulted in the formation of an impressive normal fault system, making the area an ideal natural laboratory for studying the physical processes of fault growth (Armijo et al., 1996; Roberts, 1996; Moretti et al., 2003; Bell et al., 2009; Ford et al., 2013). These properties constitute the Corinth Rift as an ideal study area for testing the applicability of NESM to earthquake physics. In addition, the results of the NESM based analysis are complemented and compared with known empirical scaling relations in geophysics, such as power-laws and exponentials or the G-R scaling relation for the earthquake size distribution. Previous studies on the temporal properties of seismicity have applied NESM on the cumulative distribution of inter-event times (i.e., the time intervals between successive earthquakes) (e.g., Abe and Suzuki, 2005; Vallianatos et al., 2012b; Papadakis et al., 2013). The present approach rather focuses on the probability density function, which is typically used in probabilistic earthquake hazard assessments. This approach is complemented by the multifractal analysis of the earthquake time series, in an attempt to elucidate the local fluctuations, the degree of heterogeneous clustering and the correlations in the temporal evolution of seismicity. Furthermore, the applicability of NESM to earthquake diffusion phenomena is investigated for two case studies of induced seismicity in the Rift, the first being related to stress diffusion caused by the occurrence of the 2010 Efpalion earthquakes (Karakostas et al., 2012) and the second 6

24 Chapter 1 Introduction to pore-pressure diffusion as the triggering mechanism of the 2001 Agios Ioanis earthquake swarm sequence (Pacchiani and Lyon-Caen, 2010). Overall, the present thesis is inspired by the words of Sornette and Werner (2009): the study of the statistical physics of earthquakes remains wide-open, with many significant discoveries to be made. The promise of a holistic approach one that emphasizes the interactions between earthquakes and faults is to be able to neglect some of the exceedingly complicated micro-physics when attempting to understand the large scale patterns of seismicity. The marriage between this conceptual approach, based on the successes of statistical physics and seismology, thus remains a highly important domain of research. 1.2 Thesis Outline The thesis is divided into 8 chapters, which address the applicability of the generalized statistical mechanics framework to earthquake physics in the Corinth Rift, present the analysis and results, discuss the implications that arise regarding fault growth, the dynamic evolution of seismicity and earthquake hazard assessments in the Rift, draw the conclusions and highlight possible directions for future research. Chapter 2 reviews the statistics of fault and earthquake populations and the connections to physical mechanisms that have been proposed in the literature. It then introduces the concept of entropy and the statistical mechanics approach to earthquake physics and provides the physical rationale for the application of generalized statistical mechanics to earthquakerelated phenomena. The associated theory, as it applies to earthquake data, is described and the various related studies on seismicity and faulting are reviewed. The generalized statistical mechanics analysis of fault and earthquake populations in the Corinth Rift and the associated results are given in Chapters 3 to 6. In particular, Chapter 3 introduces the tectonic setting and the deformation patterns in the Corinth Rift and describes the fault dataset used in the analysis. The results of the scaling properties of the fault system, as a function of increasing strain in the Rift, are then discussed in terms of fault growth and fault network evolution. The earthquake data that are used in the thesis, the analysis of the frequency-magnitude distribution of 7

25 Chapter 1 Introduction earthquakes and the related hazard assessments are discussed in Chapter 4. Chapter 5 investigates the temporal properties of the earthquake activity in the Rift, in terms of generalized statistical mechanics and multifractal analysis. Earthquake diffusion phenomena are studied in Chapter 6, for two case studies of triggered seismicity, which have been associated with stress diffusion, the first, and fluid diffusion at depth, the second. Chapter 7 discusses the outcomes of the thesis and the research questions that have been addressed, as well as possible implications that arise for earthquake hazard assessments in the Rift, earthquake physics and possible directions for future research. Finally, Chapter 8 reviews the results of the thesis and draws the main conclusions. 8

26 Chapter 2 Earthquake Statistics and Generalized Statistical Mechanics 2.1 Introduction In the present chapter the empirical scaling properties of fault and earthquake populations are reviewed and the emergence of generalized statistical mechanics in deriving these large-scale properties from the specification of the relevant microscopic states is described. Essentially, the review initiates with the definition of fractals and multifractals that will be met throughout this chapter. The concept of fractals has widely been applied to fracturing phenomena and constitutes one of the principal properties of seismicity. Some of the main theories, which have been developed to provide physical mechanisms for the empirical scaling properties that are observed in seismicity, such as critical point phenomena and self-organized criticality are briefly reviewed. The foundations of statistical mechanics and the concept of entropy as a variational principle for making predictions are then introduced. The rest of the present chapter is dedicated to describe the emergence of the generalized framework, termed as non-extensive statistical mechanics (NESM), as a consistent framework applicable to a class of complex out-of-equilibrium systems with fractal structures and/or long-range

27 Chapter 2 Earthquake Statistics and Generalized Statistical Mechanics interactions. It is exactly these properties that constitute NESM as an appropriate framework for studying fracturing phenomena and seismicity. The optimization of the generalized entropic function for deriving probability distributions of faults and earthquake populations, subject to the constraints of our limited knowledge on the physics of the process, is described and the applications of the theory to related phenomena are thoroughly reviewed. 2.2 The concept of Fractals The concept of fractals has become quite popular in Earth Sciences due to the selfsimilar properties that natural objects, like drainage networks or rock fragments, frequently present. The fractal geometry was introduced by Benoit Mandelbrot in his pioneering 1967 paper on the self-similar structure of the coast of Britain (Mandelbrot, 1967). Scale-invariant objects are thought to be self-similar, where the objects repeat themselves at multiple scales and in all directions (isotropic scale-invariance). If scaleinvariance is anisotropic, where the object scales differently in the various directions (i.e., horizontally and vertically), then the object is termed as self-affine (e.g., Malamud and Turcotte, 1999). Mandelbrot defines a fractal set in strict mathematical terms as a set for which the Hausdorff - Besicovitch dimension strictly exceeds the topological (Euclidean) dimension (Mandelbrot, 1982). The distribution of objects with fractal geometry is described by the power-law distribution, as this is the only distribution that does not include a characteristic length-scale (Mandelbrot, 1982). In geophysics, the concept of fractals has widely been applied, in order to provide quantifiable measures that characterize the complex structure of earthquake activity and fault networks (e.g., Smalley et al., 1987; Hirata, 1989). Fractality has been associated with the power-law size distributions that fragmented materials and earthquakes present, which mathematically takes the form (Turcotte, 1997): N() l Cl D, (2.1) 10

28 Chapter 2 Earthquake Statistics and Generalized Statistical Mechanics where N(l) is the number of objects (i.e., fragments) with characteristic linear dimension greater than l, C is a proportionality constant and D is the power-law exponent, known as the fractal dimension that takes non-integer values in the space E- 1 < D < E, where E is the Euclidean dimension (Turcotte, 1997). Various methods can be used for determining the fractal dimension. Two commonly used are the mass density and the box-counting methods that are illustrated in Fig.2.1 for the case of a fractured medium (from Bonnett et al., 2001). In these methods, a number of circles or boxes, of an appropriate radius or size r, are used to cover the fractal object in the medium. The mass dimension can be estimated as the total fracture length (L) included in circles of varying sizes, with radius r (Fig. 2.1a). The total length L should vary with radius r as L r D m, where Dm is the mass dimension. In the boxcounting method, the number of boxes (N) of size r that intersects the fractal object is counted for various sizes of r (Fig. 2.1b) and should vary as fractal dimension (detailed description from Bonnett et al., 2001). N r D, where D is the Figure 2.1: Fig. 3 from Bonnett et al. (2001). The classic techniques used to estimate the fractal dimension in a fractured medium: a) the mass dimension, where the total length of fractures lying in a circle of radius r is calculated; b) the box-counting method, where the medium of size Λ is covered by a regular mesh grid of size r (two grids are shown, with different mesh sizes the boxes with fractures inside them are shaded and empty boxes are open); c) the multifractal analysis derived from the boxcounting method, where each box is weighted by the total length of fractures that is included in it darker boxes have greater fracture lengths. 11

29 Chapter 2 Earthquake Statistics and Generalized Statistical Mechanics In nature though, most fractals in complex systems are expected to be heterogeneous (Mandelbrot, 1989). In this case, rather than simple fractal geometry, more complex scaling relations are needed to quantify the complex structure of the system and the application of multifractal tools becomes essential. Multifractal tools are primarily used to quantify the degree of heterogeneous clustering and intermittency in the fractal set of interest (Main, 1996). Usually, a multifractal set involves a range of scales with different individual scaling exponents (Sornette, 2006). The standard method for defining the multifractal structure of a set is to calculate the moments of order q of a probability distribution pi(r). The probability distribution pi(r) is defined as the sum of the object of interest, within an array of boxes of size r (Fig.2.1c). In the case of the fractured medium illustrated in Fig.2.1, pi(r) is defined as (Bonnett et al., 2001): Li () r, (2.2) L() r where n is the total number of boxes, Li(r) is the total fracture length in each box and the sum in the denominator gives the total cumulative length of all fractures over all boxes. The moments of order q of pi(r) are then defined as: If scaling holds for various q, then Mq(r) will scale with r as: q 1 D. (2.3) q M ( r) ~ r, (2.4) q where Dq can be considered as the generalized fractal dimension (Feder, 1988). For the various q, the range of Dq is then estimated. For a monofractal series, Dq will have the same value for the various values of q. By definition, for q=0, Dq=D, which is also known as the capacity dimension. p () r i n n1 n M q( r) pi( r) n1 For a multifractal set, the singularity spectrum f(a) can also be estimated as the fractal dimension of singularities with strength a (Feder, 1988). In practice, f(a) denotes the number N(r) of boxes having similar local scaling that is characterized by the same exponent a (Sornette, 2006). The generalized fractal dimension Dq and the singularity spectrum f(a) are considered as fundamental characteristics of a multifractal set. The i q 12

30 Chapter 2 Earthquake Statistics and Generalized Statistical Mechanics calculation of Dq and f(a) is described analytically in Chapter 5, where multifractal analysis is applied to the earthquake time series in the Corinth Rift. 2.3 Scaling in Fault and Earthquake Populations Scaling properties of fault attributes Faults are key components of the Earth system as they affect the evolution of the Earth s lithosphere and surface and are the sites of large earthquakes (e.g., Beroza et al. 2005). Fault populations exhibit scaling relations, which describe the statistics of the frequency-size distribution of fault attributes, such as trace-lengths and displacements, the relation between the length of a fault and its displacement and the spatial distribution of faults (Cowie, 1998a). In numerous studies the scaling properties of fault populations are thought to be scale-invariant over several orders of magnitude, so that the cumulative distribution of fault trace-lengths exhibits a power-law distribution of the form (King, 1983; Main et al., 1990; Sornette and Davy, 1991; Cladouhos and Marrett, 1996; Main, 1996; Turcotte, 1997): N( L) AL D, (2.5) where N(>L) is the number of faults with length greater than L, A is a constant and D is the scaling exponent. Scale-invariance is supported by the similar pattern that the traces of fault populations exhibit across a wide range of scales that may vary from the laboratory scale to plate-boundary faults (Fig. 2.2) (Main et al., 1990; Main, 1996; Sornette, 2006). However, even if scale-invariance does hold, it is restricted to a finite range of scales that are related to the finite size of the fractured system. The other argument that favors scale-invariance in fault populations is the frequency-magnitude distribution of earthquakes that is discussed in the next section ( 2.3.2). The assumption of scale-invariance in fault systems led various researchers to extrapolate the observable scaling properties to scales below the limit of resolution and estimate the contribution of small faults to the total strain (Scholz and Cowie, 1990; Walsh et al., 1991; Marrett and Allmendinger, 1991). 13

31 Chapter 2 Earthquake Statistics and Generalized Statistical Mechanics The self-similar structure across a wide range of scales (Fig. 2.2) and the power-law frequency-size distribution (Eq. 2.5) have been considered as strong indications of fractal geometries in fracture and fault populations (Turcotte, 1997). Fractal geometry has been used as a first-order approximation to describe the complex patterns of fault systems (Sornette, 2006) and physical models on fractal fault growth have been developed (Sornette and Davy, 1991; Cowie et al., 1993a; Miltenberger et al., 1993; Sornette et al., 1994). Cowie et al. (1993a) used a two-dimensional numerical model to show how fractal fault patterns emerge as the result of correlations in the fault network, induced from both short- and long-range elastic interactions in an heterogeneous material (random heterogeneity) that is deformed under constant velocity (see also Miltenberger et al., 1993; Sornette et al., 1994). Cowie et al. (1995) used a similar model and the box-counting method (Fig.2.1c) to show that as deformation progresses and strain localizes on a few large faults, a multifractal structure emerges as the superposition of a fractal distribution of fault displacements onto a fractal fault pattern. Fractal geometry and the box-counting method have been used extensively to describe the spatial distribution of faults in a region (Fig.2.1b) (e.g., Yielding et al., 1996). If scaling holds (see 2.2), then a fractal dimension of D < 2 indicates localized deformation, while for D = 2 the fault pattern is more homogeneous over the map area (Cowie, 1998a). In other natural fault systems though, the exponential function has been found to best describe N(>L) (Cowie, 1998a). The exponential function in this case is expressed as: N( L) Aexp( L L ), (2.6) 0 where A is a constant and L0 a characteristic length scale. Cowie et al. (1993b) studied fault networks in mid-ocean ridges and concluded that the frequency-size distribution of fault trace-lengths is better described by the exponential function. Gupta and Scholz (2000a) found this type of scaling for the fault network in the high-strain zone in the Afar Rift and Vétel et al. (2005) considered that fault trace-length scaling in the Turcana Rift (Northern Kenya) is more consistent with the exponential function. Furthermore, numerical models (Spyropoulos et al., 2001; Hardacre and Cowie, 2003) and laboratory experiments (Spyropoulos et al., 1999; Ackermann et al., 2001) indicated the transition from power-law to exponential scaling as a function of increasing strain. This transition 14

32 Chapter 2 Earthquake Statistics and Generalized Statistical Mechanics is further discussed in Chapter 3, where the scaling properties of the fault population in the Corinth Rift are studied. Figure 2.2: Figure 1 from Main (1996). Traces of fault populations in a range of scales, from laboratory experiments (Fig. 2.1c and d) to plate boundary faults, like the San Andreas Fault system (Fig. 2.1a) (Original data from Tchalenko [1970], Howard et al. [1978] and Shaw and Gartner [1986]). Other functions that have been used in the literature to describe the frequency-size distribution of faults are the lognormal and gamma functions (reviewed in Bonnett et al., 2001). The distribution n(w) of a fault property w (trace-length, displacement), according to the various scaling functions, is shown in Fig. 2.3 (after Bonnett et al., 15

33 Chapter 2 Earthquake Statistics and Generalized Statistical Mechanics 2001). Davy (1993) compared the performance of these functions for fitting the frequency-size distribution of the San Andreas Fault system. Davy concluded that the gamma function best fits the observed distribution and argues that although the lognormal function also provides a good fit, its physical significance is questionable since it presents a decrease in the density of small fault lengths (Fig. 2.3). This contradicts geological evidence, which indicates a large number of small faults (Davy, 1993). Figure 2.3: Fig. 1 from Bonnett et al. (2001). The plot illustrates the various functions (power-law, exponential, gamma and lognormal) that are most frequently used to describe the distribution n(w) of a fault property w. The other scaling relation that has been widely studied is the one between the tracelengths of faults (L) and their displacements (D). Several studies (reviewed by Cowie, 1998a) indicate that fault lengths and displacements are strongly correlated. Some researchers have proposed that the relationship between D and L is the power-law of the form D=cL n, with exponents of n=1.5 or n=2 (Walsh and Watterson, 1987; Marrett and Allmendinger, 1991; Gillespie et al., 1992). By compiling a global fault data set, Schlische et al. (1996) showed that the relationship between D and L is almost linear 16

34 Chapter 2 Earthquake Statistics and Generalized Statistical Mechanics over several orders of magnitude (Fig. 2.4). Schlische et al. also showed that the average D/L ratio (c) is 0.01, which implies that the maximum displacement of a fault is 1% of its length (Fig. 2.3). The strong correlation between D and L implies that the distribution of fault displacements follows similar scaling functions as L. Indeed, the scaling function that is most frequently reported in the literature is power-law (Childs et al., 1990; Pickering et al., 1996), although exponential scaling has also been found in the distribution of fault displacements (Dauteuil and Brun, 1996). Figure 2.4: a) Displacement (D) vs. length (L) on double logarithmic axes, for various published fault datasets. A family of curves, for various c and n (see text), are also shown. Abbreviations: N-normal faults, T-thrust faults, SS-strike-slip faults. b) Size distribution of fault trace-lengths for the Solite dataset and the power-law fit (linear curve) to the data (from Schlische et al., 1996) Frequency-size distribution of earthquakes 17

35 Chapter 2 Earthquake Statistics and Generalized Statistical Mechanics Since the beginning of 20 th century and the early years of modern seismology, it became apparent that small earthquakes are considerably more frequent than larger ones (e.g., Utsu, 1999). In 1944, Gutenberg and Richter established empirically a scaling relation that describes the frequency of earthquakes in California (Gutenberg and Richter, 1944). This is the well-known Gutenberg-Richter (G-R) scaling relation: log N( M) a bm, (2.7) where log is the logarithm to the base of 10, N(>M) is the number of earthquakes with magnitude M greater than M in a given region and a, b are positive constants. The constant a represents the regional level of seismicity and b, known as the seismic b- value, is the slope of the distribution that estimates the proportion of small events to large events. Earthquake magnitude is estimated from the seismic waves recorded at seismographic stations and includes a variety of measures such as moment (Mw), local (ML), surface-wave (Ms) and body-wave (mb) magnitude, depending on the part of the seismic wave that is used to determine its value (Lay and Wallace, 1995). The G-R empirical relation is valid for various earthquake sequences, scales and tectonic environments. Regional variations in b-values have been found (Schorlemmer et al., 2005), but generally b takes values close to 1 (Frohlich and Davis, 1993; Kagan, 1999). Eq.(2.7) can be alternatively written as: ( ) 10 a N M bm, (2.8) which expresses a power-law dependence between the number of earthquakes N with magnitude greater than M and M. The seismic energy (E) can be related to the earthquake magnitude (M) according to the relationship: log E cm d, (2.9) with a global average of c=1.5 (Kanamori, 1978; Hanks and Kanamori, 1979). Eq. (2.8) can then be written in terms of the seismic energy as: N( E) ~ E 1, (2.10) with β = 2 b. The latter expression has been considered as the indication of scale- 3 invariance in the earthquake magnitude distribution or the distribution of seismic 18

36 Chapter 2 Earthquake Statistics and Generalized Statistical Mechanics energies (Main, 1996; Turcotte, 1997). Furthermore, the dissipated seismic energy E, in terms of the seismic moment Mo, is related to the surface area of a fault with the relationship Mo = μδa, where μ is the rigidity of the material, A the surface area of the fault break and Δ the average slip during the earthquake (Kanamori, 1978). According to the previous definition of Mo and Eq.2.10, the number of earthquakes N(>A) with rupture areas greater than A has a power-law dependence on the area A (Fig.2.5) (Turcotte, 1997). According to the definition of a fractal set given in Eq.2.1 and for A ~ l 2 and c = 1.5, Turcotte (1997) showed that the fractal dimension D is simply twice the b-value, D = 2b. The latter indicates that the empirical G-R scaling relation (Eq.2.7) is equivalent to a fractal distribution (Aki, 1981). Figure 2.5: Fig. 4.1 from Turcotte (1997). Number of earthquakes per year N with magnitudes greater than m as a function of m, on global scale (solid line). The square root of the rupture area A is also given. The cumulative distribution of moment magnitudes m is from the Harvard Centroid Moment Tensor Catalogue, for the period January 1977 to June The dashed line represents the G-R scaling relation (Eq. 2.3) for the values of b=1.11 and a= yr

37 Chapter 2 Earthquake Statistics and Generalized Statistical Mechanics The latter relation, which indicates a positive correlation between the fractal dimension D and the b-value, has been debated in several studies (e.g., Hirata, 1989; Wyss et al., 2004; Chen et al., 2006). Wyss et al. (2004) found that the latter relation only holds for locked fault segments of the San Andreas Fault system near Parkfield and Chen et al. (2006) found results consistent to the previous relation for the aftershock sequence of the 1999 Chi-Chi, Taiwan, earthquake. However, other researchers found a negative correlation between the fractal dimension and the b-value in the Tohoku region, Japan (Hirata, 1989), in the João Câmara region, Brazil (Henderson et al., 1994) and in the Geysers geothermal area, California (Henderson et al., 1999). The latter studies suggest that the relation between the fractal dimension and the b-value is not simple, but it might change according to the regional distribution of stress (Singh et al., 2008). Figure 2.6: Fig. 1 from Kagan and Jackson (2013). Cumulative number of earthquakes with moment magnitudes (m) equal to or greater than m as a function of m (solid line), for the shallow earthquakes in the Harvard Centroid Moment Tensor Catalogue, in the Japan-Kurile-Kamchatka zone (Flinn-Engdahl zone 19). The total number of events is 425 for the threshold magnitude of mth=5.8. The G-R scaling relation is shown as the red dotted line, with β= The dashed lines represent the modified G-R scaling relation (MGR) for two corner magnitudes, mc s =8.7 (left line, cyan) and mc m =9.4 (right line, blue). 20

38 Chapter 2 Earthquake Statistics and Generalized Statistical Mechanics The extrapolation of the G-R scaling relation to larger earthquake magnitudes is uncertain due to finite energy release rates that can be related to the stress regime, the thickness of the seismogenic zone and the fault zone geometry (Bell et al., 2013). An upper bound or a taper may appear in the frequency-magnitude distribution that limits the maximum expected earthquake magnitude. This upper bound has been modeled by modifying the G-R scaling relation to include an exponential tail or a taper (Kagan and Jackson, 2013; Bell et al., 2013). The modified G-R scaling relation (MGR) then takes the form of a gamma distribution: N( E) ~ E exp( E m c ), (2.11) which exhibits an exponential tail for a characteristic or corner seismic moment mc (Bell et al., 2013). The G-R and MGR scaling relations that model the shallow ( 70 km) earthquake activity in the Japan-Kurile-Kamchatka zone for the period , are shown in Fig.2.6 (from Kagan and Jackson, 2013) Spatiotemporal scaling properties of seismicity Earthquakes are considered as a complex spatiotemporal phenomenon due to high variability in the time and space of their occurrence (e.g., Telesca et al., 2003). The most prominent feature in the spatiotemporal patterns of seismicity is the presence of clustering. Spatial clustering is exemplified by the concentration of seismicity along the tectonic plate boundaries and regional fault networks (e.g., Scholz, 2002; Utsu, 2002). Temporal clustering is best observed in aftershock sequences, where a significant increase in seismicity rate occurs immediately after the occurrence of a strong earthquake (e.g., Utsu et al., 1995). The aftershock production rate n( t) dn( t) dt (where N(t) is the number of earthquakes in time t after the major event) has been found empirically to decay as a power-law with time t according to the modified Omori formula (Omori, 1894; Utsu et al., 1995): n() t K t c p, (2.12) where K and c are constants and p is the power-law exponent. This expresses a shortterm clustering effect that can be seen in almost every seismic catalogue (Vere-Jones, 21

39 Chapter 2 Earthquake Statistics and Generalized Statistical Mechanics 1970). The background activity of main earthquakes in a seismic region is often considered as uncorrelated and statistically independent in time. Models that express randomness, like the general Poisson model, have been used to describe the temporal properties of this background seismicity (Gardner and Knopoff, 1974; Console and Murru, 2001). In other cases, long-term clustering effects and power-law scaling have been found to characterize long inter-event times (Kagan and Jackson, 1991; Mega et al., 2003). Livina et al. (2005) studied one global and five regional earthquake catalogues and found statistical dependence and clustering effects in both short and long inter-event times that according to the authors is indicative of memory in the earthquake generation process. Fractal geometry has in many cases been used to characterize the degree of clustering in seismic sequences and laboratory acoustic emissions (Kagan and Knoppof, 1980; Smalley et al., 1987; Hirata et al., 1987; Hirata, 1989). Spatiotemporal variability and heterogeneous clustering effects are quantitatively consistent with multifractal structures (e.g., Geilikman et al., 1990; Telesca and Lapenna, 2006). The multifractal structure of inter-event time series is evident in various earthquake sequences (Godano and Caruso, 1995; Godano et al., 1997; Telesca et al., 2002; Enescu et al., 2006; Telesca and Lapenna, 2006). Multifractal analysis has been used to provide second-order approximations to the correlations of seismicity and to enlighten the local clustering effects. In particular, properties like non-stationarity and intermittency are common in earthquake time series, such that the clustering degree varies with time. These variations can be studied by using multifractal approaches, which can then provide an appropriate tool for identifying the dynamical changes of seismicity. This attribute has in fact been used in various studies to detect possible temporal changes in the multifractal dimension prior to large earthquakes (Hirabayashi et al., 1992; Nakaya and Hashimoto, 2002; Kiyashchenko et al., 2003). In addition, many studies and discussions since 2002 have focused on whether interevent times follow particular distributions or exhibit universality in their scaling properties. The discussion was stimulated by the work of Bak et al. (2002), who proposed a unified scaling law for earthquakes that considers the G-R frequency-size distribution, the fractal distribution of epicenters and the Omori scaling of aftershocks. By dividing California into spatial cells of various sizes L 2 and by calculating inter- 22

40 Chapter 2 Earthquake Statistics and Generalized Statistical Mechanics event times τ in each cell for various cut-off magnitudes, the authors showed that the resulting inter-event time distributions fall onto the same curve (Fig. 2.7), if inter-event times τ (x-axis) are rescaled according to the expression and the probabilities PS,L(τ) (y-axis) according to τ a, where a is associated with the Omori exponent (Eq.2.12). According to the authors, the expression is a measure of the average number of earthquakes with magnitude greater than log (S) that occur in the time interval τ, b is the G-R b-value and df the spatial fractal dimension.this curve that expresses the unified law for earthquakes has the form: S b L d f S b L d f d f a b PSL, ( ) f ( S L ). (2.13) The latter expression (Eq. 2.13) exhibits a power-law regime for over 8 orders of magnitude (linear part in Fig.2.7) and a rapidly decaying regime, which is consistent with an exponential decaying function. The latter implies uncorrelated earthquakes for large enough inter-event distances and/or times, while the power-law regime indicates correlated earthquakes within a window of time τ and spatial size L 2 (Bak et al., 2002). The unified scaling law of Bak et al. (2002) was further studied by Corral (2003). Corral (2003) used the same data as Bak et al. (2002) to propose that inter-event time distributions f(τ) depend only on the seismicity rate and obtained the collapse of distributions onto the same curve by multiplying inter-event times and dividing the probabilities by the seismicity rate (Fig. 2.8a). When mixing different seismicity rates, Corral (2003) derived a distribution that exhibits two power-law regions (Fig. 2.8b), while for stationary periods, where the seismicity rate does not fluctuate, the distributions can be fitted by the gamma function of the form (Fig. 2.8a): 1 f( ) C exp. (2.14) In a later work, Corral (2004) used a wide variety of earthquake catalogues to show that inter-event time distributions for stationary periods fall onto the same curve and are well described by the gamma function (Eq. 2.14) for the values of γ = 0.67 ± 0.05, δ = 0.98 ± 0.05, τ0 = 1.58 ± 0.15 and C = 0.50 ± 0.10, if the distributions are rescaled according to the seismicity rate. He derived the same result for non-stationary aftershock sequences in California by rescaling in this case with the time varying event 0 23

41 Chapter 2 Earthquake Statistics and Generalized Statistical Mechanics rate. These results motivated Corral (2004) to claim universality in the temporal occurrence of earthquakes. Figure 2.7: Fig. 4 from Bak et al. (2002). The probability distribution PS,L(T) of interevent times T (for T > 38 sec) for various threshold magnitudes and spatial scales of California seismicity, rescaled as shown in the axes. The Omori exponent a = 1, the G- R b-value of b = 1 and the spatial fractal dimension of df =1.2 have been used in order to collapse the observed distributions onto the same curve. Universality in the temporal occurrence of earthquakes has been tested and questioned in several studies (Davidsen and Goltz, 2004; Molchan, 2005; Hainzl et al., 2006; Saichev and Sornette, 2006; 2007; Touati et al., 2009). Davidsen and Goltz (2004) analyzed data from California and Iceland and found an additional power-law regime for the shortest inter-event times, although their data did not fall onto one universal curve at large inter-event times. Molchan (2005) proved mathematically that if such a universal law exists it should be exponential, in strong contradiction to observations. Saichev and Sornette (2006; 2007) used the epidemic-type aftershock sequence (ETAS) model to derive the shape of inter-event time distribution and found that this is only 24

42 Chapter 2 Earthquake Statistics and Generalized Statistical Mechanics a) b) Figure 2.8: a) Fig. 2 from Corral (2003). Probability distributions Dxyt(τ) of inter-event times τ (τ 38 sec) in California, for several stationary periods and different spatial scales, after rescaling with the seismicity rate r. b) Fig. 4 from Corral (2003). Probability distributions D(τ) of inter-event times τ (τ 38 sec) for different spatial scales L and threshold magnitudes mc, with df = 1.6 and b = The straight lines indicate the double power-law behavior, with exponents 0.9 and 2.2. approximately universal and of a gamma form. Hainzl et al. (2006) analyzed both real and synthetic catalogues produced with the ETAS model and concluded that the shape 25

43 Chapter 2 Earthquake Statistics and Generalized Statistical Mechanics of the distribution is strongly dependent on the rate of background and aftershock activity. According to the authors, a Poissonian background activity combined with triggered aftershocks that scale according to the modified Omori relation are sufficient to reproduce the observed scaling. Touati et al. (2009) used a similar approach to show that the distribution exhibits a bimodal character of gamma-distributed correlated aftershocks and exponentially distributed uncorrelated activity at short and long interevent times respectively. Touati et al. showed that this distribution is not universal and is strongly affected by the size of the area, where larger areas exhibit a higher initiation rate of uncorrelated events and smaller areas exhibit highly non-random inter-event times. Scaling has also been shown in the distribution of spatial distances between successive earthquakes (inter-event distances). Davidsen and Paczuski (2005) found power-law scaling with an exponent of 0.6 at ranges of km in the inter-event distance distribution of California seismicity, regardless of the threshold magnitude, which might imply a dynamic triggering effect at distances far beyond the aftershock zone scaling (e.g., Kagan, 2002). Corral (2006) studied the inter-event distance distribution for stationary periods in California and global seismicity and found an additional power-law regime with a smaller exponent at large inter-event distances indicating bimodality, irrespective of the threshold magnitude. The crossover point between the two power-law regimes is different in the two catalogues (200 km for global and 15 km for California seismicity), implying a scale-dependent maximum triggering distance of correlated aftershocks at shorter distances and uncorrelated events at longer distances (Corral, 2006) Stochastic models of seismicity Various stochastic 2 models of seismicity have been developed in order to model the phenomenology of earthquake occurrence and to provide the basis of probabilistic earthquake hazard assessments. In these models, earthquakes are considered as a pointprocess in time and space, marked with the magnitude of the event. A conditional 2 The term stochastic refers to a random object that takes values from a given subset to model the evolution of the system in time and space. 26

44 Chapter 2 Earthquake Statistics and Generalized Statistical Mechanics intensity function 3 λ(t Ht) is used to describe the point process and estimate the probability rates as a function of time and the history of the process. Stationary Poisson models (Gardner and Knopoff, 1974) are commonly used to form the basis of any probabilistic earthquake hazard assessment. The stationary Poisson model considers the background earthquake activity in a region, where aftershock clusters have been removed, as an uncorrelated Poisson process with constant seismicity rate over time. A Poisson process considers uncorrelated and identically distributed inter-event times that are drawn from an exponential distribution until the time of the next large event. In this case, the conditional intensity function λ is constant and independent of the time or history. Other models that are frequently used in earthquake hazard assessments, like the non-stationary Poisson model and the renewal model, consist of generalizations of the Poisson process (reviewed in Zhuang et al. 2012). Another well-known stochastic model is the epidemic type aftershock sequence (ETAS) model that incorporates some of the empirical scaling relations of seismicity, like the G-R relation and the modified Omori formula (Ogata, 1988). The basic idea of the ETAS model is that each earthquake triggers its own aftershocks, which in turn can trigger their own events, resulting in a superposition of cascades of triggered events that cluster in space and time (Ogata, 1988; Helmstetter and Sornette, 2002a). The conditional intensity function λ(t Ht) is given by: t H t K exp a m m t t c 0 ti t i 0 p, (2.15) where μ (earthquakes per unit time) represents a stationary Poisson seeding rate of background activity and K0 the productivity rate. The coefficient a represents the efficiency of an earthquake of magnitude m to trigger aftershocks, m0 is the lower threshold magnitude and c and p are the parameters of the modified Omori formula (Eq. 2.12). A similar stochastic model to ETAS that instead of the productivity rate K0 uses 3 H t represents the history of the process up to time t, but not including t (Zhuang et al., 2012). 27

45 Chapter 2 Earthquake Statistics and Generalized Statistical Mechanics Bath s relation 4 for magnitude-dependent aftershock production, is the branching aftershock sequences (BASS) model (Turcotte et al., 2007a). Although stochastic models can efficiently reproduce some of the empirical patterns of seismicity, such as the G-R scaling of earthquake magnitudes, the modified Omori aftershock decay rate or Bath s relation and the spatial diffusion of aftershocks (e.g., Helmstetter and Sornette, 2002a), the physics that govern these processes are missing from the models. In contrast, the statistical physics approach to seismicity strives to derive the statistical models and empirical observations from the specification of the microscopic laws of friction, rupture, rock-water interactions, chemical reactions and so on, at the microscopic scale (Sornette and Werner, 2009). 2.4 Critical phenomena and Self-Organized Criticality The universal validity of power-law scaling and fractal or multifractal structures in seismicity has led to the development of a variety of physical models of seismogenesis as a critical point phenomenon undergoing a phase-transition (see the review from Sornette, 2006). In the context of statistical physics, the critical point is associated with power-law scaling and strong correlations acting at all scales in the system (long-range correlations); properties that are produced as the system approaches the critical point in the order-disorder phase transition (Bruce and Wallace, 1989; Main, 1996; Sornette, 2006). A characteristic example is the liquid (more ordered state) gas (more disordered state) phase transition where the two phases coexist 5 at a specific critical point, characterized by a critical temperature Tc and pressure Pc. At this critical point, correlations of the liquid-gas molecules emerge at all scales, characterized by powerlaw scaling (Bruce and Wallace, 1989). Another example is magnets (e.g., Ising models) where, below the critical temperature (Curie temperature), a more organized 4 Bath s relation describes the size of the largest expected aftershock with respect to the main shock that is thought to be constant. Generally, it is considered to be 1.2 magnitude-scales smaller (Bath, 1965). 5 At a temperature-pressure phase diagram, the critical point is the end point of the liquid-gas boundary line. Beyond the critical point, the system enters a supercritical state, where the two phases are indistinguishable. 28

46 Chapter 2 Earthquake Statistics and Generalized Statistical Mechanics state, with power-law size distribution and fractal scaling (ferromagnetic), emerges from random and chaotic (paramagnetic) behavior (Bruce and Wallace, 1989). Such critical behavior emerges as repeated long-range interactions between the microscopic elements of the system lead to a macroscopic self-similar state (Sornette, 2006). Systems that present critical behavior and similar scaling properties are thought to belong to the same universality class (Bruce and Wallace, 1989). In the previous examples, the system is driven to the critical point by the precise tuning of some external variables, such as the temperature or pressure. However, in other complex physical systems there is no external tuning and the system organizes itself to the critical point. Such mechanism is termed as self-organized criticality (SOC), introduced by Bak et al. (1987). Bak et al. (1988) used a cellular automaton model 6 of a sand pile that is supplied with new grains at a constant rate, to illustrate the SOC mechanism. When the sand pile sufficiently builds up and the pile reaches some critical angle, new added grains cause avalanches of all sizes that obey a power-law size distribution (Bak et al., 1988). The dynamic evolution of the sand pile cannot reach equilibrium, but instead remains in a statistically stationary but metastable state, which is characterized by power-law size distributions and fractal geometries (Bak et al., 1988; Bak, 1996; Main, 1996; Sornette, 2006). The fundamental properties of a particular class of out-of-equilibrium systems that exhibit SOC are (Sornette, 2006): 1) slow driving rates, 2) highly non-linear behavior, 3) sensitivity to small changes that could trigger large events, 4) globally stationary statistical properties characterized by power-law distributions and fractal geometries (including long-range correlations). The strikingly similar properties that earthquakes exhibit led various workers to consider the Earth s crust as a SOC system (Bak and Tang, 1989; Sornette and Sornette, 1989; Ito and Matsuzaki, 1990). The deformation rates of the crust involve a wide range of time scales that vary from cm/yr, at the borders of tectonic plates, to m/s during brittle failure and propagation of rupture. An earthquake occurs when the slowly accumulated stress in the crust reaches some critical threshold of failure, causing stress drops that are relatively small in comparison to the regional tectonic stress levels (Abercombie, 6 Cellular automata are mathematical models consisting of a grid of elements. These models are used in various disciplines to study the complex dynamic behavior of natural systems (e.g., Wolfram, 1984). 29

47 Chapter 2 Earthquake Statistics and Generalized Statistical Mechanics 1995). Even small stress perturbations are known to trigger earthquakes, even at relatively large distances (e.g., Stein et al., 1994; Gomberg and Davis, 1996), reflecting criticality and long-range interactions in the earthquake generation process. Additionally, earthquake sizes scale according to the G-R scaling relation that resembles a power-law size distribution ( 2.3.2). These properties make the earthquake phenomenon one of the most relevant natural paradigms of SOC. A family of slider-blocks has been used to model the behavior of seismogenic faults and their relevance to SOC (e.g., Rundle et al., 2003). This type of model has been introduced by Burridge and Knopoff (1967) and typically consists of a set of blocks that are connected to each other via springs (Fig.2.9). The blocks are pulled over a surface at a constant velocity by a driver plate, attached via springs (Fig.2.9). When the spring force reaches some critical level due to the motion of the driving plate, the blocks slip, representing the slip on faults due to the motion of tectonic plates. The blocks interact with their neighbors, so that slip on one block can trigger the propagation of slip across several blocks. Although simplistic, the slider-block model can reproduce some of the empirical properties of seismicity. Its numerical implementation, using the cellular automaton approach, exhibits critical behavior, similar to SOC, with a frequency-size distribution of slips similar to the G-R scaling relation (Bak and Tang, 1989; Carlson and Langer, 1989; Ito and Matsuzaki, 1990; Olami et al., 1992). By changing the model parameters, such as the spring stiffness or the plate-driving velocity and the friction coefficients, various different behaviors can be observed, indicating some dependence on the initial tuning of the model that does not meet the demands of true SOC (Rundle and Klein, 1993; Main, 1996). Furthermore, Huang and Turcotte (1990) showed that when the symmetry in the model is broken (i.e., unequal masses), the slider-blocks exhibit deterministic chaos 7. This result implies that earthquakes might exhibit chaotic behavior, similar to the behavior of the atmosphere or oceans (Turcotte, 1997; Rundle et al., 2003). In this case, absolute prediction of an earthquake in a deterministic sense is not possible; only probabilistic approaches can be used for describing upcoming earthquake events (Geller et al., 1997). Although there is general agreement that this type of model can be used to study the dynamical behavior of fault systems, in reality slider-blocks represent crude approximations of natural fault systems 7 Deterministic chaos refers to chaos theory that was developed to describe the dynamic behavior of systems with high sensitivity to the initial conditions. 30

48 Chapter 2 Earthquake Statistics and Generalized Statistical Mechanics and the complex interactions that occur on a wide range of scales. Beyond this, various mechanisms can be used to produce power-laws and SOC (e.g., chapter 14 in Sornette, 2006) and this property alone does not make them an appropriate model for earthquakes (Kagan, 1994). Figure 2.9: Graphic representation of the slider-block model. The blocks with mass m are interconnected via springs (spring constant kc) and are pulled across a surface by a driver plate at constant velocity V. Each block is connected to the driver plate via springs of spring constant kl (from Rundle et al., 2003). Characterization of the earthquake generation process as a critical phenomenon has been used extensively as the physical basis for the earthquake phenomenology. Furthermore, this hypothesis presents some important implications for earthquake hazard assessments (Main, 1995; 1996; Rundle et al., 2003; Sornette and Werner, 2009). Within the SOC context, the Earth s crust self-organizes into a stationary critical or near-critical state, with intermittent fluctuations of individual earthquakes of powerlaw size distributions that correspond to the G-R scaling relation ( 2.3.2). Earthquakes cluster on a fractal or multifractal system of faults ( 2.3.1) and long-range spatial and temporal correlations emerge as the result of elastic stress interactions. In a critically stressed crust, even small stress perturbations are sufficient to trigger earthquakes, consistent with the SOC hypothesis (e.g., Grasso and Sornette, 1998). All these properties contradict the Poisson model that is frequently used as the basis of probabilistic earthquake hazard assessments ( 2.3.4). Furthermore, the regional 31

49 Chapter 2 Earthquake Statistics and Generalized Statistical Mechanics variations of the power-law exponent (b-value) may indicate that the crust is globally in a state below the critical point, which has been termed as sub-criticality (Al-Kindy and Main, 2003). However, Jaume and Sykes (1999) and Sammis and Sornette (2002) suggest that criticality can be reached locally on a fault system prior to a large earthquake. The fault system then moves away from criticality until long-range stress interactions build up again. This hypothesis explains observations of accelerating moment release (Bowman et al., 1998), where an increase of intermediate-size events may occur prior to a large event (see also Zoller et al., 2001). Although accelerating moment release has been observed in various earthquake sequences, this phenomenon is not universal, in the sense that it does not occur systematically prior to large events and thus cannot be used for earthquake prediction (e.g, Turcotte et al., 2007b). 2.5 Statistical Mechanics and Information Theory Foundations of Statistical Mechanics Statistical mechanics, together with the associated concept of entropy, constitutes one of the pillars of contemporary physics that establishes the remarkable relation between the microscopic states of a system and its macroscopic description. Originally, entropy was used in thermodynamics in the mid-1800 s, in an effort to understand the kinetic theory of gases and to describe their macroscopic behavior. At that time, this had important implications in the efficiency of heat engines on producing work and consequently to the industrial revolution. In 1865, Rudolf Clausius ( ) extended the work of Sadi Carnot ( ) on heat engines and coined the term entropy (S) to describe the unique state of a system in thermal equilibrium 8 : dq ds, (2.16) T 8 Thermal equilibrium is the state, as it is specified by macroscopic variables such as temperature or pressure, in which an isolated system evolves to through irreversible processes. At this state no further physical or chemical changes occur with time (e.g., Kondepudi and Prigogine, 1998). 32

50 Chapter 2 Earthquake Statistics and Generalized Statistical Mechanics where ds is the change in entropy, dq is the heat exchange with the exterior (dq > 0 if heat is absorbed by the system and dq < 0 if it is discarded) and T is the temperature. Entropy S depends only on the initial and final states and in the ideal case of a fully reversible cycle the change in entropy is zero 9. Eq.(2.16) states that for a reversible heat flow Q under constant temperature, the change in entropy is Q/T. In the more realistic case of an irreversible cycle, a fraction of heat (Q) is converted to work and the total change of entropy is positive (ds > 0). These limitations on converting heat to work led to the seminal second law of thermodynamics that can be stated as the sum of entropy changes of a system with its surroundings can never decrease 10 (see the analytic review on the derivation of the second law by Kondepudi and Prigogine, 1998). The thermodynamic foundation of entropy is entirely macroscopic and can be defined in terms of few parameters, such as the volume, temperature or pressure. However, the mechanical description of a system, in terms of the microscopic state of particles and their behavior, would require a huge number of variables (~6 x for a gas). A solution to this problem is the statistical approach, where suitable averages of the properties of individual particles can be used to determine the behavior of the system and link the microscopic configurations to the large (macroscopic) scale. Ludwig Boltzmann ( ) was the first to adopt such an approach and link the mechanics of the different microstates with the macroscopic properties of a system and essentially thermodynamics, introducing the concept of statistical mechanics (e.g., Dugdale, 1996; Kondepudi and Prigogine, 1998). Boltzmann interpreted entropy statistically, in terms of the different microstates, in the celebrated expression: W S k p ln p B B i i i1, (2.17) 9 This expresses Carnot s theorem on the maximum efficiency of reversible engines. 10 The second law of thermodynamics has many different statements, summarized by Clausius as the entropy of the universe approaches a maximum. 33

51 Chapter 2 Earthquake Statistics and Generalized Statistical Mechanics where SB is the total entropy of the system 11, pi is the probability of the i th microstate with W i1 p 1, W is the total number of discrete microstates and kb is Boltzmann s i constant (kb = x J K -1 ). In the particular case of equal probabilities ( p 1 W, i), the entropy SB becomes: i S k lnw, (2.18) B B which is the celebrated expression of entropy, carved on Boltzmann s grave in Vienna. As Eq.2.18 indicates, entropy increases logarithmically with the number of microstates W. A larger number of available microstates in the system corresponds to higher entropy. Entropy can then be regarded as a measure of disorder in the system. In 1902, J.W. Gibbs ( ) redefined Eq.2.17 for more general systems (not just gases) by introducing the theory of generalized statistical ensembles. The significance of the ensembles theory was that it enabled the estimation of a complete set of thermodynamic parameters of a system from the purely mechanical properties of its microscopic elements (Pathria and Beale, 2011). The work of Gibbs established the notion of statistical mechanics and Eq.2.17 is frequently referred to as Boltzmann-Gibbs entropy (SBG). Since the present thesis deals with the statistical mechanics of continuous variables, the following expressions are referred to the continuous case. For a continuous variable x that takes values in the space [0, ], entropy SB takes the integral form: S k p( x)ln p( x) dx. (2.19) B B 0 Boltzmann asserted that the probabilities of the most probable state at equilibrium are those that maximize SB and satisfy the constraints of normalization of p(x) (e.g., Pressé et al., 2013): 11 The subscript B has been assigned to Boltzmann s entropy S B to distinguish it from the thermodynamic entropy S. 34

52 Chapter 2 Earthquake Statistics and Generalized Statistical Mechanics 0 p( x) d( x) 1 (2.20) and the expectation value (average value) of x: x xp( x) dx. (2.21) 0 The most probable state of p(x) is estimated by maximizing the entropy, subject to the previous constraints, using the Lagrange multipliers method and the function (e.g., Tsallis, 2009; Pressé et al., 2013): S k p( x) dx xp( x) dx, (2.22) B B 0 0 where α and β are the Lagrange multipliers. The probability distribution p(x) that maximizes Eq.2.22 is the well-known Boltzmann distribution: px ( ) 0 e e x x dx. (2.23) The term e x is often called the Boltzmann factor, with kt. The denominator 1 B of Eq.2.23 is a normalization factor that is referred to as the partition function, x Z e dx. (2.24) 0 Another frequent case that is deeply connected to statistical mechanics is that of the squared variable x 2. We might know that the average value of x is zero x 0, but the average value of the squared variable is not zero, 2 x 0. In this case, the distribution p(x) that maximizes SBG under the appropriate constraints is the Gaussian distribution: p( x) e x 2, (2.25) 35

53 Chapter 2 Earthquake Statistics and Generalized Statistical Mechanics where β is the Lagrange multiplier. This distribution is also known as the Maxwell- Boltzmann distribution in the kinetic theory of gases 12 (e.g., Brush, 1975). Expressions (2.17) and (2.23) are the milestones of statistical mechanics and have been widely and successfully applied in physics, chemistry, mathematics and elsewhere. For full reviews and historic remarks on the foundations of statistical mechanics the reader is referred to Brush (1975; 1976) and Pathria and Beale (2011) Information Theory In 1948, Claude Shannon, in his study on gain and loss of information during transmission through telecommunication lines, derived a mathematical function similar to Eq.2.17 to measure the uncertainty or the missing information associated with a probability distribution pi (Shannon, 1948). Shannon noticed the striking similarity between the classic definition of entropy and the derived equation and he named it as information entropy (H): N H k p log p, (2.26) i1 i i where k can be any positive constant, depending on the units of measure. H is positive and additive across independent variables. It increases with increasing uncertainty and diminishes with increasing information (e.g., Pressé et al., 2013). According to this property, H is monotonically increasing as a function of the size N, so that a larger number of possible states pi in the system is reminiscent of larger uncertainty. Shannon estimated the most probable state or the most likely transmitted message by maximizing H, subject to constraints, and deriving an exponential distribution of pi s, similar to Boltzmann s distribution (Eq.2.23). The issue of similarity between information entropy and statistical mechanics was addressed by E.T. Jaynes (Jaynes, 1957). Jaynes approach was to look into Gibbs work and how it is related to Boltzmann s statistical mechanics, in the light of Shannon s 12 In his H-theorem in 1872, Boltzmann stated that the Maxwell-Boltzmann distribution is the only stationary distribution that describes molecular collisions. 36

54 Chapter 2 Earthquake Statistics and Generalized Statistical Mechanics work (Dewar, 2005). Jaynes went beyond the physical meaning of statistical mechanics and treated it as a method of statistical inference under the perspective of insufficient information. Jaynes argued that the quantity N i1 p log p, given in Eq.2.17 and Eq.2.26, can describe the missing information about any system and not just the microstates of equilibrium systems (Dewar, 2005). Inferences can be drawn by maximizing the entropy (Eq.2.17) subject to constraints, which illustrates the only available information about the system. This procedure yields the same distribution as that of Botzmann (Eq.2.23). Entropy can then be used as a general principle for inferring the least biased probability distribution from limited information (Jaynes, 1957). The latter is commonly known as the maximum entropy principle (MaxEnt). A considerable advantage of Jaynes formalism is that it can be extended to a wide class of out-of-equilibrium systems (Jaynes et al., 1979). During the last decades, innovative insights into the macroscopic states of many physical, chemical and biological systems have been accomplished by applying the MaxEnt hypothesis, since only the constraints of the large-scale system need to be known, regardless of the complexity that may characterize the different microscopic configurations (e.g., Martyushev and Seleznev, 2006; Pressé et al., 2013). A wide-open field of research nowadays concerns the behavior of complex systems and whether this can be explained in terms of maximum entropy production as the driving mechanism (Dewar, 2003; 2005; Whitfield, 2005). i i Statistical Mechanics and Earthquakes Given the complexity of the earthquake phenomenon and the large number of degrees of freedom that are incorporated in the process, the application of statistical mechanics and information theory to earthquake data arose naturally. One of the first studies that applied statistical mechanics to earthquake data was that of Berrill and Davis (1980). The authors maximized entropy to derive a probability density function of earthquake magnitudes p(m) that has the form of a truncated exponential distribution: p( M ) 1 e m1 e M, (2.27) 37

55 Chapter 2 Earthquake Statistics and Generalized Statistical Mechanics where m1 is the minimum magnitude in the dataset and β the Lagrange multiplier. Berrill and Davis (1980) also used MaxEnt to derive a Poissonian distribution for the time of occurrence of earthquake events that are frequently modeled as a Poisson process ( 2.3.3). The frequency-magnitude distribution of earthquakes was further studied using statistical mechanics by Shen and Manshina (1983) and Main and Burton (1984). By maximizing SBGS using the constraints of the average magnitude m and the average seismic energy release E, a gamma distribution of earthquake energies was derived, consisting of a power-law distribution at small magnitudes that corresponds to the G-R scaling relation and an exponential (Boltzmann) tail at larger magnitudes (Shen and Manshina, 1983; Main and Burton, 1984). This distribution is similar to Eq.2.11 and has the form: p( E) ~ E e B1 E/, (2.28) where B is a scaling exponent and θ a characteristic energy that reflects the probability of occupancy of the different energy states E (Main, 1995). Main and Burton (1984) used earthquake data from the Mediterranean region and from Southern California to show that this type of distribution is more consistent with real observations. Furthermore, Main and Al-Kindy (2002) used Eq.2.28 to investigate the proximity of the global earthquake dataset to the critical point ( 2.4), characterized by the dissipated seismic energy E and the entropy S. The authors defined a sub-critical state for positive θ and a supercritical state for negative θ and concluded that global seismicity is in a near-critical state, where large fluctuations are dominated by fluctuations in θ rather than B and large seismic energy fluctuations can occur for smaller entropy fluctuations (Main and Al-Kindy, 2002). The hypothesis that the Earth s lithosphere is in a state of thermodynamically driven maximum entropy production and SOC ( 2.4) was tested by Main and Naylor (2008; 2010). These authors explored this connection using the Olami-Feder-Christensen (OFC) model (Olami et al., 1992) and real seismicity data and concluded that their observations were consistent with the hypothesis of entropy production as the driving 38

56 Chapter 2 Earthquake Statistics and Generalized Statistical Mechanics mechanism for self-organized sub-criticality in natural and model seismicity (Main and Naylor, 2008). In other studies, Shannon entropy (H) was used as a measure of disorder in earthquake sequences (Telesca et al., 2004; De Santis et al., 2011). Telesca et al. (2004) estimated H with time for the earthquake magnitudes and inter-event time series in Umbria- Marche region (central Italy) and found significant variations that correspond to the largest event in the series. De Santis et al. (2011) related H to the G-R scaling relation (Eq.2.7) and derived a relation between H and the b-value that reads as: b b max 1.2 H H, (2.29) with bmax elog e 1.2. Using Eq.2.29, De Santis et al. (2011) studied the variability of the b-value and H during two earthquake sequences in Italy and identified three dynamic regimes with respect to H variations: (i) a preparatory phase, where H increases slowly with time, (ii) the phase of occurrence of a strong earthquake, where H exhibits an abrupt increase after the occurrence of the main shock and (iii) a final diffuse phase, where H recovers normal values and seismicity spreads all over the region. 2.6 Generalized Statistical Mechanics The non-additive entropy Sq Boltzmann-Gibbs (BG) statistical mechanics has proven quite successful over the twentieth century for describing the macroscopic behavior of a wide class of physical systems with a large number of degrees of freedom from the specification of the underlying microscopic dynamics. However, there is an important class of complex out-of-equilibrium systems in nature that violate some of the essential properties of BG statistical mechanics (e.g., Bouchaud and George, 1990; Gell-Mann, 1994; Bak, 1996; Sornette, 2006; Tsallis, 2009a). One such property of SBG is extensivity, namely its directly proportional relationship with the number of elements, N, in the system (Eq.2.18). If strong interactions among the different microstates of a system exist, the 39

57 Chapter 2 Earthquake Statistics and Generalized Statistical Mechanics probabilities pi of the available microstates i are interrelated and the occurrence of one microstate strongly depends on the occurrence of another. Some microstates are favored, while some others are neglected. For instance, water molecules in the presence of a whirlpool do not take any path but only those paths that resemble a vortex, due to correlations in the molecules motions (Cartwright, 2014). In this case, the number of possible microstates, W, no longer increases exponentially with N, so SBG is no longer proportional to N (see Eq.2.18) and extensivity is violated. Another essential property of BG statistical mechanics is ergodicity. An ergodic system dynamically explores the accessible phase space with equal probability over the course of time and phase space averages correspond to time averages. However, in other systems ergodicity is broken and the time averages do not coincide with phase space averages 13 (Lutz and Renzoni, 2013). Complex systems that violate BG statistical mechanics involve properties such as longrange interactions, long-range microscopic memory or (multi) fractal structures and are characterized by broad probability distributions with asymptotic power-law behavior (Tsallis, 1999; Sornette, 2006). Some characteristic examples can be drawn from longrange interacting systems such as 3-D gravitation, ferromagnetism or spin-glasses and anomalous diffusion phenomena among others (see Tsallis, 2001 and Tsallis, 2009a for a comprehensive list of such systems and corresponding references). In these systems, BG statistical mechanics has limited applicability and another type of statistical mechanics seems more appropriate to describe their macroscopic properties (e.g., Zaslavsky, 1999; Gell-Mann and Tsallis, 2004; Sornette, 2006). In 1988, Constantino Tsallis, inspired by the probabilistic description of multifractal geometries, proposed a generalization of SBG to overcome some of the limitations that BG statistical mechanics presents (Tsallis, 1988). Tsallis named his generalized entropic function as non-additive entropy Sq (also known as Tsallis entropy) that for the discrete case reads as: S q W 1 pi i1 k q 1 q, (2.30) 13 Non-ergodicity can in fact be the generic case for many complex systems (Gell-Mann, 1994). 40

58 Chapter 2 Earthquake Statistics and Generalized Statistical Mechanics where k is a positive constant, such as Boltzmann s constant kb or the constant k=1 frequently used in information theory, pi is the set of probabilities, W is the total number of microscopic configurations and qq is the entropic index that expresses the degree of non-extensivity of the system (Tsallis, 1999). Note that the probabilities are similar to the moment order q of pi(r) in the definition of multifractality ( 2.2, Eq.2.3). However, in multifractals q is a running index that is used to describe the different scalings, whereas in the definition of Sq q is a fixed index that characterizes the system. Since the probabilities pi take values between zero and unity, for q < 1, p q i p and for q > 1, i events, which are close to zero (Tsallis, 2001). p q i p. Therefore, q < 1 enhances the probabilities of rare i For equal probabilities p 1 W, i, Sq obtains its maximum value: i q p i S k ln W, (2.31) q q where the q-logarithmic function lnqw is defined as: 1 q x 1 ln q x x 0;ln1 x ln x 1 q. (2.32) In the limit of q 1, Sq precisely recovers SBG and Eq.2.30 and Eq.2.31 recover Eq.2.17 and Eq.2.18 respectively (Tsallis, 1988). The entropic function Sq is the hallmark of the generalized statistical mechanics framework introduced by Tsallis, commonly known as non-extensive statistical mechanics (NESM). Sq (q>0) shares a variety of properties with SBG, such as non-negativity, concavity (related to thermodynamic stability), continuity (related to experimental robustness) and finiteness of entropy production per unit time (related to the exploration of the available phase space) (Gell-Mann and Tsallis, 2004). However, SBG is additive, whereas Sq q 1 is non-additive 14. According to this property, for any two probabilistically independent systems A and B, i.e. if the joint probability satisfies ( ) ( ),, p A B p A p B i j, SBG satisfies: ij i j 14 Tsallis named S q as non-additive entropy after this property. 41

59 Chapter 2 Earthquake Statistics and Generalized Statistical Mechanics ( ) ( ) S A B S A S B, (2.33) BG BG BG whereas Sq satisfies: S A B S ( A ) S ( B ) S 1 ( A ) S ( B ) q q q q q q. (2.34) k k k k k The origin of non-additivity of Sq comes from the last term on the right hand side of this equation (Tsallis, 1999). For q > 1, S A B S ( A) S ( B) ( ) ( ) q q q and for q < 1, q q q S A B S A S B. These cases are referred to as sub-additivity and superadditivity respectively. However, if the systems A and B are correlated, then a q-value might exist such that S A B S ( A) S ( B). In this case, Sq is extensive for q 1 q q q (Tsallis, 2009a). According to Tsallis, entropy must always remain extensive to be consistent with the laws of thermodynamics and the entropic function to be used in each case should vary accordingly (Tsallis, 2009b; Cartwright, 2014) Optimizing Sq The present thesis concerns the applicability of NESM to fracturing phenomena and earthquakes. Similar to BG statistical mechanics and information theory, the optimization of the non-additive entropy Sq under the appropriate constraints is applied, in order to derive probability distribution functions that can describe the macroscopic properties of fault and earthquake populations. Let s consider a continuous variable X with probability distribution p(x) (0 p(x) 1). The X variable may represent the seismic moment (Mo), the inter-event times (τ) or distances (r) between successive earthquakes or the fault trace-lengths (L) in a given region. I focus on the continuous case since these variables can take any real value from a finite range of values. In this case, the non-additive entropy Sq is expressed through the integral formulation: S q k q 1 p ( X ) dx 0. (2.35) q 1 42

60 Chapter 2 Earthquake Statistics and Generalized Statistical Mechanics In the following, k is set equal to unity for the sake of simplicity and without losing generality. In order to obtain the distribution p(x) that optimizes Sq, subject to constraints, the Lagrange multipliers method is applied. The first constraint refers to the normalization condition of p(x): p( X ) dx 1. (2.36) 0 The second constraint refers to the following average value, which is referred to as the q-expectation value (or q-mean value) Xq (Tsallis et al., 1998): X ( ) q X XPq X dx, (2.37) q 0 where Pq(X) is the escort probability distribution that is defined as: q p ( X) P ( X ), with P ( X ) dx 1. (2.38) q 0 q p ( X ) dx 0 q In the limit q 1, Xq recovers the standard average value of X, X. The reason for which the escort probability Pq(X) is used in the second constraint (Eq.2.37) instead of the physical probability p(x) is further discussed below. At this stage, Sq is optimized with the constraints given in Eq.2.36 and Eq Using the Lagrange multipliers method, the following functional is optimized: Sq a p X dx X ( ) * q 0, (2.39) 0 where α and β * represent the Lagrange multipliers. Optimization of Eq.2.39 yields the optimal physical probability: 1 q 1 q q X q q X 1 1 exp px ( ). (2.40) Z Z q q 43

61 Chapter 2 Earthquake Statistics and Generalized Statistical Mechanics The probability distribution p(x) can be normalized only for q < 2. The denominator of Eq.2.40 is the q-partition function defined as: Z q q q 0 exp X dx. (2.41) The term βq is associated with the Lagrange multiplier β* and the second constraint (Eq.2.37) as: q * q, with Cq p ( X ) dx. (2.42) * Cq 1 q X q 0 The numerator of Eq. (2.40) is the q-exponential function defined as: 1 1q exp q( X ) 1 1 q X, for 1 (1 q) 0 exp ( X) 0, for 1 (1 q) 0 q. (2.43) The inverse of the q-exponential function is the q-logarithmic function given in Eq In the limit q 1, the q-exponential and q-logarithmic functions lead to the ordinary exponential and logarithmic functions respectively. If q > 1, the q-exponential function exhibits asymptotic power-law behavior with slope 1 q 1, whereas for 0 < q < 1 a cut-off appears (Abe and Suzuki, 2003; 2005). The q-exponential function for various values of q is shown in Fig In various applications, the cumulative distribution function P(X) is preferred to the probability distribution p(x). In the NESM framework it has been proposed that P(X) should be obtained upon integration of the escort probability Pq(X) rather than p(x) (Abe and Suzuki, 2003; 2005). Following this approach and by using Eq.2.38, P(X) is derived as: X P( X ) Pq( X ) dx expq X. (2.44)

62 Chapter 2 Earthquake Statistics and Generalized Statistical Mechanics q=1.5 q=0.5 q=1 a) q=1.5 q=0.5 q=1 b) Figure 2.10: The q-exponential function expq(x) (Eq.2.41) for various values of q in a) log-linear axes and b) log-log axes. We see that P(X) is also given by a q-exponential function. X0 is always positive, has the units of X and is defined as 1 X 0 1 q X q (Abe and Rajagopal, 2000). q 45

63 Chapter 2 Earthquake Statistics and Generalized Statistical Mechanics Eq.2.44 can alternatively be written as PX ( ) 1 1 X 1 q X0 q. The latter equation implies that after the estimation of the appropriate q-value that describes the distribution of X, the q-logarithmic function (Eq.2.32) is linear with X, with slope 1 X 0 (Vallianatos and Sammonds, 2011). The latter is expressed in mathematical form as ln ( X) q PX 1 q ( ) 1 1 X. 1 q X0 An interesting issue of debate in the NESM theory is which distribution shall be compared with observations. The common approach is to introduce the escort probability in the second constraint (Eq.2.37) and optimize Sq as described earlier. The other forms that have been used in the theory are thoroughly described in Tsallis (2009a) and are discussed in detail in Wada and Scarfone (2005) and Ferri et al. (2005). According to Tsallis (2009a), the escort probability is preferred because it solves a number of difficulties that are related to the connection between the theory and thermodynamics (see Tsallis, 2009a for a full description). However, in practice the different forms are all correct and can be transformed one into the other with simple operations redefining the values of q and X0. For instance, the cumulative distribution P(X), as obtained upon integration of the escort probability Pq(X), was given in Eq If P(X) is estimated by the integration of the physical probability p(x) given in Eq.2.40, instead of the escort probability Pq(X), then: X P( X ) ( ) exp q p X dx ' ' X, (2.45) 0 0 ' with q' 1 2 q and X X q (Picoli et al., 2009). By applying these transformations, Eq.2.45 becomes: 2 q 1q P( X X ) 1 1 q X 0. (2.46) In addition, Sq can be optimized in terms of the squared variable X 2. In this case, the optimization process leads to the q-gaussian distribution that has the form: 46

64 Chapter 2 Earthquake Statistics and Generalized Statistical Mechanics q 1 X p( X ) 1 1 q Zq X 0. (2.47) In the limit q 1, Eq. (2.47) recovers the Gaussian distribution (Eq.2.25). For q > 1, the q-gaussian distribution displays power-law tails with slope 2 q 1. The q-gaussian distribution is shown in Fig.2.11 for various values of q. The introduction of NESM more than 25 years ago has proved to be quite important, as it can provide a physical generative principle for producing a range of power-law to exponential type distributions that characterize the behavior of a wide class of natural systems. As in BG statistical mechanics, by knowing only the mean and variance of a variable, we can draw inferences about the macroscopic behavior of a system by optimizing the non-additive entropy Sq. The solutions in this case are q-exponential or q-gaussian distributions, which may act as attractors for a large class of out-ofequilibrium systems in nature (Tsallis, 2013). Then a natural question arises: can Sq be determined by the microscopic dynamics of the system? Or in other words: is the q- index related to the first principles of a system? In an ongoing debate, detractors of the theory claim that the q-index is just a fitting parameter for systems that are not well understood (see the related discussion in Cartwright, 2014). However, Caruso and Tsallis (2008) calculated the q-index directly from first principles for a chain of particle spins in a transverse magnetic field, where values of q 1 emerge as strong correlations between the spins are formed. Another example in the microscopic scale is the anomalous transport of cold atoms in dissipative optical lattices that cannot be described within BG statistical mechanics (Lutz and Renzoni, 2013). Lutz (2003) demonstrated theoretically that in the anomalous regime the momentum distribution of cold atoms is q-gaussian, with q-values being solely related to the depth of the optical potential. This was experimentally verified later on (Douglas et al., 2006). 47

65 Chapter 2 Earthquake Statistics and Generalized Statistical Mechanics a) b) Figure 2.11: The q-gaussian distribution for various q-values, in a) linear axes and b) log-linear axes. In the limit of q 1, the q-gaussian recovers the Gaussian distribution. Applications of the theory have expanded across physics, chemistry, biology, medicine, economics and more, providing new insights into the complex behavior that is frequently observed (see Tsallis, 2009a for an extensive list of applications). In 48

66 Chapter 2 Earthquake Statistics and Generalized Statistical Mechanics earthquake phenomena, the application of NESM arose naturally (see the following section), as the theory addresses properties such as fractal geometries, long-range interactions and criticality, properties that seem quite important for understanding earthquake physics (Sornette, 2006). Since the microscopic dynamics that are incorporated in the deformation of the Earth s crust are widely unknown, the q-index is determined by the observations. This is the standard procedure of fitting the mathematical forms that emerge in the theory (i.e., q-exponentials or q-gaussians) to the observational data to obtain the appropriate q-values that best describe the observations Applications to Seismicity NESM theory and the corresponding functional forms have been successfully applied to a variety of earthquake-related variables, such as the seismic energies or magnitudes, the inter-event times or distances between successive earthquakes or the distribution of fault-trace lengths in a tectonic region. These variables are characterized by fractal or multifractal geometries, clustering effects and power-law distributions among other properties ( 2.3), making NESM an appropriate framework for studying their macroscopic properties. The results of various studies that apply NESM to earthquakes are briefly summarized in the following paragraphs Spatiotemporal description The first applications of NESM to earthquake data were those of Abe and Suzuki (2003; 2005), who studied the spatiotemporal properties of seismicity in California and Japan and showed that the cumulative distributions of inter-event distances P(>r) and times P(>τ) between successive earthquakes are well described by the q-exponential distribution (Eq.2.44) for q-values of qr < 1 and qτ > 1, respectively. In particular, Abe and Suzuki (2003; 2005) found q-values of qr = 0.773, qτ = 1.13 for California and qr = 0.747, qτ = 1.05 for Japan and suggested the spatiotemporal duality of earthquakes, where q q r 2. 49

67 Chapter 2 Earthquake Statistics and Generalized Statistical Mechanics Table 2.1: Spatiotemporal scaling properties of seismicity according to the q- exponential distribution for various earthquake catalogues and tectonic environments. Region Time period M0* qτ X0τ qr X0r Reference California sec km Abe and Suzuki, 2003; 2005 Japan sec km Abe and Suzuki, 2003; 2005 Global km Vallianatos and Sammonds, 2013 Iran Darooneh and Dadashinia, 2008 HSZ** ± ±0.02 Papadakis et al., 2013 Greece ±0.054 Antonopoulos et al., 2014 Santorini 2012 unrest Vallianatos et al., 2013 AAS*** ± days 0.53 Vallianatos et al., 2012b * Minimum magnitude of the catalogue; **Hellenic Subduction Zone; *** Aigion aftershock sequence The results of Abe and Suzuki were further verified through laboratory experiments (Vallianatos et al., 2012a), numerical models (Hasumi, 2007; 2009) and regional seismicity (see the full list of references in Table 2.1). Hasumi (2007; 2009) used the 2-D slider-block model ( 2.4; Fig.2.9) to study the cumulative distributions of interevent times and distances between successive slip events and, by varying the model parameters, they found results consistent with q-exponential scaling. Vallianatos et al. (2012a) investigated the scaling properties of acoustic emissions (AE), produced during triaxial deformation of basalts in the laboratory and found that the q-value triplet q, q, q 1.82,1.34,0.65 M r characterizes the cumulative distribution functions of scalar moments (M), inter-event times (τ) and distances (r) of AE, which also take the form of q-exponentials. Further studies on regional and global earthquake catalogues 50

68 Chapter 2 Earthquake Statistics and Generalized Statistical Mechanics showed that the cumulative distributions of inter-event times and distances are also consistent with q-exponential statistics. The results of these studies, including the regions and the time period of the catalogues, as well as the corresponding NESM parameter estimates, are summarized in Table (2.1). Figure 2.12: The cumulative distribution of inter-event times P(>τ) for the Aigion aftershock sequence in double logarithmic axes. The solid line represents the q- exponential distribution for the values of qτ = 1.58 ±0.02 and τ0 = ± days. Inset: the q-logarithmic distribution lnq(p(>τ)), exhibiting a correlation coefficient of ρ = The straight line corresponds to the q-exponential distribution (Vallianatos et al., 2012b). The application of NESM to aftershock sequences was tested by Vallianatos et al. (2012b) 15, in a case that is related to the Corinth Rift seismicity and the last major 15 Vallianatos, F., Michas, G., Papadakis, G., Sammonds, P. (2012b). A non-extensive statistical physics view to the spatiotemporal properties of the June 1995, Aigion earthquake (m6.2) aftershock sequence (West Corinth Rift, Greece). Acta Geophysica, 60,

69 Chapter 2 Earthquake Statistics and Generalized Statistical Mechanics earthquake that occurred in the area; the 1995 Aigion earthquake (Ms = 6.2, Bernard et al., 1997). In particular, we studied the spatiotemporal scaling properties of the Aigion aftershock sequence and we found results consistent with the q-exponential distribution (Table 2.1). The cumulative distribution of inter-event times and the corresponding q- exponential fit (Eq.2.44) is shown in Fig The temporal scaling properties of volcanic seismicity and in particular during the unrest at the Santorini volcanic complex were studied by Vallianatos et al. (2013) 16. In the latter study we found that when the volcano-related seismicity takes swarm-like character, complex correlations of seismicity emerge that are characterized by a q-exponential inter-event times distribution. The effect of aftershock sequences on the temporal scaling properties of seismicity in Greece (time period ) was investigated by Antonopoulos et al. (2014) 17. In this work, we used q-statistics to study the scaling properties of interevent times for the entire dataset and the declustered one, where the aftershock sequences have been removed. Our analysis indicated that both distributions deviate from an ordinary exponential and are better described by q-exponential distributions, with decreasing q-values that approach unity for the declustered dataset. This result implies the loss of temporal correlations and close proximity to Poissonian (random) behavior, once the aftershock sequences are removed from the dataset (Antonopoulos et al., 2014) Earthquake magnitudes NESM theory was first applied to the frequency-size distribution of earthquakes by Sotolongo-Costa and Posadas (2004), who developed a physical model for earthquake dynamics based on the small-scale processes that take place in fault zones. Consistent with the idea of stick-slip frictional instability in fault zones (e.g., Scholz, 1998), Sotolongo-Costa and Posadas considered that the triggering mechanism of an earthquake involves the interaction between the irregular surfaces of the fault planes 16 Vallianatos, F., Michas, G., Papadakis, G., Tzanis, A. (2013). Evidence of non-extensivity in the seismicity observed during the unrest at the Santorini volcanic complex, Greece. Natural Hazards and Earth System Sciences, 13, Antonopoulos, G., Michas, G., Vallianatos, F., Bountis, T. (2014). Evidence of q-exponential statistics in Greek seismicity. Physica A, 409,

70 Chapter 2 Earthquake Statistics and Generalized Statistical Mechanics and the fragments of various sizes and shapes that fill the space between them (Fig.2.13). When the accumulated stress exceeds a critical value in a fault zone, the fault planes slip, displacing the fragments and breaking asperities that hinder their motion, releasing energy. Sotolongo-Costa and Posadas considered that the released seismic energy is related to the size of the fragments and based on the fragment-size distribution they used the NESM formalism to establish an energy distribution function (EDF) for earthquakes. The derived formula was then applied to the frequency-size distribution of earthquakes in Southern California and Iberian Peninsula with results consistent with observations. The incorporation of NESM in fragmentation phenomena is further supported by the scale-invariant properties of fragments (Krajinovic and Van Mier, 2000), the presence of long-range interactions among the fragmented materials (Sotolongo-Costa and Posadas, 2004) and the deficiency of BG statistical mechanics to account for the presence of scaling in the fragmentation process (Englman et al., 1987). Figure 2.13: Fig. 1 from Sotolongo-Costa and Posadas (2004). Illustration of the relative motion of two irregular fault planes (a, b) and the fragments of size r that fill the space between them. In the following, the mathematical formulations and the derivation of the fragmentasperity interaction model of Sotolongo-Costa and Posadas, which was later revised by Silva et al. (2006) and Telesca (2011; 2012), are described. 53

71 Chapter 2 Earthquake Statistics and Generalized Statistical Mechanics The non-additive entropy Sq, in terms of the probability p(σ) of finding a fragment of area σ, is expressed as (Sotolongo-Costa and Posadas, 2004): S q q 1 p d k. (2.48) q 1 In what follows, k is set equal to unity for the sake of simplicity. Sotolongo-Costa and Posadas (2004) obtained the probability p(σ) by maximizing Sq under the constraints of normalization of p(σ) (Eq.2.36) and the q-mean value of p q (σ). Silva et al. (2006) introduced the condition about the q-expectation value in the second constraint (σq instead of Xq in Eq.2.37) and by using the Lagrange multipliers method, they maximized the function given in Eq.2.39 and derived the following expression for the fragment size distribution function p(σ): 1 1 q 1 q q 2 q p 1. (2.49) The proportionality between the released earthquake energy E and the size of the fragments r is introduced as E ~ r 3 (Silva et al., 2006), in accordance with the standard definition of seismic moment scaling with rupture length (Lay and Wallace, 1995). The proportionality between the released relative energy E and the 3-D size of the fragments r 3 now becomes: E q E 23. (2.50) In the last equation, σ scales with r 2 and αe is the proportionality constant between E and r 3 that has the dimension of volumetric energy density. By using the latter equation, the energy distribution function (EDF) can be written on the basis of the relationship between density functions of correlated stochastic variables (Telesca, 2011): q 1 E d 1 q E p( E) p q 1 de E de 2 q E d where the term dσ/de can be obtained by differentiating Eq. (2.50):, (2.51) 54

72 Chapter 2 Earthquake Statistics and Generalized Statistical Mechanics 1 3 d 2 E de. (2.52) de E The EDF now becomes (Silva et al., 2006; Telesca, 2011): 1 3 CE 1 pe ( ) CE 2 1 q1, (2.53) 2 with C E and C 1 q qe. In the latter expression, the probability of the energy is p(e) = n(e)/n, where n(e) corresponds to the number of earthquakes with energy E and N is the total number of earthquakes. A more viable expression can now be obtained by introducing the normalized cumulative number of earthquakes given by the integral of Eq.2.53: N E E N th Eth p E de, (2.54) where N(E > Eth) is the number of earthquakes with energy E greater than the threshold energy Eth and N the total number of earthquakes. Substituting Eq.2.53 in Eq.2.54 the following expression is derived: N E Eth 1 q E E 1 N 2 qe E 2qE 2 1qE 3. (2.55) The latter expression can be written in terms of the earthquake magnitude M, if we consider the relation between E and M. Silva et al. (2006) considered that M and E are related as M 1 ln 3 E, whereas Telesca (2011) considered the relation (Eq.2.9; Kanamori, 1978) and from Eq.2.55 he derived the equation: 2 M ~ log E 3 55

73 Chapter 2 Earthquake Statistics and Generalized Statistical Mechanics M N M 1 q E N 2 qe E 2qE 1qE. (2.56) Later on, Telesca (2012) proposed that the threshold magnitude M0, i.e. the minimum magnitude M0 of the earthquake catalogue, has to be taken in account, as the cumulative distribution assumes values of 1 for M = M0. Eq.2.56 should then be slightly changed to (Telesca, 2012): M 1 q 10 E 1 23 N M 2 qe E M0 N 1 q E qe E 2qE 1qE. (2.57) The fragment-asperity interaction model, in the form of Eq.2.56 or Eq.2.57, or in the previous forms of Sotolongo-Costa and Posadas (2004) and Silva et al. (2006), has been applied to various regional earthquake catalogues and diverse tectonic environments. The results of these studies indicate that the model can successfully reproduce the frequency-magnitude distribution of earthquakes and can then be used in regional earthquake hazard assessments. An example is shown in Fig.2.14, where the model (Eq.2.57) is applied to the regional seismicity of Greece for the period and provides a good fit for the entire dataset and the de-clustered one (Antonopoulos et al., 2014). The various qe and αe values found in regional earthquake catalogues and the corresponding references are summarized in Table 2.2. In comparison to the G-R scaling relation (Eq.2.7), the fragment-asperity model provides a good description of the observed earthquake magnitudes over a wider range of scales, while for values above some threshold magnitude, the G-R relation can be derived as a particular case, for the value of b = (2-qE) / (qe-1) (Telesca, 2012). In another recent study, Vallianatos and Sammonds (2013) used NESM to study the effect of the Sumatra (2004) and Honshu (2011) mega-earthquakes (Mw 9) on the global frequency-magnitude distribution. The authors used a cross-over formulation of NESM 18 to interpret thermodynamically the deviation of greater magnitudes from a 18 A cross-over NESM formulation considers two or more different scaling regions with different q-values within the same probability distribution (Tsekouras and Tsallis, 2005). 56

74 Chapter 2 Earthquake Statistics and Generalized Statistical Mechanics pure power-law (also dicussed in 2.3.2) and found that the distribution of greater earthquakes (Mw 7.6) changes from an exponential to another power-law prior to the two mega-events, implying a global organization of seismicity as the two mega-events were approached (Vallianatos and Sammonds, 2013). Increases in the q-values of the fragment-asperity model prior to strong events have been observed by Telesca (2010c) and Papadakis et al. (2015). Telesca (2010c) found an increase from qe=1.48 to qe=1.74 some days before the 2009 L Aquila earthquake (ML=5.8), indicating an increase in the out-of-equilibrium state of the seismogenic zone surrounding the epicentral area prior to the strong event. A similar increase in the q-values, some months prior to the 1995 Kobe earthquake (M=7.2) was found by Papadakis et al. (2015). In both L Aquila and Kobe case studies, the q-value increases were associated with the occurrence of moderate-size events prior to the main shock, indicating the initiation of a preparatory phase leading to a strong earthquake. These observations are in accordance with accelerating moment release, where the rate of moderate size events increases prior to the main shock ( 2.4). Figure 2.14: Fig. 1 from Antonopoulos et al. (2014). a) Normalized cumulative distribution of earthquake magnitudes in Greece ( , 3523 events) and the corresponding fit according to Eq.2.57 (dashed line), for the values of qe = ±0.018 and ae = ± b) Normalized cumulative magnitude distribution for the de-clustered dataset (2153 events) and the corresponding fit according to Eq.2.57 (dashed line), for the values of qe = 1.46 ±0.018 and ae = ±

75 Chapter 2 Earthquake Statistics and Generalized Statistical Mechanics Table 2.2: Analyses of the frequency-magnitude distribution of earthquakes according to the q-exponential distribution and the fragment-asperity model, for various earthquake catalogues and tectonic environments. Region Time M0 qe αε Model Reference Global q-exponential Vallianatos and Sammonds, 2013 Greece Telesca (2012) Antonopoulos et al., ±0.018 ± Italy Silva et al. (2006) Telesca, 2010b Vesuvius Silva et al. (2006) Telesca, 2010b Etna Silva et al. (2006) Telesca, 2010b Santorini 2012 unrest ± ±78 Telesca (2011) Vallianatos et al., 2013 HSZ* Telesca (2012) Papadakis et al., 2013 Taiwan Silva et al. (2006) Telesca and Chen, 2010 Southern California Telesca (2011) Telesca, 2011 California SCP (2004)** Sotolongo-Costa and Posadas, 2004 California Silva et al. (2006) Darooneh and Mehri, 2010 Iran Silva et al. (2006) Darooneh and Mehri, 2010 Iberian Peninsula SCP (2004) Sotolongo-Costa and Posadas, 2004 New Madrid Silva et al. (2006) Silva et al., 2006 Anatolian Fault Silva et al. (2006) Silva et al.,

76 Chapter 2 Earthquake Statistics and Generalized Statistical Mechanics Samambaia, Brazil Silva et al. (2006) Silva et al., 2006 San Andreas Silva et al. (2006) Vilar et al., 2007 Javakheti Silva et al. (2006) Matcharashvili et al., 2011 Mexico Silva et al. (2006) Valverde-Esperanza et al., 2012 * Hellenic Subduction Zone; ** Sotolongo-Costa and Posadas (2004) The probability distribution of incremental earthquake energies (i.e., the energy differences between successive earthquakes) in real earthquake data and in the dissipative OFC model (Olami et al., 1992) was studied by Caruso et al. (2007) and was found to follow the q-gaussian distribution (Eq.2.47). In particular, Caruso et al. (2007) showed that in the critical regime (small-world lattice), the incremental earthquake energies in the OFC model exhibit a q-gaussian probability distribution, while in the non-critical regime (regular lattice) the probability distribution is close to Gaussian (Fig.2.15a). Caruso et al. (2007) repeated their analysis on real earthquake data and found that the probability distribution of incremental earthquake energies in global seismicity and Northern California also follows the q-gaussian distribution (Fig.2.15a), providing further evidence for SOC, intermittency and long-range interactions in seismicity. These results also imply that although long-range temporal and spatial correlations exist in seismicity, together with a certain degree of statistical predictability, it is not possible to predict the magnitude of the events (Caruso et al., 2007). The probability distribution of incremental earthquake energies in volcano seismicity was further investigated by Vallianatos et al. (2013). In this study we found that the probability distribution of incremental earthquake energies during the unrest at the Santorini volcanic complex follows the q-gaussian distribution for the q-value of q = 2.24, a value that is in agreement with the Ehrenfest dog-flea SOC model (Bakar and Tirnakli, 2009). 59

77 Chapter 2 Earthquake Statistics and Generalized Statistical Mechanics a) b) Figure 2.15: a) Probability distribution function (PDF) of incremental avalanche sizes x, normalized to the standard deviation σ, for the OFC model on a small world topology (critical state; open circles) and on a regular lattice (non-critical state; filled circles). The first curve is fitted with the q-gaussian distribution (solid line) for q = 2 ±0.1. The Gaussian distribution (dashed line) is also shown for comparison. b) PDF of incremental earthquake energies for the Northern California earthquake catalogue (open circles) and the corresponding fit with the q-gaussian distribution (solid line) for q = 1.75 ±0.15. The Gaussian distribution (dashed line) is also shown for comparison (from Caruso et al., 2007). 60

78 Chapter 2 Earthquake Statistics and Generalized Statistical Mechanics Fault populations The applicability of NESM to the scaling properties of fault sizes was examined by Vallianatos et al. (2011a) and Vallianatos and Sammonds (2011). These studies showed that the cumulative distribution of fault lengths can well be approximated with the q- exponential distribution (Vallianatos et al., 2011a; Vallianatos and Sammonds, 2011). In particular, Vallianatos et al. (2011a) used the NESM formalism to derive a q- exponential distribution for fault lengths and showed that the fault length distribution Figure 2.16: Fig. 2 from Vallianatos (2013). Normalized cumulative distribution function P(>L) of fault lengths in Mars and the corresponding fit according to the q- exponential distribution (solid line), for a) compressional faults with q=1.114 and b) extensional faults with q=

79 Chapter 2 Earthquake Statistics and Generalized Statistical Mechanics in Crete, in the front of the Hellenic Arc, is q-exponential with q=1.16. Vallianatos and Sammonds (2011) used a similar approach to study the scaling properties of linked and independent faults in an extraterrestrial fault system; the Valles Marineris extensional province, Mars, and showed that the fault length distributions follow a q-exponential, with q=1.75 for linked faults and q=1.10 for the independent ones. These results indicate the strong mechanical correlations of linked faults and the weaker ones for the independent faults, with a q-value that approaches unity and BG statistical mechanics. Furthermore, Vallianatos (2013a) studied the scaling properties of thrust (compressional) and normal (extensional) faults in Mars and found that the former are described by the q-exponential distribution for the value of q=1.114 and the latter for q=1.277 (Fig.2.16). 2.7 Summary In the present chapter, the phenomenology that characterize the collective properties of fault and earthquake populations, such as clustering effects and power-laws have been reviewed and some of the methods and theories, which have been developed to quantify or explain these properties, such as fractal geometries, critical phase transitions and self-organized criticality, have been discussed. Stochastic models of seismicity and simplified mechanical models like the slider-block model have been used to reproduce earthquake statistics, based on the empirical observations and the associated scaling relations that have been developed. However, earthquake physics in the microscopic scale is inaccessible and is often missing from these approaches. Since our knowledge of the physical processes and the complex dynamics that take place in fault zones at depth is really limited, statistical mechanics can be used to derive the macroscopic properties of seismicity from the specification of the microscopic elements. During the chapter, the foundations of statistical mechanics, the concept of entropy and the advance of information theory as a variational principle for making predictions, for systems beyond classic thermodynamics, were briefly introduced. Although powerful, BG statistical mechanics presents limitations and generalized statistical mechanics has been developed, namely non-extensive statistical mechanics (NESM), to account for a class of out-of-equilibrium systems that exhibit (multi) fractal geometries, long-range interactions, high fluctuations and anomalous diffusion phenomena, among other 62

80 Chapter 2 Earthquake Statistics and Generalized Statistical Mechanics properties. These properties can lead to power-law distributions with fat tails and enhanced probabilities of extreme events; an issue that presents important implications for earthquake physics, earthquake hazard and the occurrence of large earthquakes. It is the scope of the present thesis to apply the concept of NESM to fault and earthquake population in a highly active seismic zone and one of the fastest extending continental rifts in the world, the Corinth Rift. In the following chapters, the NESM concept, supplemented with other methods such as multifractal analysis and the empirical scaling relations of seismicity, is applied to fault and earthquake populations in the Corinth Rift, aiming to provide new insights into the physics of the earthquake generation process and the evolution of earthquake activity and the fault network in the Rift, contributing to earthquake hazard assessments. 63

81 Chapter 3 Statistical Mechanics of the Fault Network 3.1 Introduction The scaling properties of fault systems are studied in order to gain insights into the physical mechanism of fault growth and evolution (Cowie, 1998a). Based on the scaling properties of faults, a variety of fault array evolution models have been developed and employed in various fields of geology, geophysics and enginnering (Davy et al., 1995, Berkowitz et al., 2000; Bell and Jackson, 2015). In 2.3.1, it has been discussed that in many case studies the distribution of fault lengths has been approximated by fractal geometries and power-law distributions (Eq.2.5) (e.g., Main et al., 1990; Turcotte, 1997; Bonnett et al., 2001). Fractal geometries are typically characterized by the absence of any characteristic length scale in the system ( 2.2) that implies a scaleinvariant fault growth process. However, in other studies, fault-length distributions are better described by the exponential function (Eq.2.6) (Cowie et al., 1993b; Vetel et al., 2005). The transition from power-law to exponential scaling has been observed in regional tectonic settings (Gupta and Scholz, 2000a), in numerical models (Spyropoulos et al., 2002; Hardacre and Cowie, 2003) and analogue laboratory experiments (Spyropoulos et al., 1999; Ackermann et al., 2001) and has been associated with the total amount of strain that the fault system has been subjected to. This transition is associated with the rates of fault nucleation, growth and coalescence, which define

82 Chapter 3 Statistical Mechanics of the Fault Network the main stages of fault population evolution (Fig.3.1). The models indicate that at the very early stages of deformation, where few new faults have nucleated, their spatial organization is random and the trace-length distribution exhibits exponential scaling (Ackermann et al., 2001; Spyropoulos et al., 2002). As strain accumulates, more faults are nucleated and grow in size and number and start to coalesce forming larger faults (Fig.3.1). At this stage, where growth and coalescence start to dominate, faults display a power-law size distribution with exponents that decrease as strain increases, indicating the increased importance of large faults in accommodating strain (Cowie et al., 1995; Ackermann et al., 2001). At later stages of deformation, strain starts to localize along few large faults that span the mechanical layer and the number of active faults decreases (Fig.3.1; see also Fig.3.24). With increasing strain, the fault population reaches saturation and the size distribution turns to exponential (Spyropoulos et al., 1999; 2002; Ackermann et al., 2001; Hardacre and Cowie, 2003). Figure 3.1: The fraction of active sites as a function of increasing strain and the main stages of fault population evolution (from Spyropoulos et al., 2002). Strain is normalized by disorder and thicker lines correspond to smaller disorder. In the numerical model of Spyropoulos et al. (2002), disorder is used as a measure of heterogeneity of the deformed material. 65

83 Chapter 3 Statistical Mechanics of the Fault Network In addition, other factors such as the regional stress field, crustal rheology, elastic stress interactions and the boundary condition of the finite brittle layer may all affect how the fault system evolves (Hardacre and Cowie, 2003; Cowie et al., 2005). All these factors contribute to the complexity of the fault network evolution, so that in many cases simple power-law or exponential distributions may not account for the full-range of the observed scaling behavior (Davy, 1993; Cladouhos and Marrett, 1996; Vetel et al., 2005; Soliva and Schultz, 2008; Vallianatos et al., 2011a). To resolve some of the limitations arising from the description of fault trace-length distributions using empirical statistical distributions, Vallianatos et al. (2011a) and Vallianatos and Sammonds (2011), based on the non-extensive statistical mechanics theory, introduced a statistical mechanics model that describes the cumulative distribution of fault sizes ( ). In the present chapter, I use this model to study the scaling properties of the fault network in the Corinth Rift. The results are compared for different strain settings and are discussed in terms of fault growth and the evolution of the fault network in the Rift. A brief overview of the regional geodynamics in the broader Hellenic region is initially provided in The physiography of the Rift is introduced in and the geology and deformations patterns are analytically discussed in and 3.2.3, in order to support the variations of strain in different zones of the Rift. The deformation patterns are discussed in terms of geodetic extension rates, seismicity, accumulated sediments in the basin, uplift rates and fault displacements, which provide evidence for different deformation patterns in time and space. In 3.3, the fault dataset and the method that was used to compile it are analytically described. In 3.4, the NESM formalism, as it applies to a fault system of various fault trace-lengths is described, followed by the analysis of the fault dataset and the results for the various strain settings, in 3.5. In 3.6, the results of the analysis and the various implications that arise for fault growth and the fault network evolution in the Rift are discussed. The present chapter is based on the article Satistical mechanics and scaling of fault populations with increasing strain in the Corinth Rift, by G. Michas, F. Vallianatos and P. Sammonds, published in Earth and Planetary Science Letters. 66

84 Chapter 3 Statistical Mechanics of the Fault Network 3.2 Tectonic setting Regional Geodynamics The area of Greece is the most seismically active area in Europe due to its location on an active tectonic plate boundary, at the convergence of the Eurasian and African lithospheric plates (e.g., McKenzie, 1970; Le Pichon and Angelier, 1979). Across the South Hellenic Subduction Zone (SHSZ, Fig.3.2) the convergence of the subducting African plate and the Aegean is occurring at rates ~35 mm/yr (McClusky et al., 2000; Reilinger et al., 2006; Floyd et al., 2010), while further NW the subducting Apulia plate converges towards Northern Greece across the North Hellenic Subduction Zone (NHSZ, Fig.3.2) at lower rates of the order ~5-10 mm/yr (Hollenstein et al., 2008). The SHSZ and NHSZ are offset by the strike-slip Kephalonia Transform Zone (KT, Fig. 3.2), which exhibits a dextral motion of ~25 mm/yr (Le Pichon et al., 1995; Hollenstein et al., 2008). The regional geodynamics of the Hellenic region are mainly controlled by these kinematics, together with the westward propagation of Anatolia towards the Aegean across the right lateral strike-slip North Anatolia Fault (NAF, Fig.3.2) at rates of ~25 mm/yr (McClusky et al., 2000), (e.g., Le Pichon et al., 1995). Significant continental extension has been taking place in the back-arc region of the Aegean since the Oligocene Miocene (Mercier, 1989). Extensive deformation, from almost arc-parallel in the Oligocene - Late Miocene, has gradually shifted in the Pliocene Quaternary into a series of extending grabens such as the North Aegean Trough (NAT, Fig.3.2), the Evia graben and the Corinth Rift (Papanikolaou and Royden, 2007). Seismicity correlates with regional tectonics and is mainly distributed along these zones, parallel to SHSZ and KT (Fig.3.2; see also Fig.4.1). The regional geodetic strain rates, the distribution of active faulting and the computed focal mechanisms indicate that active extension in the Hellenic region mainly takes place along these zones (Goldsworthy et al., 2002) Physiography and Geological setting The Corinth rift is located in central Greece and consists of a ~ 130 km long x 30 km wide high-strain zone of active deformation (Fig.3.2). Its central part is below sea level 67

85 Chapter 3 Statistical Mechanics of the Fault Network forming the Gulf of Corinth, which separates central continental Greece to the north from Peloponnese to the south (Fig.3.3). The Gulf reaches the maximum depth of ~900 m at its central part and is separated from Patras Gulf to the west by the 60 m deep Rio straits (Stefatos et al., 2002). To the east, the Gulf is divided by the Perachora Peninsula into two smaller-scale gulfs, the Alkyonides to the north and the Lechaio to the south (Fig.3.3). The Gulf of Lechaio is further separated from Saronikos Gulf to the east by the Corinth Isthmus area. Figure 3.2: The broader area of Greece and the main tectonic setting (SHSZ: South Hellenic Subduction Zone, NHSZ: North Hellenic Subduction Zone, KT: Kephalonia Transform Fault, NAF: North Anatolian Fault, NAT: North Aegean Trough). The rectangle indicates the Corinth Rift. The Corinth Rift superimposes the NNW-SSE Hellenides thrust belt that forms the prerift basement (e.g., Doutsos et al., 1993). The thrust sheets are mainly comprised of thick (up to 3 km) Mesozoic carbonates and Miocene flysch sequences that overthrust the underlying Zaroucla unit of metamorphic Phyllite-Quartzite series (Dornsiepen et 68

86 Chapter 3 Statistical Mechanics of the Fault Network al., 2001; Skourlis and Doutsos, 2003). Beginning of extension in the area has been estimated in the Pliocene (~5 Myr) (Ori, 1989; Roberts, 1996; Leeder et al., 2008). Geologic evidence about the age and evolution of the Rift are found in the Pliocene- Quaternary syn-rift sequences of sedimentary deposits that are exposed on the south coast of the Rift (Ori, 1989; Armijo et al., 1996; Rohais et al., 2007) (Fig.3.4). The synrift sediments are elevated up to m above sea level and their mean thickness can be greater than 2 km in the Akrata - Derveni region (Rohais et al., 2007). Active sedimentation occurs in the offshore basin and the syn-rift deposits reach the maximum thickness of > 2 km in the central part of the Gulf (Fig.3.5) (Bell et al., 2009; Taylor et al., 2011). Figure 3.3: Topography of the Corinth Rift region. Topography and bathymetry were extracted from ETOPO1 Global Relief Model ( 69

87 Chapter 3 Statistical Mechanics of the Fault Network a) b) c) d) Figure 3.4: a) Simplified geology and the tectonic framework of the Rift, showing the major active and inactive faults according to Bell et al. (2009). Major active faults have throws > 500 m and intermediate active faults < 500 m. Geology is after EPPE and ECPFE (1996; 1997), Ford et al. (2013), IGME geologic maps (1:50000) and Tsodoulos et al. (2008). The numbers in boxes indicate the locations of the uplift rates 70

88 Chapter 3 Statistical Mechanics of the Fault Network shown in Fig.2c. b) Geodetic horizontal extension rates across the red line shown in a), according to Clarke et al. (1998). c) Late Quaternary uplift rates according to 1) Houghton et al. (2003), 2) De Martini et al. (2004), 3) Leeder et al. (2003), 4) Armijo et al. (1996), 5) Roberts et al. (2009), 6) and 7) Collier et al. (1992). d) Summation of upper-crust extension (black) and whole-crust extension derived from crustal thinning (dark red) and basement subsidence (light grey) along three transects in the Corinth rift (Bell et al., 2011). Figure 3.5: The total sediment thickness in the Gulf of Corinth, estimated from seismic reflection profiles by Taylor et al. (2011). Thin contours are for every 0.25 km and thick contours for every 1km (from Taylor et al., 2011). Deformation in the Rift is accommodated by a system of S- and N-dipping normal faults of an E-W to WNW-ESE general direction, located both onshore and offshore (Fig.3.4a) (Armijo et al., 1996; Moretti et al., 2003; Bell et al., 2009). The faults present maximum lengths of about km and steep dips of ~ 50 o (Roberts, 1996). Late 71

89 Chapter 3 Statistical Mechanics of the Fault Network Quaternary uplifted terraces and marine notches along the south coast of the Gulf indicate an active uplifting south coast (Roberts, 1996; Pirazzoli et al., 2004) (Fig.3.4c). In the north coast of the Gulf such features are absent and the coast appears more sinuous, suggesting subsidence (Stefatos et al., 2002; Bell et al., 2009). The structure of the Rift has been interpreted as an asymmetric half-graben due to a dominant N- dipping south-margin normal fault system that controls the deformation patterns and sedimentation in the Rift, although offshore seismic reflection profiles reveal several N- and S- dipping active faults that indicate a more complex basin structure (Moretti et al., 2003; Bell et al., 2009) Deformation patterns in the Rift The deformation patterns and the active faults in the Rift have undergone several spatiotemporal changes that were associated with distinct phases of strain localization and strain acceleration (e.g., Rohais et al., 2007; Bell et al., 2009; Roberts et al., 2009; Taylor et al., 2011; Ford et al., 2013). The various indications, regarding the currently active (short-term) and Quaternary (~ 2 Myr present) (long-term) deformation patterns in the Rift, are further discussed in the following two paragraphs Short-term deformation Geodetic strain rates estimated by Global Positioning System (GPS) data show a rapid N-S active extension in the Rift (Clarke et al., 1998; Briole et al., 2000; Chousianitis et al., 2013). The estimated GPS velocities indicate a faster moving Peloponnese, with respect to central Greece, towards the SSW (Fig.3.4a). Active extension is focused offshore, as there is little or no displacement relative to one another between the GPS stations in north Peloponnese or in the north margin of the Rift (Briole et al., 2000; Chousianitis et al., 2013). Extension rates have been estimated by 5- to 10-yr GPS velocities and 100-yr triangulation GPS velocities and indicate higher rates of >15 mm/yr at the western zone, decreasing gradually eastwards to <5-10 mm/yr (Fig.3.4b) (Clarke et al., 1998; Briole et al., 2000). Greatest rates are calculated in the area west of Aigion, although variability in the estimated extension rates appears between the 72

90 Chapter 3 Statistical Mechanics of the Fault Network various studies (20-23 mm/yr, Clarke et al., 1998; mm/yr Briole et al., 2000). According to Briole et al. (2000), active deformation is strongly localized in a narrow zone of km in the west offshore zone of the Rift, whereas to the east, deformation seems to be more diffuse in the area between Corinth and Thiva. In the central zone, localization of deformation is less well defined due to the wider extent of the Gulf and the lack of GPS stations offshore. Figure 3.6: Evolution of the Corinth Rift since onset of distributed extension in the Pliocene (from Leeder et al., 2012). ΩERP and ΩPRP represent the early and present rotation poles of the Peloponnese block respectively, with an estimated rotation rate (Ω) of 6.75 /myr, according to Goldsworthy et al. (2002). The current configuration of active faulting was established between ~ Myr and present and it is localized in the offshore zone of the Rift and along the south coast of the Gulf (Fig.3.6) (Bell et al., 2009; Roberts et al., 2009; Leeder et al., 2012; Ford et al., 2013). In the central and western zones, the localization of strain is evident from the footwall uplift of the syn-rift Pliocene-Quaternary sediments that are preserved in northern Peloponnese; in the hanging-walls of the fault system further south that seems inactive at present (Fig.3.4a) (e.g., Armijo et al., 1996; Goldsworthy and Jackson, 2000; Rohais et al., 2007). These observations provide evidence for a wider deforming zone 73

91 Chapter 3 Statistical Mechanics of the Fault Network during the early-stages of rifting (Roberts et al., 2009; Leeder et al., 2012), although the narrowing of the Rift with time in its western part is limited (Fig.3.6) (Bell et al., 2009; Ford et al., 2013; Bell and Jackson, 2015). Similar observations were made to the east, where active faulting is currently localized in the Perachora peninsula and the Alkyonides Gulf and once occupied a wider zone that was expanding as far as Nemea to the south (Fig.3.6) (Goldsworthy and Jackson, 2000; Leeder et al., 2012; Charalampakis et al., 2014). At the eastern end of the Rift, the NW-SE Megara basin, filled with ~1 km thick Plio-Pleistocene sediments (Bentham et al., 1991), is considered inactive and is currently uplifting in the footwall of the active normal fault system that bounds the Alkyonides Gulf to the south (Fig.3.4a, Fig.3.6) (Leeder et al., 2008). In addition, average late-quaternary and Holocene uplift rates along the south coast of the Gulf give further evidence of an active south margin normal fault system (Fig.3.4c) (Collier et al., 1992; Armijo et al., 1996; Houghton et al., 2003; Pirazzoli et al., 2004; De Martini et al., 2004). Localization of active deformation offshore is further supported by earthquake activity, where most of the earthquakes are located in the offshore zone, at depths ranging between 6-11 km in the west and 4-13 km in the east (Hatzfeld et al., 2000; Lyon-Caen et al., 2004; Lambotte et al., 2014). Although fault geometry at depth is not clear, seismicity has mainly been associated with the active N-dipping faults that bound the south margin of the Rift, like the Psathopyrgos, Aigion and Helike faults in the western, the offshore Xylokastro fault in the central and the Pisia, Schinos and Psatha faults in the eastern zone of the Rift (Fig.3.4a) (Hatzfeld et al., 2000; Bernard et al., 2006; Lambotte et al., 2014). Some antithetic S-dipping faults are also seismically active, like the onshore Kapareli fault in the eastern end of the Rift, which along with the Pisia and Schinos faults in the Perachora peninsula ruptured at the surface during the 1981 Alkyonides earthquake sequence (Jackson et al., 1982) Long-term deformation The Quaternary (long-term) deformation rates in the Rift have been estimated by Bell et al. (2011). By summing the fault heaves for both onshore and offshore faults along three profiles that intersect the Rift, Bell et al. (2011) estimated the values of total N-S 74

92 Chapter 3 Statistical Mechanics of the Fault Network upper-crustal and whole-crust extension during the Quaternary (~2 Myr present) (Fig.3.4d) and found greater values of extension (~11 21 km whole-crust extension) in the central zone, in the area between Akrata and Kiato, decreasing to the west (~5 13 km whole-crust extension) in the area of Aigion (Fig.3.4d). Greater long-term extension rates in the central zone is further supported by the wider extent and larger thickness of Pliocene-Quaternary syn-rift sediments (Ghisetti and Vezzani, 2005), the thickest accumulated sediments within the Gulf of Corinth (Fig.3.5) (Bell et al., 2009; Taylor et al., 2011), the greater Late Quaternary (~0.4 Myr present) fault displacements estimated from fault slip-rates (Bell et al., 2009) and the greater offshore horizontal extension estimated by summing the fault heaves of offshore faults (Taylor et al., 2011). This pattern of long-term extension is in contrast to geodetic extension rates that are increasing to the west (Fig.3.4b). This implies that the current extension pattern might be a recent phenomenon (Bell et al., 2011; Ford et al., 2013). At the eastern zone, in the area between Corinth and Nemea (Fig.3.4a), the thickness of the Pliocene-Quaternary syn-rift sediments are less than the central and western zones, indicating lower rates of long-term extension (Armijo et al., 1996). Further evidence for greater long-term extension in the central zone is provided by the estimated long-term average extension rates. Leeder et al. (2008) estimated ~ mm/yr extension rate in the Alkyonides Gulf due to displacement along the southern margin normal fault system in the last ~2.2 Myr, in comparison with the greater average rate of ~3.5 mm/yr in the central (Leeder et al., 2008) and ~ mm/yr in the western zone of the Rift (Ford et al., 2013). In addition, Late Quaternary coastal uplift rates follow a similar pattern. Tectonic uplift at the south coast of the Alkyonides Gulf has been estimated as ~ mm/yr (Leeder et al., 1991) and at the Perachora peninsula as ~ mm/yr (Collier et al., 1992; Roberts et al., 2009), increasing to ~1.3 mm/yr (Armijo et al., 1996) in the central zone and to ~1-1.2 mm/yr further west (De Martini et al., 2004) (Fig.3.4c). All the previous data support greater extension in the central and western zones of the Rift in comparison with the eastern zone. 3.3 Fault dataset Fault map and fault trace-lengths 75

93 Chapter 3 Statistical Mechanics of the Fault Network The fault dataset used in the present thesis is based on the integration of the existing literature on the seismotectonic framework of the Rift. Onshore faults have been mapped in the field and are presented in published geologic maps (scales ranging from 1:5000 to 1:100000), while offshore faults have been identified and mapped from highresolution seismic reflection profiles. In particular, information on surface geology and faulting were taken from various published maps that cover the entire area of the Rift. The spatial coverage of each map that has been used to compile the fault dataset is displayed in Fig.3.7 and the associated number corresponds to the following studies: 1) Geologic map, 1:50000 (IGME, 1984; Nafpaktos sheet). 2) Valkaniotis (2009). Seven geologic maps that cover the area between Nafpaktos and Neochori, in the north onshore zone of the Rift. 3) Papanikolaou et al. (2009). Geologic map for the Itea-Amfissa region. 4) Neotectonic map, 1: (EPPE and ECPFE, 1997; Leivadia sheet). 5) Tsodoulos et al. (2009). Geologic map for the Thiva basin. 6) Morewood and Roberts (2001). Geologic map for the Kapareli region. 7) Bentham (1991). Geologic map for the Megara basin. 8) Geologic map, 1:50000 (IGME, 1984; Kaparelion sheet). 9) Roberts and Gawthorpe (1995). Geologic map for the northern part of the Perachora Peninsula. 10) Morewood (2000). Geologic map for the Perachora Peninsula. 11) Geologic map, 1:50000 (IGME, 1984; Sofiko sheet). 12) Neotectonic map, 1: (EPPE and ECPFE, 1996; Korinthos sheet). 13) Leeder et al. (2012). Geologic map for the Xylokastro - Akrata region. 14) Rohais et al. (2007). Geologic map for the Derveni - Akrata region. 15) Ford et al. (2013). Geologic map for the Derveni - Kalavrita region. 16) Ghisetti and Vezzani (2005). Geologic map for the Xylokastro - Aigion region. 17) Palyvos et al. (2005). Geologic map for the Aigion region. 18) Palyvos et al. (2007a). Geologic map for the Psathopyrgos region. 19) Palyvos et al. (2007b). Geologic map for the Psathopyrgos region. 20) Geologic map, 1:50000 (IGME, 1984; Chalandtritsa sheet). 21) Flotte (2005). Geologic map for the Patra region. 76

94 Chapter 3 Statistical Mechanics of the Fault Network These studies were supplemented by the geologic maps of Moretti et al. (2003) and Jolivet et al. (2010) that cover the entire region of the Rift. Figure 3.7: Spatial coverage of the published geologic maps that were used to compile the fault dataset in the Corinth Rift. Each map was geo-referenced in the Google Earth platform. The numbers correspond to the published studies listed in the text. In geologic maps faults are represented as crude thick lines. In order to achieve high spatial resolution of fault-traces on the surface, each map was geo-referenced in a geographical information system interface (Google Earth) (Fig.3.7). Following this procedure a 3-D visualization of the surface geology is achieved. The high spatial resolution 3-D topography in Google Earth and the 3-D surface geology can be used to locate the fault-trace more accurately. According to this method, the scarps of the faults that have been mapped in the field can be identified and the fault-trace coordinates can be precisely defined within a few meters (Roberts, 2008; Faure Walker, 2010). This method was followed to accurately locate the trace of each fault that was previously mapped in the field and presented in the geologic maps. The fault trace was then mapped by using the path tool in Google Earth. Following this method, the actual trace path that is seldom linear can be directly measured, rather than measuring the linear 77

95 Chapter 3 Statistical Mechanics of the Fault Network distance between the tips of the fault that is the common practice in the literature. Snapshots of the mapped fault traces in Google Earth are shown in Fig.3.8. a) b) 78

96 Chapter 3 Statistical Mechanics of the Fault Network c) Figure 3.8: a) Panoramic view of the north Perachora Peninsula from the north-west and the mapped fault traces (AspF/, Asprokambos Faults 1, 2 and 3; AlkF, Alkiona Fault; SchF, Schinos Fault; PisF, Pisia Fault). b) Panoramic view from the north of the Xylokastron (XF) and Korfiotisa Faults (KrF). c) Panoramic view of the Delphi valley from the east and the mapped traces of the Delphi (DelF) and Chriso Faults (ChF). In the insets, the location of each image with respect to the Corinth Rift is shown with the white rectangle. Information on the structure and the tectonic setting of the offshore zone of the Rift is provided by various offshore geophysical surveys that are summarized in Table 3.1. The spatial coverage and the seismic lines of each survey are shown in Fig.3.9. The seismic reflection profiles gained in each survey provide a rather good spatial coverage of the entire offshore area (Fig.3.9). The seismic profiles were interpreted in terms of bathymetry, stratigraphy and tectonic structure (e.g., Fig.3.10) by various research groups and were published in the studies listed in Table 3.1. Faults that offset the resolved stratigraphy were identified in the seismic profiles and mapped (Fig.3.10). The published data were used to compile the fault network offshore. In each case, the location and size of the fault were primarily adopted from the study that uses seismic profiles with better spatial coverage in that particular area of the Rift (Fig.3.9; Table A.1). 79

97 Chapter 3 Statistical Mechanics of the Fault Network Figure 3.9: Map showing the seismic lines of each survey listed in Table 3.1. The seismic reflection profiles gained were used in each study (Table 3.1) to interpret the fault network offshore. Following the previously described procedure, a total number of 391 faults were mapped in the Rift and are presented in Fig The dataset includes only the basement displacing faults. The seismic reflection profiles revealed several minor offshore faults that do not displace the surface and their geometry has not been accurately defined (Bell et al., 2009; Taylor et al., 2011; Charalambakis et al., 2014; Beckers et al., 2015). These faults are not shown in Fig.3.11 and are not included in the dataset. The code name, length, dip direction and spatial distribution of each fault shown in Fig.3.11 are presented in Table A.1 (Appendix A), along with the corresponding references. The majority of the mapped faults are located onshore (290 faults), 90 faults are located offshore and 11 have both onshore and offshore extensions. The minimum fault trace-length is 0.34 km and the maximum 29.3 km. The histogram of fault trace-lengths versus their frequencies shows that the majority of faults present trace-lengths below 5 km (Fig.3.12), with an average of 5.04 km. 80

98 Chapter 3 Statistical Mechanics of the Fault Network Table 3.1: Summary of the marine geophysical surveys and the corresponding studies, which were used to compile the fault dataset offshore. Survey Source References R/V Maurice Ewing (EW0108) 2001 Multi-channel air-gun Taylor et al., 2011 RRS Shackleton 1982 Single-channel air-gun Stefatos et al., 2002; Bell et al., 2009 M.V. Vasilios 1994 Single-channel Sparker Stefatos et al., 2002 M.V. Vasilios 1996/1 Single-channel Sparker Stefatos et al., 2002 M.V. Vasilios 1996/2 Single-channel Sparker Leeder et al., 2002 M.V. Vasilios 2003 Multi-channel Sparker McNeill et al., 2005; Bell et al., 2008; survey Single-channel air-gun Sakelariou et al., 2007 Hellenic Center for Marine Research (HCMR) 2004 Hellenic Center for Marine Research (HCMR) survey Single-channel Pinger and Sparker Single-channel air-gun Charalampakis et al., 2014 Single-channel Pinger Charalampakis et al., 2014 Charalampakis et al., 2014 R/V ALKYON 2011/2012 Single-channel Sparker Beckers et al., Possible errors in the dataset Estimation of the actual fault trace-length is subjected to errors that can be attributed to the mapping technique and the geological processes during faulting (Bonnet et al., 2001). When faults rupture and displace the surface, the slip surfaces are usually semielliptical (Dawers et al., 1993). If the fault is well exposed, its trace can be accurately resolved at the meter scale in the field, depending on the scale of the background map and the accuracy of GPS instruments used to map its trace on surface. Around the tips of the fault, where displacement dies to zero, erosion and/or vegetation may hinder the actual tip. Erosion is even more pronounced when alluvial rivers form gorges that 81

99 Chapter 3 Statistical Mechanics of the Fault Network crosscut the fault surfaces. These factors may produce an underestimation of the tracelength that is usually less than few meters or tens of meters. Figure 3.10: a) Uninterpreted and b) Interpreted seismic profile (M.V. Vasilios survey 2003) obtained from the western Gulf of Corinth (from Bell et al., 2008). Figure 3.11: Simplified geology and the fault network in the Corinth Rift (next page). The code names, trace-lengths and the corresponding references for each fault are given in Table A.1. 82

100 Chapter 3 Statistical Mechanics of the Fault Network 83

101 Chapter 3 Statistical Mechanics of the Fault Network In the case where satellite imagery is used to trace a fault, the relative errors are associated with the spatial resolution of the imagery, below which the fault displacement cannot be detected. For the digital elevation profile (DEM) used in the Google Earth platform the relative error is normally less than few tens of meters. In addition, some large faults in the Rift are segmented due to erosion and gorges that crosscut the displacement surfaces (e.g., the Pirgaki-Mamousia fault zone and the Helike faults). In these cases, the various segments were counted as one fault. The overall error in the fault trace-lengths due to these identified factors was estimated to be less than 10% of their length for the onshore faults Frequency Fault Length (km) Figure 3.12: Histogram of frequency versus fault trace-lengths of the fault dataset. For the offshore faults, the possible errors are related to the density of seismic lines in each survey (Fig.3.9; Table 3.1) and the resolution of the seismic reflection profiles. Spacing of seismic lines is not uniform but varies between few hundreds meters in some parts of the Gulf to 1-2 km in some others (Fig.3.9). Hence, the undersampling of offshore faults with lengths less than at least 1 km might be expected. In each case, the fault trace-length was adopted from the study that has better spatial coverage in that particular part of the Gulf. The offshore faults and the related references are listed in 84

102 Chapter 3 Statistical Mechanics of the Fault Network Table A.1 (Appendix A). The error in the estimated trace-length for the offshore faults is greater than that for the onshore faults, reaching up to 20% of their total length. The possible underestimation of fault trace-lengths due to limited resolution at the tips of the fault has been treated in the literature by adding a constant value to the length of each fault (Pickering et al., 1995; Gupta and Scholz, 2000a). This value corresponds to the spatial resolution of the mapping technique. In the present case, the various techniques used in each case to map the trace of the fault on the surface, does not allow for a single estimation of the possible error. 3.4 Generalized Statistical Mechanics Formalism The generalized statistical mechanics formalism, as it applies to a fault network that present various fault trace-lengths L, was introduced by Vallianatos et al. (2011a) and Vallianatos and Sammonds (2011). Since L can take any real value between the minimum (Lmin) and maximum (Lmax) fault trace-length in the area under study, the entropy Sq is expressed in the integral form: S q dl 1 pl ( ) k q 1 q, (3.1) where σ is a positive scaling factor, k a positive constant, q the entropic index and p(l)dl the probability of finding L in the interval [L, L+dL]. The optimization procedure of Sq, under the appropriate constraints and by using the Lagrange multipliers method was described in detail in By setting k=1 and under the constraints of the normalization of p(l), L q q L p( L) p( L) q dl dl 1 p( L) exp q L L q Z q Lmax * q exp q q 0 Z dl L L p( L) dl 1 and the q-expectation value of the mean,, Sq is optimized by the stationary distribution, where Zq is the partition function, expq(x) is the q-exponential distribution that has 85

103 Chapter 3 Statistical Mechanics of the Fault Network * been defined in and Eq.2.43 and, where λ is the standard Lagrange multiplier (Vallianatos and Sammonds, 2011). The upper bound Lmax corresponds to a theoretical infinite fault trace-length value, but practically it does not exceed the size of the area under study. For convenience in the calculations, a minimum fault-length of zero was used by Vallianatos and Sammonds (2011) as the lower bound in the integration, since Lmin << Lmax. The cumulative distribution function P(>L) is obtained after integrating the escort probability distribution Pq(L), L P( L) Pq( L) dl expq L, (3.2) L 0 1 where L0 is a positive scaling parameter (q > 1) given by L0 1 q L. q * According to Telesca (2012), Eq. 3.2 should be slightly changed to be consistent with real observations, to the form: L expq L0 P( L), (3.3) L min expq L0 where Lmin is the minimum fault-length in the dataset. Lmax 0 dl pl q 3.5 Analysis of the fault dataset The cumulative distribution function The scaling properties of fault trace-lengths are most frequently described in the literature by using the cumulative distribution function N(>L) (e.g., Cladouhos and Marret, 1996; Bonnet et al., 2001), which describes the number of faults with length greater than L. This function is preferred to the probability distribution function n(l) because it can be easily computed without binning the data. The choice of the bin size 86

104 Chapter 3 Statistical Mechanics of the Fault Network can be critical, since it can alter the number of faults that fall into each bin and hence alter the smoothing trend of the distribution (Bonnet et al., 2001). In the fitting process, this might produce larger errors in the estimated exponents or scaling factors. In addition, most fault datasets are low-numbered so the cumulative distribution function will produce smoother distribution trends. In the following analysis, N(>L) is used to describe the scaling properties of fault tracelengths. In practice, N(>L) is estimated by subtracting incrementally the frequency of L. For a fault dataset of total number N and length values of L1, L2,..., LN, where the values of L are in ascending order, the number of faults with length greater than L1 will be N-1, greater than L2, N-2 and so on. Hence, the cumulative distribution N(>L) will have maximum value equal to N and minimum equal to one. Since Eq.3.3 assumes probabilities between 0 and 1, P(>L) is multiplied by the total number of faults in each case in order to fit the observed N(>L). The nlinfit tool in Matlab is used to fit Eq.3.3 to the data. The nlinfit tool applies the Levenberg Marquardt (LM) nonlinear leastsquare algorithm (Levenberg, 1944; Marquardt, 1963) to the data and is further discussed in The corresponding Matlab script for performing the analysis is given in Appendix C Possible biases in the fault-length distribution Possible biases in the shape of the cumulative distribution of fault trace-lengths have been reported as being the result of scale limitations in the dataset. These are known as truncation and censoring effects and are related to short and long fault-lengths respectively (Pickering et al., 1995; Bonnet et al., 2001). The population of short faults can be incomplete in scales below the resolution limit of the mapping technique used. This is known as truncation effect and has been associated with the convex upward shape that the cumulative distribution frequently presents. This causes deviations of the observed distribution from a perfect power-law that plots as a straight line on double logarithmic axes. This part of the distribution is simply excluded from the analysis in some studies, by setting a lower threshold below which small faults are thought to be undersampled (Pickering et al., 1995; Ackermann and Schlische, 1997). However, in other studies a convex upward distribution has been recognized as a true property of the fault dataset (Vetel et al., 2005). Another possible reason for deviation from the 87

105 Chapter 3 Statistical Mechanics of the Fault Network power-law trend is the existence of a natural lower cut-off at small scales that causes the fall-off in the slope of the distribution (Odling, 1997; Bonnett et al., 2001). Censoring refers to finite-size effects that may appear in larger faults, if their size is larger than the sampling area. This causes the underestimation of the actual trace-length and a fall-off in the distribution may appear at large scales. Censoring effects are not expected to bias the fault dataset used in the present thesis, as the sampled area of the Rift is much larger than the estimated maximum fault length. Furthermore, the following analysis is restricted to faults with length 1 km, in order to account for possible incompleteness in small faults. Below this value, the frequency of the mapped faults decreases significantly (Fig.3.12), indicating that the population might be incomplete Scaling properties of the fault population Initially, the scaling properties of the complete fault dataset are analyzed, with 367 faults for L 1 km. The cumulative number of fault trace-lengths N(>L) and the corresponding fit according to Eq.3.3 are shown in Fig Eq.3.3 provides a good fit over the full-range of scales for the parameter values of q=1.03 ±0.011 and L0=4.21 ± In the limit of q 1, Eq.3.3 reduces to the exponential function, so the observed q-value close to unity indicates an almost exponential scaling and the proximity towards a Poissonian organization of the fault network in the Rift. In the inset of Fig.3.13, the corresponding q-logarithmic distribution lnq[n(>l)] is shown that approaches linearty with the correlation coefficient of Scaling properties for the short-term strain rates Possible variations in the scaling properties of the fault population are investigated for the short-term deformation patterns in the Rift. In , it has been discussed that the currently active Rift zone is confined to the narrow offshore zone and the Perachora Peninsula to the east. According to this pattern, the current high strain rate setting includes most of the offshore faults in the Gulf of Corinth and the Alkyonides Gulf, the onshore faults in the Perachora Peninsula and the active Kapareli, Psatha, Marathias, 88

106 Chapter 3 Statistical Mechanics of the Fault Network Psathopyrgos, Aigion, Helike, Labiri, Selianitika, and Fassouleika faults (the last three comprise the active Neos Erineos fault zone; Palyvos et al., 2005) (Fig.3.14). In the cuurent low strain rate setting, all the remaining faults that are mainly located onshore are included (Fig.3.14). Figure 3.13: Cumulative number of fault lengths N(>L) (filled circles) for the complete fault dataset in the Corinth Rift (L 1 km). The corresponding fit is according to Eq.3.3 for the values of q=1.03 and L0=4.21 (solid line). Inset: the corresponding q-logarithmic distribution lnq[n(>l)] that presents the correlation coefficient of The dataset in the current high strain rate zone (active Rift zone) is comprised of 133 faults for L 1 km and presents the average trace-length of Lavg=6.09 km. In the low strain rate zone (inactive Rift zone), the dataset is comprised of 234 faults (L 1 km) and presents a lower average trace-length of Lavg=4.88 km. The cumulative number N(>L) for the two datasets is shown in Fig The figure shows that although both datasets share similar numbers of faults with trace-lengths greater than ~8 km, the lowstrain setting has a greater number of small-size faults leading to the lower average trace-length. 89

107 Chapter 3 Statistical Mechanics of the Fault Network The scaling behavior for the two strain rate settings is shown in Fig.3.16, where the data are fitted according to Eq.(3.3). In the high strain rate setting, the q-value approaches unity, so that the observed N(>L) exhibits exponential scaling (Fig.3.16a). In the low strain rate setting, N(>L) is better described by Eq.3.3 for the values of q=1.13 ±0.019 and L0=3.22 ±0.055 (Fig.3.16b). In the insets of Fig.3.16a, b, the corresponding semi-log and semi q-log functions are shown that approximate linearity with the correlation coefficients of and respectively. These results indicate the transition from asymptotic power-law to exponential scaling from the currently inactive to the currently active Rift zone. Figure 3.14: Simplified geology and the fault network in the Corinth Rift, indicating the current high and low strain rate settings with red and black colors respectively. Vallianatos (2013a) showed that for two subsystems of total number N1 and N2 that are described by different q-values, the distribution of the composite system will exhibit the q-value of q = (q1lnn1+q2lnn2) / (lnn1+lnn2). By substituting in the previous equation the values of q1=1, N1=133, q2=1.13 and N2=234 found for the high and low strain rate settings respectively, the q-value of 1.07 is estimated for the entire fault population that is similar to the q-value of 1.03 found by fitting the data. 90

108 Chapter 3 Statistical Mechanics of the Fault Network Cumulative Number N (>L) High strain rate setting Low strain rate setting Fault Length (km) Figure 3.15: Cumulative number of fault trace-lengths N(>L) for the high and lowstrain rate settings, according to the short-term deformation pattern in the Rift. a) 91

109 Chapter 3 Statistical Mechanics of the Fault Network b) Figure 3.16: a) Cumulative number of fault trace-lengths N(>L) for the current high strain rate setting and the corresponding fit according to the exponential function. Inset: N(>L) in semi-log axes exhibiting the correlation coefficient of b) N(>L) for the current low strain rate setting and the corresponding fit according to Eq.3.3 for the values of q=1.13 and L0=3.22. Inset: the corresponding q-logarithmic distribution lnq[n(>l)] that presents the correlation coefficient of Scaling properties for long-term deformation The scaling properties of fault trace-lengths are further analyzed for the long-term (Quaternary; ~ 2 Myr - present) deformation patterns in the Rift. In it was discussed that the central and western zones of the Rift experienced greater extension during the Quaternary than the eastern zone. The area of the Rift is divided into the three zones by taking into account the longitudes of long-term extension rates (Fig.3.4d) as estimated by Bell et al. (2011), the Late-Quaternary tectonic uplift rates along the Rift (Fig.3.4c) and the Late Quaternary fault displacements and accumulated sediments within the Gulf of Corinth (Fig.3.5) (Bell et al., 2009; 2011). The eastern zone includes 92

110 Chapter 3 Statistical Mechanics of the Fault Network the Alkyonides and the Lechaio Gulfs, the Perachora Peninsula and the Megara, Thiva, and Corinth basins (Fig.3.17). The central zone covers the area between Akrata and Kiato in the south coast of the Gulf, the area between Galaxidi and Livadia to the north and the central area of the Gulf of Corinth, where the Gulf reaches greater depths and the accumulated sediments are thickest. The area west of Akrata and Galaxidi and up tο the Rion straits constitutes the western zone (Fig.3.17). Figure 3.17: The western, central and eastern zones of the Rift considered as the longterm strain regimes. The dataset in the eastern zone has 162 faults, in the central 120 faults and in the western 85 faults, for L 1 km. Wherever the line that is drawn to divide the different zones of the Rift (Fig.3.17) intersects a fault trace, this fault is considered as part of the zone where its greater length lies. The cumulative number N(>L) of fault trace-lengths for the three datasets is shown in Fig The similarity in the shape of N(>L) of the western and central zones is observed. In contrast, N(>L) in the eastern zone intersects the other two, with a greater number of small-size faults and a lower number of intermediate and large-size faults in comparison with the central zone. The average fault trace-length in the eastern zone is Lavg=4.41 km, in comparison with Lavg=6.3 km and Lavg=5.66 km in the central and western zones respectively. 93

111 Chapter 3 Statistical Mechanics of the Fault Network 10 2 East Central West Cumulative Number N (>L) Fault Length (km) Figure 3.18: Cumulative number of fault trace-lengths N(>L) for the eastern (black circles), central (blue squares) and western zones (brown diamonds) of the Rift respectively. The scaling behavior of N(>L) for the three zones is shown in Fig.3.19, where N(>L) is fitted according to Eq.3.3. In both the western and central zones, the q-value approaches unity indicating exponential scaling (Fig.3.19a, b). The exponential function is fitted to the data, providing a good fit to the observed N(>L). In the insets of Fig.3.19a, b the corresponding semi-log plots are shown, where N(>L) approaches linearity with the correlation coefficients of for both western and central zones. In contrast, N(>L) of the eastern zone is best described by Eq.3.3 for the values of q=1.22 ±0.045 and L0=2.57 ±0.012 (Fig.3.19c). In the inset of Fig.3.19c, the corresponding q-logarithmic distribution lnq[n(>l)] for the eastern zone is shown that approaches linearity with the correlation coefficient of These results indicate the transition from asymptotic power-law scaling in the low-strain eastern zone to exponential scaling in the highstrain central and western zones. Therefore, a similar transition with increasing strain is observed as previously, when the short-term strain rates in the Rift were considered. 94

112 Chapter 3 Statistical Mechanics of the Fault Network a) b) 95

113 Chapter 3 Statistical Mechanics of the Fault Network c) Figure 3.19: a) Cumulative number of fault trace-lengths N(>L) in the western zone (diamonds) and the corresponding fit (solid line) according to the exponential function. Inset: N(>L) in semi-log axes that exhibits the correlation coefficient of b) N(>L) in the central zone (squares) and the corresponding fit (solid line) according to the exponential function. Inset: N(>L) in semi-log axes that exhibits the correlation coefficient of c) N(>L) in the eastern zone (circles) and the corresponding fit (solid line) according to Eq.3.3 for the values of q=1.22 and L0=2.57. Inset: the corresponding q-logarithmic distribution lnq[n(>l)] that presents the correlation coefficient of Robustness of the results The results obtained for the various strain zones seem robust, in the sense that they are not affected by the particular classification of single faults in one strain zone or another that was followed here. For instance, if the faults in the Perachora Peninsula are considered in the current low strain rate zone rather than the high strain rate zone (Fig.3.14), the scaling behavior of the fault trace-lengths in each of the two zones will 96

114 Chapter 3 Statistical Mechanics of the Fault Network still be similar. The same occurs if the lines that were drawn to divide the western, central and eastern zones (Fig.3.17) are slightly moved to include different faults. In addition, the fault dataset used in the analysis might be incomplete due to missing faults that have not been mapped. To further test the robustness of the results and their sensitiveness to missing data, synthetic fault-length distributions are generated that follow the q-exponential type distribution of Eq.3.3. From each distribution data are randomly removed and the scaling behavior of the remaining data is estimated. This procedure is repeated several times and the scaling variations that appear from the original q-exponential distribution that the data were generated from are estimated. The method that was followed to generate the synthetic datasets is the Markov chain Monte Carlo (MCMC) method and the Metropolis-Hasting algorithm (Newman and Barkema, 1999). MCMC methods and particularly the Metropolis-Hasting (M-H) algorithm (Metropolis et al., 1953; Hastings, 1970) are extensively used in the simulation of nonstandard, complex distributions. The basic idea behind sampling a probability distribution p(x) by using the M-H algorithm is to use a Markov chain to generate a sequence of x values, which asymptotically reach the stationary distribution π(x) in such a way that ( x) p( x). This problem is approached by using a proposed probability density q(x,y), with q x, y dy 1, to reach the target distribution p(x). The Markov process is started at an arbitrary value x0 and includes a large number of iterations j (j=1, 2,,N). In each iteration j a new candidate value xj is generated from the proposed density q( x j x j 1) depending on the previous state of the Markov chain xj-1. Then the candidate value xj is accepted or rejected according to the acceptance probability A( x j x j 1) given by: q( x j 1 x j ) ( x ) j A( x j x j1) min 1,, ( x j1) q( x j x j1) 0 q( x j x j 1) ( x j 1) A( x x ) 1, ( x ) q( x x ) 0. j j1 j1 j j1 The candidate value xj is accepted if the acceptance probability is greater than or equal to a random number between 0 and 1, generated by the uniform distribution U(0, 1) and is rejected otherwise (e.g., Chib and Greenberg, 1995): 97

115 Chapter 3 Statistical Mechanics of the Fault Network x j x j if U A( x j x j1) and. x x if U A( x x ) j j1 j j1 The previous steps are repeated many times until the N number of x values is obtained. In the implementation of the M-H algorithm, the appropriate selection of the proposed probability density can be crucial (Rosenthal, 2010). To efficiently sample the q- exponential distribution (Eq.3.3), the t (student) distribution: 1 2 x f( x) , with ν=2, was chosen as the proposed one. The t-distribution presents heavy tails and in comparison to the normal distribution is more prone to generating values that fall away from the mean. The mhsample code in Matlab was used to apply the M-H algorithm and to generate the synthetic datasets of the q-exponential type. Ten datasets, each comprised of 100,000 L values that follow Eq.3.3 for the values of q=1.15 and L0=2.9 (Fig.3.20) were generated and another ten datasets for the values of q=1.25 and L0=2.1 (Fig.3.21). Initially, 10 subsets of 10 4 values each were randomly selected from each dataset and the corresponding q and L0 were estimated. The cumulative distributions P(>L) for each subset are plotted in Fig.3.20b and Fig.3.21b and the estimated q-values and the respective errorbars in Fig.3.20e and Fig.3.21e, for the two cases of different q-values respectively. Although the subsets include only 10% of the initial values, the estimated q and L0 values remain quite stable, with q deviations of the order of ±0.03 (Fig.3.20e,f and Fig.3.21e,f) and for L0 ±0.1. The previous analysis is repeated for 20 subsets of 10 3 L values each and 50 subsets of 10 2 L values each, which are randomly selected from the original datasets. The cumulative distributions P(>L) for each subset are plotted in Fig.3.20c and Fig.3.20d respectively for the datasets that follow Eq.3.3 with q=1.15 and in Fig.21c and Fig.21d respectively for those with q=1.25. The estimated q-values and the respective errorbars are shown in Fig.20e and Fig.21e for the two cases respectively. The boxplots of the estimated q-values versus sample size of the subsets are shown in Fig.20f and Fig.21f for the two cases respectively. The boxplots indicate the median value, the 25% and 75% quartiles and 98

116 Chapter 3 Statistical Mechanics of the Fault Network the minimum and maximum estimated q-values for each case. As it might be expected, the variability of q and L0 increases with decreasing number of L values (Fig.3.20e,f and Fig.3.21e,f). However, even though the 99% or the 99.9% of the initial L values were removed, the remaining data still follow Eq.3.3. In the case of 0.1% remaining L values, the variability of the estimated q and L0 values from the original q and L0 values that the data were generated from reaches ±0.1 and ±0.8 respectively. These results give support to the robustness of the results found previously from the analysis in the various strain regimes, even in the case where faults are missing from the dataset. a) b) c) d) 99

117 Chapter 3 Statistical Mechanics of the Fault Network e) f) Figure 3.20: a) Ten synthetic datasets, each comprised of L values that follow the cumulative distribution function given in Eq.3.3, for the values of q=1.15 and L0=2.9 (black line). b) From each of the initial datasets, 10 subsets of 104 L values each (100 total subsets) were randomly selected and their cumulative distribution is plotted. All subsets follow Eq.3.3 for the values of q=1.15 ±0.03 and L0=2.9 ±0.1. c) For 20 randomly selected subsets of 103 L values each (200 total subsets). All subsets follow Eq.3.3 for the values of q=1.15 ±0.05 and L0=2.9 ±0.3. d) For 50 randomly selected subsets of 102 L values each (500 total subsets). All subsets follow Eq.3.3 for the values of q=1.15 ±0.1 and L0=2.9 ±0.8. In each plot, the black solid line indicates the initial distribution that the data were generated from and the black dashed lines the ± standard deviation. e) The estimated q-values and the respective error bars versus the sample size of the various subsets. The solid line indicates the q-value of the original distribution that the subsets were generated from. f) The boxplots of the estimated qvalues for the various subsets. a) b) 100

118 Chapter 3 Statistical Mechanics of the Fault Network c) d) e) f) Figure 3.21: a) Ten synthetic datasets, each comprised of L values that follow the cumulative distribution function given in Eq.3.3, for the values of q=1.25 and L0=2.1 (black line). b) From each of the initial datasets, 10 subsets of 104 L values each (100 total subsets) are randomly selected and their cumulative distribution is plotted. All subsets follow Eq.3.3 for the values of q=1.25 ±0.02 and L0=2.1 ±0.1. c) For 20 randomly selected subsets of 103 L values each (200 total subsets). All subsets follow Eq.3.3 for the values of q=1.25 ±0.05 and L0=2.1 ±0.2. d) For 50 randomly selected subsets of 102 L values each (500 total subsets). All subsets follow Eq.3.3 for the values of q=1.25 ±0.1 and L0=2.1 ±0.7. In each plot, the black solid line indicates the original distribution that the data were generated from and the black dashed lines the ± standard deviation. e) The estimated q-values and the respective error bars versus the sample size of the various subsets. The solid line indicates the q-value of the original distribution that the subsets were generated from. f) The boxplots of the estimated qvalues for the various subsets. 101

119 Chapter 3 Statistical Mechanics of the Fault Network 3.6 Discussion Discussion of the results Analysis of the complete fault population and the various subsets showed that the generalized statistical mechanics model (Eq.3.3) describes well the scaling behavior of fault trace-lengths in the Corinth Rift, for various q-values in the range of 1 (exponential) to For q > 1, the distribution exhibits an asymptotic power-law behavior that reduces to the exponential in the limit of q 1. The q-value indicates how far or close to exponential and Poissonian behavior the distribution is and it may then be considered as an indication of how strong the interactions in the system are. In the case of only short-range interactions, q-value approaches unity and Sq reduces to the standard SBGS. Analysis of the fault-length distributions using the generalized statistical mechanics approach offers an underlying generative principle and a unified framework that produces a range of power-law to exponential distributions that are both observed in natural fault systems. In addition, it provides robust results even in the case of missing data and a more rigorous way to study both qualitatively and quantitatively the scaling properties of fault populations, without making assumptions about the distribution that best fits the data or the range of observations that follow this particular scaling Implications for fault growth The most prominent feature revealed from the analysis is the transition from asymptotic power-law in the eastern low-strain zone, to exponential scaling in the central and western high-strain zones in the Rift. If the currently active and inactive Rift zones are considered, a similar transition from asymptotic power-law in the current low strain rate zone, to exponential scaling in the high strain rate zone is observed. The transition from one scaling regime to the other becomes apparent as the q-value, from values greater than one in the low-strain settings, approaches unity in the high-strain settings and the cumulative distribution turns to exponential. These results provide quantitative evidence for such crustal processes in natural fault systems and in a single tectonic setting and are in agreement with the results of Gupta and Scholz (2000a), who made a qualitative interpretation of their fault dataset, without fitting the fault-length 102

120 Chapter 3 Statistical Mechanics of the Fault Network distributions, to show a similar transition with increasing strain in the fault populations of the Afar Rift. The observed properties in the distinct strain zones of the Rift are summarized in Fig In the low-strain zones the number of faults is greater, the average fault tracelength is less and the cumulative distribution displays asymptotic power-law scaling. This is in contrast to the high-strain zones, where the number of faults is less, the average fault trace-length is greater and exponential scaling appears in the cumulative distribution. These properties, in analogy to the stages of fault population evolution as defined in numerical models and laboratory experiments, suggest different stages of spatial and temporal evolution in the distinct strain zones of the Rift. In the high-strain zones, exponential scaling suggests the maturity of the fault network and fracture saturation, where faults grow by coalescence to create longer faults rather than growth. In the low-strain zones, the greater number of small-size faults suggests that nucleation and growth still dominate over coalescence leading to scale-invariant fault growth and asymptotic power-law scaling. Previous studies on the scaling properties of the fault network in the Corinth Rift made similar suggestions regarding maturity and fracture saturation, although subsets and not the entire fault network were analyzed (Poulimenos, 2000; Zygouri et al., 2008). Poulimenos (2000) studied the onshore SW part of the Rift and suggested fracture saturation, although the high- and low-strain settings were both interpreted using power-law distributions. Poulimenos (2000) estimated higher displacement: length ratios in the high-strain zone, similar to the results of Gupta and Scholz (2000a) for the Afar Rift. Such results suggest that when saturation is achieved, faults grow by displacement rather than length to accommodate increasing strain, causing the displacement rates to increase (Nicol et al., 1997). Such increases in the displacement rates of faults were reported for the Late Quaternary (~ 0.7 Myr present) by Ford et al. (2013) for the SW part of the Rift and by Roberts et al. (2009) for the Perachora Peninsula. 103

121 Chapter 3 Statistical Mechanics of the Fault Network Figure 3.22: Summary of the observed properties in the different strain zones of the Rift. Zygouri et al. (2008) used a different dataset to study only active faults in the Rift and suggested the maturity and saturation of the active fault system. Zygouri et al. divided the dataset into onshore and offshore faults and interpreted both fault trace-length distributions as bi-fractal, where the scaling behavior was described by two powerlaws with different exponents. The authors suggested that both populations follow similar growth patterns for intermediate and large-size faults, while for small-size faults nucleation is more prominent in the offshore than the onshore zone. By contrast, the analysis of the entire fault population for the short-term strain rates obtained in the present thesis, suggests that nucleation and growth is more pronounced in the onshore low strain rate zone rather than the offshore high strain rate one. 104

122 Chapter 3 Statistical Mechanics of the Fault Network Implications for the fault network evolution in the Rift The progression of the fault trace-length distribution towards exponential scaling suggests the suppression of long-range fault interactions with increasing strain. Fault interactions are created as the fault system grows in size and faults start to interact through their stress fields (Fig.3.23) (e.g., Sornette et al., 1990; Cowie, 1998b; Gupta and Scholz, 2000b). According to Sornette et al. (1990), brittle failure along a fault causes the redistribution of strain perturbations in the medium that decay algebraically with distance according to the equations of elasticity, producing interactions in the far field. In the near field, stress interactions around larger faults, where strain localizes, produce a combination of stress screening and enhancement effects (e.g., Cowie, 1998b; Gupta and Scholz, 2000b). This effect is shown in Fig.3.23, where a slip event on a normal fault causes a stress increase along the fault strike and a stress drop transverse to strike (Gupta and Scholz, 2000b). In earthquake triggering, this mechanism is quite important as it may accelerate or retard upcoming events on neighboring faults (King et al., 1994). In fault growth, the stress perturbations around larger faults favor deformation in some areas, while other areas remain relatively undeformed. Hence, optimally oriented faults in stress increase zones continue to grow, while other faults in stress drop zones may cease activity. Figure 3.23: The stress field around a normal fault. The stress field produces an area of stress drop around the main body of the fault and stress increase near the tips of the fault (from Gupta and Scholz, 2000b). 105

123 Chapter 3 Statistical Mechanics of the Fault Network Figure 3.24: The main stages of fault network evolution according to the numerical model of Cowie (1998b). A stress feedback mechanism has a dominant role in the model. a) Stage 1: nucleation of many isolated faults. b) Stage 2: enhanced growth of optimally oriented faults (W, X, Y and Z). Activity is suppressed at those faults that are not optimally located. c) Stage 3: localization of deformation at a few large faults that grow by linkage (Y and Z). In intervening regions faults become inactive (X and W). Stress interactions are incorporated in various models of fault growth (Cowie, 1998b; Ackermann et al., 2001; Soliva and Schultz, 2008). Cowie (1998b) developed a numerical model to explain the spatiotemporal variations in the growth rates of faults. The model is based on a stress-feedback mechanism that is related to stress interactions on neighboring faults. During a seismic rupture on a fault, the surrounding stress field 106

124 Chapter 3 Statistical Mechanics of the Fault Network is perturbed and seismic ruptures on neighboring faults that are optimally oriented are accelerated while the stress levels on other faults are relaxed. Cowie showed how this mechanism could lead to the rapid localization of strain and the increase of displacement rates in optimally oriented faults in the stress increase zones (Fig.3.23). The main stages of fault growth, according to the stress-feedback mechanism, are shown in Fig During the initial stage of deformation, several isolated faults are nucleated in the weaker parts of the crust (stage 1, Fig. 3.24a). As deformation progresses (stage 2), displacements rates are enhanced on those faults that are optimally oriented in stress increase zones (W, X, Y and Z, Fig. 3.24b). At later stages (stage 3), strain localizes at some more optimally oriented faults that grow by linkage to form larger structures (Y and Z, Fig.3.24c), while other faults (X and W) may become inactive if they are located in stress drop zones. In addition, fault growth and interactions are scale-dependent on the thickness of the brittle layer within which the faults are confined (e.g., Cowie, 1998a; Soliva et al., 2006). The vertical growth of large faults that span the brittle layer is restricted and faults can only grow along strike to accommodate increasing strain (Ackermann and Schlische, 1997). Ackermann et al. (2001) investigated the effect of the brittle layer s thickness in laboratory experiments and found that in thin models fracture saturation was achieved more rapidly with increasing strain and exponential scaling emerged, compared with thicker models where these properties were achieved with higher amounts of total strain. In the ductile zone, where the mechanical resistance of the crust is weaker than the upper brittle layer, deformation processes may be controlled by aseismic viscous flow rather than frictional stick-slip failure (Cowie et al., 2013). The effect of the brittle-ductile rheology on fault growth was investigated in the experimental models of Sornette et al. (1993) and Davy et al. (1995). In these models, deformation of the brittle-ductile system was represented as the uniaxial compression of a brittle layer of dry sand superimposed on a ductile layer of silicon putties with Newtonian viscosities. In the experimental model of Sornette et al. (1993), the deformation localized along a few large faults that spanned the brittle layer reaching the brittle-ductile transition with increasing strain. Davy et al. (1995) showed that localization of strain depends on the strength ratio between the brittle-ductile layers in continental regions. The strength ratio depends on the thickness and density of the brittle layer and on the thickness, the viscosity and the imposed strain rate of the ductile 107

125 Chapter 3 Statistical Mechanics of the Fault Network layer (Cowie, 1998a). In the experimental models of Davy et al. (1995), deformation localized along a few large faults for large strength ratios between the brittle and ductile layers (>5-10), while for smaller strength ratios, deformation was more distributed. According to Cowie (1998a), the brittle-ductile strength ratio of the lithosphere varies with heat flow and strain rates, so that different styles of fault network evolution are expected in different regions. In the Corinth Rift, although fault geometry at depth is not clear (Lambotte et al., 2014), the seismogenic layer in the western zone is confined to a lower depth than the eastern zone (Hatzfeld et al., 2000). In the west, earthquake activity is constrained in a zone gently dipping to the north at depths of 8-11 km (Fig.3.25) (Rigo et al., 1996; Lambotte et al., 2014). In an ongoing debate, this zone has been related to the brittle-ductile transition in the crust (Hatzfeld et al., 2000) or to a low-angle fault or a detachment zone that controls the kinematics and rheology in the western part of the Rift (Rigo et al., 1996; Sorel et al., 2000) (Fig.3.25). Whether the brittle-ductile transition or a lowangle detachment, this zone constrains the depth to which a fault can grow in the western zone of the Rift. In synergy with stress interaction effects and in analogy to the experiments of Ackermann et al. (2001), localization of strain and fracture saturation can be achieved more rapidly as a function of increasing strain in this part of the Rift. During the Late Quaternary, initially distributed extension in the Corinth Rift has progressively localized into the narrow offshore zone and the Perachora Peninsula (Roberts et al., 2009; Leeder et al., 2012; Ford et al., 2013) (Fig.3.6). Such localization has also been observed in other extensional rifts such as the Timor Sea (Meyer et al., 2002), the Gulf of Suez (Gawthorpe et al., 2003) and the North Sea (Cowie et al., 2005). The mechanism of strain localization in continental rifts, although not fully understood, has been attributed to crustal rheology, heat fluxes or changes in regional strain rates (Cowie et al., 2005). In the broader region of the Corinth Rift there is no evidence that such processes took place during the Quaternary. The results of the present analysis, summarized in Fig.3.22, rather suggest that fault growth processes in the upper crust control the fault network evolution and the localization of strain. In the eastern zone, the lower amounts of strain during the last 2 Myr, the thicker brittle layer and the 108

126 Chapter 3 Statistical Mechanics of the Fault Network Figure 3.25: Cross-section indicating the geometry and interactions of faults at depth in the western zone of the Corinth Rift (MPF: Pirgaki-Mamousia Fault zone; EEF East Eliki Fault; WCF: West Channel Fault; SEF: South Eratini Fault; NEF: North Eratini Fault). In the inset, the location of the cross-section is shown and the locations of Ms > 5 earthquakes, after Ambraseys and Jackson (1990) and Bernard et al. (1997). Faults have been projected with dips of 60 (dashed lines), down to the probable depth of the brittle-ductile transition. The position of the low-angle detachment zone, proposed by Sorel (2000) and Rigo et al. (1996), has also been projected (from Bell et al., 2008). asymptotic power-law behavior of the fault-length distribution, suggest that the fault network has not reached saturation and the activity is more diffuse. In the central and western zones of the Rift, fault growth, due to the higher strain during the last 2 Myr, led to fracture saturation, the suppression of fault interactions and exponential faultlength scaling, and ultimately to the localization of strain in the narrow offshore zone (Fig.3.6). Such mechanisms might be more pronounced in the western zone, where the britlle-ductile transition depth is shallower than in the eastern part, leading more rapidly to fracture saturation and exponential scaling. It further suggests that once fracture saturation is achieved, the displacement rates increase for some faults and/or new faults are nucleated to accommodate increasing strain. In support of the latter stage of 109

127 Chapter 3 Statistical Mechanics of the Fault Network evolution, Poulimenos (2000) reports such an increase in the displacement rates of the fault system in the SW part of the Rift, in the area south of Aigion (Fig.3.3). For the same area, in agreement to the results of Poulimenos (2000), Ford et al. (2013) report accelerations in the displacement rates of large faults during the period ~ 0.7 Myr present and further observe that once the critical fault-length of km is achieved, faults cease activity and activity migrates to the faults further north. In addition, the 1995 Ms6.2 Aigion earthquake and the two 2010 Mw5.1 Efpalion earthquakes that occured in the western part of the Rift, are thought to have ruptured blind faults (Ganas et al., 2013; Lambotte et al., 2014), giving support to the existence of young active structures in the Rift that have not yet displaced the surface and have not been mapped. This evolution pattern of the fault network in the distinct strain zones of the Rift is consistent with the stress feedback mechanism proposed by Poulimenos (2000), Cowie and Roberts (2001) and Roberts et al. (2009) to explain the slip rate variations in the fault system during the Quaternary (~ 2 Myr present). 110

128 Chapter 4 Earthquake-Size Distribution 4.1 Introduction In Chapter 4 the earthquake-size distribution (ESD) in the Corinth Rift is studied. The ESD describes the relative occurrence of small to large earthquakes and is typically used in earthquake hazard assessments in order to define the expected seismicity rates, based on the recorded past activity. In most cases the ESD has been found to follow the empirical G-R scaling relation, which was described in (Eq.2.7 and Eq.2.8). An alternative to the G-R scaling relation is the NESM formulation of the fragmentasperity (F-A) model. The F-A model, based on the interaction between the fragments and asperities that fill the space between the fault planes, can provide a physical model for the observed ESD in a given region. The analytic derivation of the F-A model by optimizing the non-additive entropy Sq was described in and in the present chapter it is used to study the ESD in the Corinth Rift. The results are compared to the G-R scaling relation and the obtained scaling functions are used to calculate the longerterm expected seismicity rates in the Rift.

129 Chapter 4 Earthquake-Size Distribution 4.2 Earthquake activity in the Corinth Rift Greece represents the most seismically active region in Europe (e.g., Tsapanos, 2008) due to its location on the active tectonic plate-boundary, at the convergence of the African and Eurasian lithospheric plates in the Eastern Mediterranean Zone (see and Fig.3.1). The earthquake activity in the region is more than 60 per cent of the European seismicity, with maximum expected earthquake magnitudes up to Mw = 8.2 (Papazachos, 1990). The majority of the earthquake activity is concentrated along the Hellenic Subduction Zone (HSZ), the Kephalonia Transform Fault (KT), the North Aegean Trough (NAT) and the Corinth Rift (Fig.4.1 and Fig.3.1 for the major tectonic features). Figure 4.1: The earthquake activity in Greece for Mw 4, according to the NOA bulletins (see 4.3 and 4.4). The rectangle marks the Corinth Rift region. 112

130 Chapter 4 Earthquake-Size Distribution The high earthquake activity in the Corinth Rift has been known since historic times, when several large events caused extended catastrophes in ancient cities, such as Delfi, Corinth and ancient Helike, which was submerged by an earthquake and a subsequent tsunami in 373 BC (Papazachos and Papazachou, 2003). In the last 50 years the area has suffered six earthquakes of magnitude Ms 6, the 1965 Eratini earthquake (Ms = 6.4), the 1970 Antikyra earthquake (Ms = 6.2), the 1981 Alkyonides earthquakes (Ms = 6.7; Ms = 6.4; Ms = 6.2) and the 1995 Aigion earthquake (Ms = 6.2). Fault plane solutions for these events indicate almost pure normal faulting of an E-W general direction and N-S extension (Fig.4.2) (Taymaz et al., 1991; Baker et al., 1997), consistent with the tectonic characteristics of the area and the direction of extension obtained from GPS data (see also 3.2.3). In the west zone of the Rift, fault planes dip to the north at shallow angles of 30, while in the east zone the dips are steeper (45-50 ) (Baker, et al., 1997; Hatzfeld et al., 2000). Historic and instrumental records for the area indicate 29 events of magnitude M 6 since 1694 (Fig.4.2) (Ambraseys and Jackson, 1997). Even though these events are not regularly spaced in time (Fig.4.2), the data imply that an earthquake of M 6 occurs every ten years in the area. In addition, the typical length of large faults in the Corinth Rift (of the order of km) and the lack of discontinuity between the different fault segments imply that the maximum expected earthquake magnitude in the area is less than 7 (Roberts and Jackson, 1991). This maximum threshold is in accordance with the maximum observed earthquakes that hardly exceed magnitudes of 6.8 (Ambraseys and Jackson, 1997; Papazachos and Papazachou, 2003). Another characteristic pattern of the seismicity in the Rift is the frequent occurrence of earthquake swarm sequences, which involve several small to moderate size events occurring over a few days. Such earthquake swarms occurred in 1991 (Rigo et al., 1996), 2001 (Pacchiani and Lyon-Caen, 2010) and during and (Bourouis and Cornet, 2009). The last three sequences have been associated with fluid diffusion at depth (Bourouis and Cornet, 2009; Pacchiani and Lyon-Caen, 2010), a hypothesis that is further discussed in Chapter 6. The main characteristics of microseismicity are similar to those of larger events (Bernard et al., 2006). The majority of events are concentrated in the narrow offshore zone of the Gulf of Corinth (Fig. 4.7), at depths ranging between 5-11 km in the west and 4-13 km in the east (Hatzfeld et al., 2000; Lambotte et al., 2014). The computed focal mechanisms of micro-earthquakes 113

131 Chapter 4 Earthquake-Size Distribution indicate E-W striking faults and N-S extension (Rigo et al., 1996; Hatzfeld et al., 2000; Godano et al., 2014), in accordance with larger events and regional geodynamics. Figure 4.2: Historic earthquakes of magnitude greater than 6 in the Corinth Rift since 1694 AD (after Ambraseys and Jackson, 1997), shown by stars. Instrumental earthquakes of Ms > 6 are shown by black dots, along with their corresponding focal mechanisms (after Taymaz et al., 1991; Baker et al., 1997; Bernard et al., 1997). 4.3 Seismic Networks and Earthquake Catalogues The first seismograph in Greece was installed in Athens, in 1898 (Bath, 1983). About six decades later, the first WWSSN (World Wide Seismograph Station Network) 3- component (3-D) station began operating at the National Observatory of Athens (NOA) in 1962, followed three years later by the installation of another four stations and of the Wood-Anderson seismograph in Athens, which marked the estimation of the respective local magnitude (ML) that has been used ever since (Bath, 1983; Chouliaras, 2009). Since then, the seismic network operated by NOA has been continuously expanded and upgraded (Fig.4.3), consisting nowadays of 45 permanent digital stations, equipped 114

132 Chapter 4 Earthquake-Size Distribution with broadband, 3-D component seismometers ( Since 1964, the bulletins of the Institute of Geodynamics of NOA have provided information on earthquake activity in the area of Greece (34-42 N and E) and are used to compile the earthquake catalogues for the region ( Figure 4.3: Fig. 1 from Mignan and Chouliaras (2014) time series of a) the number of seismic stations of the NOA network, b) the annual rate of events, c) a proxy to the completeness magnitude Mc of the regional catalogue obtained by the median-based analysis of the segment slope (MBASS) technique (see Mignan and Chouliaras, 2014). 115

133 Chapter 4 Earthquake-Size Distribution Other networks operating in Greece are those of the Seismological Laboratory of the University of Athens (UOA), the Laboratory of Geophysics of the Aristotle University of Thessaloniki (AUTH) and the Seismological Laboratory of the University of Patras (UOP). In 2005 a national project was launched under the name Hellenic Unified Seismological Network (HUSN) that intended to unify the NOA network with the other three networks operate by the Greek Institutions. HUSN currently consists of more than 125 stations and has substantially increased the detectability of earthquakes in Greece, producing more accurate and detailed earthquake catalogues since the beginning of 2008 (Fig.4.3) (Papanastassiou, 2010). In Fig.4.4 the more than 20 HUSN stations that are operating in the Corinth Rift are shown. Figure 4.4: Seismic stations of HUSN operating in the Corinth Rift. The network s performance was evaluated by D Alessandro et al. (2011) and Mignan and Chouliaras (2014). D Alessandro et al. (2011) applied the Seismic Networks Evaluation through Simulation (SNES) method to evaluate the detectability of the network, the epicentral and hypocentral errors and the magnitude of completeness (Mc) of the recorded events. Mc indicates the lower magnitude above which all earthquakes 116

134 Chapter 4 Earthquake-Size Distribution Figure 4.5: Fig. 3 from Mignan and Chouliaras (2014). Spatial variations of Mc according to the BMC method, for four periods of the regional earthquake catalogue produced by NOA and HUSN (see Fig.4.3). The observed Mc corresponds to the mean of Mc values obtained from 200 bootstrap frequency-magnitude distribution samples. The predicted Mc corresponds to the default model of BMC, calibrated to the Greek data. The estimated Mc according to the BMC method is the average of the observed and the predicted, weighted according to their respective uncertainties (see Mignan and Chouliaras, 2014). 117

135 Chapter 4 Earthquake-Size Distribution are detected. The analysis showed that the area of the Corinth Rift is well covered by the network, with Mc in the range of (ML). The analysis also indicated that the epicenter of an event with ML=2.5 can be estimated with errors of 3 km or less in central Greece, while the errors in the hypocentral depth are generally higher than 4 km (D Alessandro et al., 2011). These results regarding Mc are in agreement with those of Mignan and Chouliaras (2014) (Fig.4.5). The latter authors used the Bayesian Magnitude of Completeness (BMC) method to estimate the spatial variations of Mc and the results they obtained are shown in Fig.4.5 for the four main periods of the regional earthquake catalogue of NOA and HUSN (Fig.4.3c). During the period , the Mc in central Greece was approximately (ML), while after the HUSN started to operate Mc dropped below 2.5 (ML) (Fig.4.5). A software upgrade in 2011 further improved the analysis of the recorded events and Mc dropped to 1.5 (ML) (Fig.4.3 and 4.5). Figure 4.6: The CRL network, composed of 11 stations equipped with 2 HZ seismometers (triangles in light green, orange or rose) and 9 stations equipped with broadband seismometers recording continuously at 100 Hz (triangles in violet, dark green or orange). CNRS (France) runs 14 stations (triangles in violet and rose), University of Patras runs 4 stations (triangles in orange) and the University of Athens 2 stations (green triangles). Full circles show permanent GPS stations and stars the location of strain and tilt-meters (from 118

136 Chapter 4 Earthquake-Size Distribution In the Corinth Rift Laboratory Network (CRLN) was developed in the western zone of the Corinth Rift to complement the existing seismic stations of UOA and UOP (Fig.4.6). The objective of this joint project was to provide a detailed continuous monitoring of the earthquake activity in the western zone of the Rift and to constrain the active faults at depth (Lyon-Caen et al., 2004). Initially, 12 short-period seismic stations were installed in the area and in 2015 CRLN comprises more than 30 stations that include short-period and broadband seismometers, accelerometers, GPS stations, strain-meters, tide-gauges, tilt-meters, water-levels in wells and meteo stations (Fig.4.6) ( CRL produces earthquake catalogues based on automatic pickings that are available for at (last accessed on May 2015). 4.4 Implementing the G-R and F-A models to earthquake data Estimating the a and b values in the G-R scaling relation For estimation of the a and b values of the G-R scaling relation (Eq.2.7) and of their uncertainties the Maximum Likelihood Estimation (MLE) procedure is commonly deployed in the literature (e.g., Aki, 1965; Utsu, 1966; Wiemer and Wyss, 1997; Marzocchi and Sandri, 2003). MLE is preferred to the least squares technique, as the latter can produce strong biases in the estimated b-value and its uncertainties (Shi and Bolt, 1982; Sandri and Marzocchi, 2007). For a continuous distribution of magnitudes M, the cumulative distribution of M according to the G-R scaling relation (Eq.2.7) corresponds to the probability distribution function p(m): b M M 0 p( M) b10 log e, (4.1) where M0 is the minimum earthquake magnitude in the dataset and M0 << Mmax (Aki, 1965; Marzocchi and Sandri, 2003; De Santis et al., 2011). For Eq.4.1 the MLE of the b-value yields (Aki, 1965): log e b M M 0, (4.2) 119

137 Chapter 4 Earthquake-Size Distribution where M is the mean value of all possible magnitudes in the dataset. The corresponding uncertainty in the estimated b-value is (Aki, 1965): b b N, (4.3) where N is the number of earthquakes. In practice, the magnitudes M are grouped into some binning interval M that for instrumental records is usually M 0.1. To account for the use of finite binning intervals, Utsu (1966) suggested a slight modification to the Eq.4.2 formula: log e b M M M 0 2. (4.4) Furthermore, Shi and Bolt (1982) proposed an alternative estimation of the uncertainty of the b-value: 2.30b b N 2 i1 M i M N N 1 2, (4.5) which seems more reliable in the presence of possible temporal and/or spatial variations of the b-value (Shi and Bolt, 1982; Marzocchi and Sandri, 2003). After the estimation of the b-value from Eq.4.2 or Eq.4.4, the a-value (Eq.2.7) can be estimated from the expression (De Santis et al., 2011): a log N bm 0 (4.6) and the associated uncertainty by the Gauss error propagation (De Santis et al., 2011): a M b M 0 b 2 2. (4.7) In the following, Eq.4.4 and Eq.4.5 are used to estimate the b-value and its uncertainty and Eq.4.6 and Eq.4.7 the a-value and its uncertainty, respectively. 120

138 Chapter 4 Earthquake-Size Distribution Estimating the ae and qe values in the fragment-asperity model The derivation of the cumulative distribution function of earthquake magnitudes, according to the fragment-asperity model and the NESM formulation, was described in The expression given in Eq.2.57 is used to study the ESD in the Corinth Rift. This expression can be alternatively written as: M 1 q 10 E 1 23 N M 2 q 2 qe E E log log. (4.8) M0 N 1 q E 1 q 10 E qe E The ae and qe values that best describe the cumulative distribution of earthquake magnitudes N(>M) in a tectonic region can be estimated by fitting Eq.4.8 to the observed distribution. The fitting procedure that is followed here is the Levenberg Marquardt (LM) nonlinear least-square method (Levenberg, 1944; Marquardt, 1963) that is well-known for its efficiency in solving least-square problems for functions that are not linear in their unknown parameters. The LM method uses an iterative procedure to minimize the square of errors S(p) (or residuals) between the observed data y(xi) and the nonlinear model f(xi, p), where p is the vector of unknown parameters. Given a set of initial values for p, the unknown parameters are re-estimated at each step of the iterative procedure, till the convergence to the minimum of the following relation is achieved (e.g., Seber and Wild, 2003; Pujol, 2007). N 2 i i p S( p) y( x ) f ( x, ) i1 (4.9) The LM method has previously been used to estimate the ae and qe values in various tectonic regions with quite stable results (Telesca and Chen, 2010; Telesca, 2010a; 2010b; Matcharashvili et al., 2011; Papadakis et al., 2013; Vallianatos et al., 2013). The stability of the LM method and the sensitivity of the estimated ae and qe values to the initial a0 and q0 guesses were further tested by Telesca and Chen (2010) and Telesca (2010b). Providing reasonable initial estimates for q0 and a0 in the intervals [1 2] and [ ] respectively, which include all the estimated values in the literature (Table 2.2), the ae and qe values returned by the LM method appear quite stable (Telesca and Chen, 2010; Telesca, 2010b). An alternative approach for estimating the ae and qe 121

139 Chapter 4 Earthquake-Size Distribution values, which is based on the maximum likelihood estimation (MLE) method, was provided by Telesca (2012). In the following analysis, the LM method is implemented on the data by using the nlinfit function in Matlab. Given the set [a0 q0] of initial guesses for the ae and qe values, the nlinfit function returns the vector p of the estimated parameters that best fit the observed data, as well as the residuals and the Jacobian matrix that are further used to estimate the 95% confidence intervals of the estimated parameters. The Matlab script for fitting Eq.4.8 to the cumulative distribution of earthquake magnitudes according to the LM method is provided in Appendix C. 4.5 Analysis of the earthquake-size distribution Dataset In earthquake hazard assessments and in order to reliably estimate the expected seismicity rates in a region, the use of homogeneous earthquake catalogues is quite important. Earthquake catalogues are produced by seismic networks that frequently use different seismograph calibrations or different magnitude scales to define the size of an event. In addition, network expansions and upgrades may alter the detectability of small size earthquakes and consequently the magnitude of completeness of the catalogue (Fig.4.3 for the NOA network). These heterogeneities can produce artifacts and appropriate calibrations are needed in order to achieve homogeneity. For defining the long-term activity, catalogues that cover longer periods are more adequate, although for instrumental seismicity these catalogues hardly exceed the period of 40 to 50 years. The benefit to use such datasets is that they are more representative of the long-term seismicity and more reliable estimates can be made. On the other hand, the essential selection of a high cut-off magnitude in order to ensure completeness of the reported seismicity has a cost on small size events and consequently to the total number of available data. 122

140 Chapter 4 Earthquake-Size Distribution Figure 4.7: Seismicity map for the shallow earthquakes (depth 30 km) in the Corinth Rift, according to the NOA bulletins. In the analysis of ESD in the Corinth Rift, the 50-years NOA earthquake catalogue is considered more representative of the long-term activity and is preferred to the 7-years HUSN or the 13-years CRL catalogues. The dataset comprises the earthquake activity as it is reported in the bulletins of NOA 19. The year of 1967 was chosen as the initiation point of the dataset, as Mc becomes more stable at that time, with almost similar values for (Fig.4.3 and Fig.4.5). For that period (up to November 2007), the local magnitudes (ML) reported by NOA are estimated from the maximum trace amplitudes of the two horizontal components of the Wood-Anderson (WA) seismograph installed in Athens and are expected to be homogeneous (Roumelioti et al., 2010). After November 2007, the ML estimation is based on synthetic WA seismograms. To homogenize the reported magnitudes in the dataset, local magnitudes (ML) are converted to moment magnitudes (Mw). The Mw scale (Hanks and Kanamori, 1979) is 19 Available at (last accessed on April 2015). 123

141 Chapter 4 Earthquake-Size Distribution based on the seismic moment M0 rather than the amplitude of a particular seismic phase and in many cases is preferred to the other magnitude scales due to its uniformity over all magnitude ranges and the advantage of non-saturation for large magnitudes (Lay and Wallace, 1995). The relation between Mw and the other magnitude scales has been accurately defined and for a specific magnitude range (ML 6.5) Mw = ML (Heaton et al., 1986). The latter relation between Mw and ML is theoretically expected for the NOA catalogue and is the one that is found for the reported ML after 2007 and the use of synthetic WA seismograms (Roumelioti et al., 2010; Scordilis et al., 2013). However, prior to this period, systematic biases in the estimation of ML have been reported and have been attributed to the calibration of the WA seismograph in Athens (Papazachos et al., 1997; Margaris and Papazachos, 1999). These biases are almost constant during and compared to Mw cause the underestimation of ML by magnitude units (Kiratzi and Papazachos, 1984; Papazachos et al., 1997; Margaris and Papazachos, 1999; Papazachos et al., 2002). Furthermore, Roumelioti et al. (2010) used the source parameters of 576 earthquakes in Greece to define the relation between Mw and ML reported by NOA and reached the simple relation: M W M (4.10) L The latter relation is used to estimate Mw from ML for the period (up to the end of November 2007) and from December 2007 Mw is considered equal to ML as reported in the NOA bulletins. The analyzed dataset corresponds to shallow earthquakes (depth 30 km) that occurred in the Rift, in the area shown in Fig.4.7. According to Mignan and Chouliaras (2014), the magnitude of completeness Mc for the considered period in the Corinth Rift is 3.5 (ML) (Fig.4.5). The value of Mc changes to 3.8 if the moment magnitudes Mw are considered, according to the relation between ML and Mw given in Eq The cumulative number of events for the entire dataset and for Mw 3.8 is shown in Fig.4.8a and Fig.4.8b respectively. The number of recorded events increased substantially after 2008 and the operation of HUSN that increased the detectability of smaller size events (Fig.4.8a). If only the events with Mw 3.8 are considered, then the cumulative number increases at almost constant rates, apart from the periods where large events occurred in the area (Fig.4.8b). Such events are followed by aftershock sequences and can be seen in Fig.4.9, where the earthquake magnitude rate is plotted in time. The larger 124

142 Chapter 4 Earthquake-Size Distribution events in the dataset are the 1981 Alkyonides earthquakes, which were followed by a large number of moderate-size aftershocks (Fig.4.9). This causes the abrupt increase of the cumulative number of events during 1981 (Fig.4.8b). a) b) Figure 4.8: Cumulative number of events for , a) for the entire and the declustered dataset, b) for Mw

143 Chapter 4 Earthquake-Size Distribution Magnitude Time Figure 4.9: Rates of earthquake magnitudes (Mw) over time for In addition, the seismic moment Mo of an event can be estimated from Mw according to the relation (Hanks and Kanamori, 1979): log (4.11) M0 M w According to the latter relation, the released seismic moment (Mo) of an earthquake of Mw = 6 is almost 32 times larger than that of an earthquake of Mw = 5. By estimating Mo from Eq.4.11 and plotting the cumulative seismic moment release in the Rift for Mw 3.8, it is seen that a great proportion of the seismic energy was released during the 1981 Alkyonides earthquakes (Fig.4.10) Seismicity Declustering The declustering of earthquake catalogues is widely used in earthquake hazard assessments and earthquake prediction models. The aim of the declustering analysis is to separate the background activity in a region from the triggered earthquakes (e.g., aftershocks). The background activity consists of independent earthquakes caused by 126

144 Chapter 4 Earthquake-Size Distribution the tectonic loading of faults, whereas triggered earthquakes depend on mechanical processes, such as static or dynamic stress changes, or seismically-activated fluid flows that are controlled by previous earthquakes (van Stiphout et al., 2012). By removing the clusters of triggered (dependent) earthquakes from an earthquake catalogue, the remaining independent earthquakes, for large enough tectonic regions, are expected to be homogeneous in time, i.e., to follow a stationary Poisson process ( 2.3.4) (van Stiphout et al., 2012). Figure 4.10: Cumulative seismic moment (in dyn cm) over time. Aftershocks are identified on the basis of their spatio-temporal proximity to larger events and by the fact that they substantially increase the seismicity rate in a region in comparison with the averaged long-term seismicity rate (e.g., Fig.4.8b). A simple way of distinguishing between mainshocks and aftershocks is the window method (Gardner and Knopoff, 1974). According to this method, for each earthquake with magnitude M in the catalogue, the subsequent events are considered as aftershocks if they occur within a specified time interval T(M) and a distance interval L(M) (van Stiphout et al., 2012). The time-space windows are thus dependent on the magnitude of the largest event in a sequence (Fig.B.1, Appendix B). Other declustering methods (reviewed in 127

145 Chapter 4 Earthquake-Size Distribution van Stiphout et al., 2012) consider spatio-temporal interaction zones (Reasenberg, 1985) or stochastic models, such as the ETAS model (Zhuang et al., 2002). The window method (Gardner and Knopoff, 1974) is used to decluster the earthquake dataset in the Corinth Rift and to separate the background activity from aftershocks. The window parameter settings proposed by Uhrhammer (1986) were further used to define the space-time intervals, which read as: M M L( M) e [km], T( M) e [days] (4.12) The latter space-time intervals, in comparison with the parameter settings proposed by Gardner and Knopoff (1974), provide more conservative estimations regarding the expansion of the aftershock zone in time and space (for M < 6.4) (see Fig.B.1 in Appendix B) and may be considered more representative for the size of the fault segments in the Corinth Rift. In addition, the results of the declustering analysis were further compared with those of the method proposed by Reasenberg (1985), which yielded similar results regarding the number of earthquake clusters and the number of events that are considered as mainshocks. The ZMAP toolbox, which includes a set of scripts written in Matlab (Wiemer, 2001), was used to perform the declustering analysis. According to the space-time intervals defined in Eq.4.12, a number of events were identified as aftershocks (41.34% of the dataset). The remaining (declustered) dataset has events that are considered as the background activity in the almost 50-years earthquake dataset. The cumulative number of events for the declustered dataset and for the one with Mw 3.8 is shown in Fig.4.8a and Fig.4.8b respectively. The depletion of aftershocks from the dataset results in a smoother and almost constant rate of events over time in the dataset (Fig.4.8b) Earthquake-size distribution in the Corinth Rift The ESD in the Corinth Rift is studied in terms of the cumulative frequency magnitude distribution N(>M) that represents the cumulative number of events with magnitudes greater than M. In Fig.4.11 N(>M) for (according to the dataset 128

146 Chapter 4 Earthquake-Size Distribution described previously) is shown, along with the frequency of the observed magnitudes, counted in equal bins (ΔΜ) of 0.1 magnitude units. In the latter, a roll-off in the frequencies of magnitudes below 3.4 appears due to the limited detectability of the network during The large number of small size events recorded after 2007 causes the observed frequencies to increase at magnitudes 1 2 and fall again for even smaller magnitudes. For larger magnitudes (M 3.4), an almost linear relation between N(>M) and M appears that corresponds to the G-R scaling regime (Eq.2.7) N (>M) 10 2 b = Magnitude Figure 4.11: Cumulative frequency magnitude distribution (squares) and the noncumulative (incremental) frequency magnitude distribution (triangles) on a log-linear scale. The solid line represents the G-R scaling relation for b = 1.18 and a = The a and b values of the G-R scaling relation and their corresponding uncertainties are estimated according to the MLE method ( 4.5.1) and Eq.4.4 Eq.4.7 respectively. The appropriate selection of the minimum magnitude M0 in the calculations is crucial, as it can produce biases in the estimated values. The effect of M0 on the estimated b values is shown in Fig.4.12, where strong variations appear for M0 3.7 due to the incompleteness of the dataset in events smaller than this magnitude. For M0 = 3.8, the 129

147 Chapter 4 Earthquake-Size Distribution estimations yield the values of b = 1.18 ±0.043 and a = 7.39 ± The G-R scaling relation for these values provides a good fit to the observed N(>M) for moderate and large size events (Fig.4.11). Figure 4.12: The estimated b-values and their uncertainties as a function of the minimum magnitude M0 for the considered dataset. The corresponding fit of the fragment asperity model (Eq.4.8) to the observed N(>M) is shown in Fig.4.13 for various values of M0. In Fig.4.13 the data and the model are presented on semi-log axes and in the non-normalized form of Eq.4.8, in order to be consistent with Fig.4.11 and the G-R fit to the data. The good agreement between the F-A model and the data can be seen in Fig Regardless of the initial selection of M0, the model provides almost identical results and a good fit to the observed distribution, particularly for moderate and large size events. For M0 = 3.8 the non-linear least-square algorithm according to the LM method returns the values of qe = ±0.013 and ae = The range of the estimated qe-values and their corresponding uncertainties are shown in Fig.4.14 as a function of M0. The estimated qe-values vary between ±0.012 (for M0 = 3.3) and ±0.024 (for M0 = 4.5). The small variations in the second decimal degree of the order of denote the stability of qe, regardless of the initial selection of M0. Similar results, regarding the variations 130

148 Chapter 4 Earthquake-Size Distribution of qe with M0, have been found by Telesca (2010c) in the analysis of the 2009 L Aquila seismicity. The stability of qe with M0 is quite important in earthquake hazard assessments, as reliable estimates of qe can be made even in cases where the catalogue presents possible uncertainties in the magnitude of completeness and there are potential variations of Mc in time and space. This is the main advantage of the F-A model over the G-R scaling relation, which appears quite sensitive to the initial selection of M0. The other advantage is that the model can describe even smaller size events, in the part of the distribution where the roll-off appears and the distribution starts to deviate from the G-R scaling regime (Silva et al., 2006; Vallianatos et al., 2013). To check the relevant performance of the F-A model and the G-R scaling relation for fitting the data, the misfit is estimated as a function of M0. The misfit is calculated as the averages of the absolute values of the residuals y y fit between the observations (y) and the predicted values (yfit) (Telesca, 2011). The results of this analysis are shown in Fig.4.15 for the M0 range of As expected, for both the F-A model and the G-R scaling relation the misfits decrease with M0, as y also decreases and less data are included in the calculations. For small values of M0, the G-R scaling relation does not provide a good fit to the data and the misfit is generally larger. For values of M0 3.3, the G-R scaling relation presents lower misfits and seems to perform slightly better than the F-A model (Fig.4.15). This result can be realized by the qualitative inspection of Fig.4.11 and Fig.4.13, in which the two models are fitted to the data. The G-R scaling relation provides a better fit at magnitude ranges of , where y is larger and approximates a straight line (Fig.4.11), resulting in smaller misfits. However, for large size events (M 5.6), where y is smaller, the F-A model better describes the observations (Fig.4.13), an effect that is not captured in the misfit. In addition, the G-R scaling relation and the F-A model were fitted to the declustered dataset, where the aftershocks have been removed. For the G-R scaling relation and for M0 = 3.8, the estimations yield the values of b = 1.16 ±0.06 and a = 7.02 ±0.116 (Fig.B.2), whereas for the F-A model the values of qe = 1.48 ±0.02 and ae = (Fig.B.3), which are similar to the parameter values found for the entire dataset. The corresponding figures are provided in Appendix B. 131

149 Chapter 4 Earthquake-Size Distribution Figure 4.13: The cumulative frequency magnitude distribution and the corresponding fits according to the F-A model for various values of the minimum magnitude M0. Figure 4.14: The estimated qe values (squares) and their uncertainties (95% confidence intervals presented as errorbars) as a function of the minimum magnitude M0. 132

150 Chapter 4 Earthquake-Size Distribution Figure 4.15: The misfits between the observed distribution and the G-R scaling relation (triangles) and the fragment asperity (F-A) model (circles), for various values of M Expected seismicity rates The importance of the ESD originates from the fact that it can directly be used in probabilistic earthquake hazard assessments by estimating the expected seismicity rates in a region. In terms of the probability P(>M) of an earthquake of magnitude greater than M to occur in the considered period, the G-R scaling relation can be rewritten as: P( M) 10 abm dt, (4.13) where dt is the observation period. If the F-A model is considered instead, then the probability P(>M) can be calculated from Eq.2.57 as: 2qE 1qE M 1 q 10 E qe ae P( M ) N dt, (4.14) M0 1 q E qe ae 133

151 Chapter 4 Earthquake-Size Distribution where N is the number of the observed earthquakes with M M0 in the considered period dt and ae and qe the estimated parameter values of the F-A model. The annual probability of an earthquake of magnitude M occurring in the region can be estimated by using Eq.4.13 or Eq Substituting into Eq.4.13 the a and b values of a = 7.39 and b = 1.18 ±0.043 found previously for the (dt = 47 years) earthquake activity in the Corinth Rift for M0 3.8, we estimate that annually 0.66 ±0.38 earthquakes of magnitude 5 or ±0.018 earthquakes of magnitude 6 can be expected. From Eq.4.14 and for the values of M0 = 3.8, N = 802 events, qe = ±0.013 and ae = 0.001, the annual probability of an M = 5 earthquake becomes 0.82 ±0.14 and of an M = 6 earthquake ± The annual probability of earthquakes in the magnitude range 5 7 according to the two models is shown in Fig Figure 4.16: The annual probability of an earthquake of magnitude M occurring in the Corinth Rift, according to the G-R scaling relation and the F-A model (solid lines) and their corresponding uncertainties (dotted lines). 134

152 Chapter 4 Earthquake-Size Distribution The recurrence time of an earthquake of magnitude M can be estimated from Eq.4.13 or Eq.4.14 as 1 P( M). Thus, the recurrence time of an M = 5 earthquake according to the G-R scaling relation is 1.52 years, of an M = 6 earthquake 23 years and of an M = 6.8 earthquake 202 years (Fig.4.17). The F-A model predicts the recurrence times of 1.22 ±0.18 years for an M = 5 earthquake, 15.3 ±3.8 years for an M = 6 earthquake and ±36.9 years for an M = 6.8 earthquake (Fig.4.17). Comparing these results with the recurrence time of 11 years estimated by the M > 6 earthquakes since 1694 AD (Fig.4.2), it is seen that the G-R scaling relation underestimates the expected rates of M = 6 events, while the rate estimated by the F-A model for M = 6 earthquakes is almost similar to the 320 years rate. Figure 4.17: The recurrence time in years of an earthquake of magnitude M in the Corinth Rift, according to the G-R scaling relation and the F-A model (solid lines) and their corresponding uncertainties (dotted lines). 135

153 Chapter 4 Earthquake-Size Distribution Overall, the F-A model can be used to reliably estimate the expected seismicity rates in the Corinth Rift. However, the expected rates are not equally spaced in time and temporal clusters of large events may occur, like the 1981 Alkyonides earthquakes. Large events can be followed by long periods of relative quiescence, such as the current 20 year period since the last major event in the Rift in 1995; the Aigion earthquake. 136

154 Chapter 5 Temporal Properties of Seismicity and the Inter-event Time Distribution 5.1 Introduction Earthquake time series refer to the time of occurrence of earthquake events marked by their magnitude and are widely studied in order to gain insights into the physical mechanism of seismogenesis and to predict future activity based on past events. These two objectives are approximated by various scaling relations and models that are used to understand the temporal properties of seismicity and to efficiently forecast future events. The standard temporal models of seismicity were reviewed in and vary between complete randomness (Poisson models), semi-randomness, in which Poissonian background activity is interspersed by correlated aftershock sequences (e.g., ETAS model, 2.3.4) and models indicating correlated seismicity at all timescales (e.g., Livina et al., 2005; Lennartz et al., 2008). All these models refer to a probability distribution function that describes the structure of the earthquake time series and is further used to estimate the recurrence rates of seismicity. The probability distribution function presents only a first-order approximation to the temporal properties of seismicity and multifractal approaches are essential to

155 Chapter 5 Temporal Properties of Seismicity and the Inter-event Time Distribution investigate the local fluctuations (see 2.3.3). Multifractal analysis has been used in various studies as a second-order approximation to the temporal structure of seismicity, in order to study the range of correlations and the degree of heterogeneous clustering (e.g., Geilikman et al., 1990; Godano et al., 1997; Telesca and Lappena, 2006). These two approximations, the probability distribution function and multifractal analysis, are used in the present Chapter to study the temporal structure of seismicity in the Corinth Rift. The present Chapter is based on the published articles: Michas, G., Vallianatos, F., and Sammonds, P. (2013), Non-extensivity and long-range correlations in the earthquake activity at the West Corinth rift (Greece), Nonlin. Processes Geophys. 20, and Michas, G., Sammonds, P., and Vallianatos, F., (2015). Dynamic multifractality in earthquake time series: Insights from the Corinth rift, Greece. Pure and Applied Geophysics, 172, In Michas et al. (2013), we studied the probability distribution function of the earthquake time series and the range of temporal correlations in the West Corinth Rift for In the present Chapter, the analysis is extended to include the and the earthquake activity in the western part and the entire Corinth Rift respectively. Likewise, the multifractal analysis, which was performed for the earthquake activity in the Corinth Rift in Michas et al. (2015), is here applied to both datasets. 5.2 Datasets The earthquake time series in the Corinth Rift are studied in terms of the inter-event times (or waiting times) that represent the time intervals between successive earthquake events. If ti is the time of occurrence of the ith event, then inter-event times τ are defined as τi = ti+1 ti, with i = 1, 2,...,N-1 with N being the total number of events. To study the probability distribution of inter-event times p(τ) and the multifractal structure of the inter-event time series τi, a sufficient number of events is required. In Chapter 4, the NOA earthquake catalogue was considered as the appropriate one to use, in order to study the long-term earthquake size distribution in the Corinth Rift and to estimate the recurrence times of large earthquakes. This catalogue has 802 events for Mw 3.8 that was considered as the lowest magnitude for which the catalogue is 138

156 Chapter 5 Temporal Properties of Seismicity and the Inter-event Time Distribution complete. The analysis is here performed for the earthquake activity in the Rift, according to the HUSN catalogue and for the earthquake activity in the West Corinth Rift, according to the CRLN catalogue. These two catalogues are complete for lower earthquake magnitudes (see 4.3) and have several thousands of events, thus producing more reliable results about the temporal structure of seismicity The HUSN catalogue The Hellenic Unified Seismological Network (HUSN) has been operating since 2008 and has significantly increased the detectability of smaller size earthquakes in Greece (see 4.3). The HUSN stations that are operating in the area of the Corinth Rift are shown in Fig.4.4. The magnitude of completeness Mc of the catalogues produced by HUSN (available at is approximately Mc = 2 for central Greece and drops down to Mc = 1.5 after 2011 (Fig. 4.5). In Fig.5.1 the crustal (depth 30 km) earthquake activity in the Corinth Rift is shown. For ML 2 the catalogue has 8320 events, with the majority of earthquakes occurring in the west part of the Rift and within the narrow zone of the Gulf of Corinth (Fig.5.1) that is currently experiencing higher geodetic extension rates ( and Fig.3.4b). Two other spatial clusters can be recognized in the central offshore part and in the northeastern end of the Rift (Fig.5.1). The spatial distribution of events is similar to the spatial distribution of events in the Rift that is shown in Fig.4.7, indicating consistency in the spatial patterns of seismicity over the last 50 years. By removing the aftershocks from the dataset using the declustering technique described in 4.5.2, a total number of 4074 events remains that can be considered as the background activity during this period. In Fig.5.2 the seismicity rate and the magnitude rate per day are shown for The largest earthquakes during this period are the two 2010 Mw5.1 Efpalion earthquakes (Karakostas et al., 2012; Ganas et al., 2013), which are highlighted with red in the map of Fig.5.1. After the occurrence of the first event on January 18, 2010 a sudden increase in the seismicity rate occurs (Fig.5.2b) due to the production of numerous aftershocks. The seismicity rate continues to fluctuate till the beginning of 2011, where a period of low earthquake activity and a rather constant seismicity rate per day initiates in the Rift 139

157 Chapter 5 Temporal Properties of Seismicity and the Inter-event Time Distribution (Fig.5.2b). The period of low seismicity rate lasts for more than two years and is interrupted in mid-2013 by two earthquake swarms that occurred during 22/5/13 26/6/13, both of which lasted approximately two weeks. The first one occurred near Aigion (Kapetanidis et al., 2015) and the second one at the northeastern end of the Rift, near Kapareli. Figure 5.1: The earthquake activity in the Corinth rift for ML 2. Solid black lines represent major faults in the broader area. The inset map shows the main tectonic features in the broader area of Greece (SHSZ - South Hellenic Subduction Zone, NHSZ - North Hellenic Subduction Zone, KT - Kephalonia Transform Fault). The rectangle marks the area of study (from Michas et al., 2015). The inter-event time series τi for and for ML 2 is shown in Fig The series presents high fluctuations and varies over almost 6 orders of magnitude (Fig.5.3). The mean inter-event time of the series, t t N terms of the mean seismicity rate per day, is days. In N 1 1 R 1, the latter value of indicates a mean value of R 3.82 earthquakes per day (for ML 2) in the Rift. 140

158 Chapter 5 Temporal Properties of Seismicity and the Inter-event Time Distribution Magnitude a) Time b) Figure 5.2: a) Magnitude (ML) rate per day and b) seismicity rate per day and the cumulative number of events for the entire (solid line) and the declustered (dashed line) dataset, for the earthquake activity (ML 2) in the Rift according to the HUSN catalogue. The double arrow indicates the time period of stationary activity in the Rift, where the mean seismicity rate is approximately constant. 141

159 Chapter 5 Temporal Properties of Seismicity and the Inter-event Time Distribution Figure 5.3: The inter-event time series τi (in days) versus their index number i on semilog axes, for the earthquake activity in the Rift (ML 2) The CRLN catalogue The other catalogue that is further used in the analysis is the CRLN catalogue that reports the earthquake activity in the western part of the Corinth Rift (Fig.4.6) for ( 4.3). The CRLN catalogue includes more than events for the 13 years period and provides a detailed dataset for the small size earthquakes in the currently more active western part of the Rift. The magnitude of completeness Mc of the catalogue was estimated by Pacchiani (2006) to be Mc = 1.4. The spatial distribution of crustal seismicity (depth 30 km) for M 1.4, according to the CRLN catalogue, is shown in Fig.5.4. In its great majority, the earthquake activity is concentrated in a narrow 61 km x 55 km zone, highlighted with the rectangle in Fig.5.4 (area between N and E). For M 1.4, the selected dataset has events. By declustering the catalogue using the window method described in and then removing the afterhocks, a total number of events remains that can be considered as the background activity. 142

160 Chapter 5 Temporal Properties of Seismicity and the Inter-event Time Distribution Figure 5.4: Spatial distribution of seismicity (depth 30 km; M 1.4) in the West Corinth Rift according to the CRLN catalogue. The rectangle marks the selected earthquakes that are further used in the analysis. The magnitude rate and the seismicity rate per day are shown in Fig.5.5. The largest earthquake in the dataset is the 2010 M = 4.65 Efpalion earthquake (Fig.5.5a). The magnitude for this event given by CRLN is 0.45 magnitude units less than the one reported by HUSN (Mw5.1), indicating that the magnitudes in the CRLN catalogue might be underestimated. After the occurrence of the Efpalion earthquakes, the seismicity rate increases substantially, reaching almost 700 events per day (for M 1.4) (Fig.5.5b). Other increases in the seismicity rate are associated with the 2001 Agios Ioannis earthquake swarm (Pachianni and Lyon-Caen, 2010), the 2007 earthquake swarm, which occurred in the offshore western part of the Rift (Bourouis and Cornet, 143

161 Chapter 5 Temporal Properties of Seismicity and the Inter-event Time Distribution 2009) and the mid-2013 Aigion earthquake swarm (Kapetanidis et al., 2015), also reported in the HUSN catalogue Magnitude a) Time b) Figure 5.5: a) Magnitude rate per day and b) seismicity rate per day and the cumulative number of events for the entire (solid line) and the declustered (dashed line) dataset, for the earthquake activity (M 1.4) in the West Corinth Rift, according to the CRLN catalogue. The double arrows indicate stationary periods, where the seismicity rate is approximately constant. 144

162 Chapter 5 Temporal Properties of Seismicity and the Inter-event Time Distribution The inter-event time series τi for the CRLN dataset is shown in Fig.5.6. As for the HUSN dataset (Fig.5.3), the series presents high fluctuations and varies over almost eight orders of magnitude. The mean inter-event time is 0.13 days and the mean seismicity rate is R 7.7 earthquakes per day (M 1.4). Figure 5.6: The inter-event time series τi (in days) versus their index number i on semilog axes, for the earthquake activity in the western part of the Rift (M 1.4). 5.3 The Inter-event Times Distribution Inter-event Times Histograms The histogram of inter-event times τ is constructed to study their probability distribution p(τ) and structure. The initial choices that are made in the construction of the histogram, such as linear versus logarithmic bin widths on the x-axis, and linear or logarithmic scaling on the y-axis, can result in entirely different representations of the same dataset (Naylor et al., 2010). The appropriate representation is a key part of the analysis and is crucial for understanding the structure of the series. Since a wide range of scales is involved in the inter-event time series (Fig.5.3 and Fig.5.6), the representation on 145

163 Chapter 5 Temporal Properties of Seismicity and the Inter-event Time Distribution double logarithmic axes is preferred. This type of representation has also the advantage of the simple identification of power-law regimes that plot as straight lines in double logarithmic axes. In this case logarithmic binning is more straightforward as it plots uniformly over the logarithmic space. Another representation that in many studies is preferred (e.g., Corral, 2003; Hainzl et al., 2006; Touati et al., 2009) is to plot the probability density function by counting the number of τ that fall into each logarithmically spaced bin and then normalize this number by dividing it by the bin width, in order to be proportional to the probability of occupation of that bin (Naylor et al., 2010). To obtain the probability density of the series, the counts are further divided by the total number of counts so that the probabilities of occupation sum to one. The effect of using different representations for the same dataset is shown in Fig.5.7, where the histogram of inter-event times τ for the HUSN dataset is plotted. Inter-event times are shown in seconds and for τ 60 sec. For τ lower than this value, the dataset is expected to be incomplete due to overlapping of the successive events in the seismograms (Bak et al., 2002; Corral, 2003). This creates a shadow effect that hinders the detection of events that follow at short times. Such events represent less than 2% of the total number of events in the dataset and their exclusion does not alter the results of the analysis. In the first case (Fig.5.7a) the frequency of the series is counted into 50 linear bins that uniformly cover the range of τ values and plotted on linear axes. A large spike in the first bin is observed that decays rapidly for larger τ. From this graph we cannot understand much about the structure of the series, apart from the information that most inter-event times belong to the interval seconds. This image is slightly improved by plotting the frequencies on log-log axes (Fig.5.7b). In this case the form of the decay can better be seen, which approximates a straight line and powerlaw scaling with exponent -1.6 up to 10 5 seconds (Fig.5.7b). For larger τ the data are scattered and their structure is obscured. In the next graph (Fig.5.7c) the data are counted into 50 logarithmic bins and plotted on log-log axes. The logarithmic binning of the data results in a uniform representation over the logarithmic space and the structure of the series over the entire range of values is better seen. In this case, the frequency of short and intermediate τ is ascending as a straight line and a power-law with exponent 0.39 up to ~10 4 sec, where the maximum frequency is observed, and then is decaying rapidly for τ > 10 5 sec (Fig.5.7c). The probability density of the series on log-log axes is shown in the next graph (Fig.5.7d). In this case the frequency is 146

164 Chapter 5 Temporal Properties of Seismicity and the Inter-event Time Distribution normalized to the bin width and the total count. The power-law regime (straight line) for short and intermediate τ is clearly observed up to ~10 4 sec. The power-law regime presents the exponent of 0.56, one unit lower than the exponent in the histogram of frequencies of Fig.5.7b (Bonnett et al., 2001). For larger τ the decay is smoother than Fig.5.7c and approximates another power-law in the tail of the distribution. The last two representations (Fig.5.7c, d) provide a better description of the structure of the series and are the ones that are further studied. Figure 5.7: The histogram of inter-event times τ for the HUSN dataset, in a) linear bin widths and linear axes, b) linear bin widths and log-log axes, c) logarithmic bin widths on log-log axes, d) logarithmic bin widths on log-log axes, with each frequency normalized by the bin width and for total sum of frequencies equal to one. The histogram and the probability density of inter-event times τ according to the CRLN dataset are shown in Fig.5.8a and Fig.5.8b respectively. In both graphs the power-law 147

165 Chapter 5 Temporal Properties of Seismicity and the Inter-event Time Distribution scaling in short and intermediate τ, in the range of sec, is apparent. For larger τ (> 10 5 sec), another power-law regime appears. Figure 5.8: The histogram of inter-event times τ for the CRLN dataset, in a) logarithmic bin widths on log-log axes, b) logarithmic bin widths on log-log axes, with each frequency normalized by the bin width and for total sum of frequencies equal to one. The probability density is further studied for the rescaled inter-event times T. In this case, τ scales to the mean seismic rate as T R, or equivalently to the mean interevent time as T. The seismic rate acts as a weighting factor, as a higher rate in a given region corresponds to a higher number of earthquakes that are produced and contribute to the distribution (Corral, 2003). By rescaling τ with R, it has been shown that the probability densities, for various earthquake catalogues, threshold magnitudes and spatial scales, fall onto a unique curve (Corral, 2004; see the discussion in 2.3.3). For stationary periods, where the seismic rate does not fluctuate, this curve approximates the gamma distribution (Corral, 2004; Hainzl et al., 2006): 1 f ( T) CT exp( T ). (5.1) Using synthetic datasets generated by the ETAS model (Hainzl et al., 2006; Touati et al., 2009), this type of scaling has been interpreted as a mixed distribution of correlated aftershocks that scale as a power-law at short and intermediate inter-event times and uncorrelated (Poissonian) background activity that decays exponentially at long interevent times. 148

166 Chapter 5 Temporal Properties of Seismicity and the Inter-event Time Distribution However, for non-stationary time series in California and Japan, long inter-event times scale as another power-law (Corral, 2003; Talbi and Yamazaki, 2010), similar to the scaling behavior of the inter-event times observed in the HUSN and CRL datasets (Fig.2.8b). Asymptotic power-law scaling has also been shown in a series of studies, in which the q-exponential distribution was used to describe the cumulative distribution of inter-event times (Abe and Suzuki, 2005; Darooneh and Dadashinia, 2008; Vallianatos et al., 2012b, Papadakis et al., 2013; Antonopoulos et al., 2014; see also ). Taking in account these observations and the non-extensive statistical mechanics theory, I propose a scaling relation for non-stationary earthquake time series that presents two power-law regimes in both short and long inter-event times. This relation generalizes the gamma distribution and has the form of a q-generalized gamma distribution: f T CT T, (5.2) 1 ( ) exp q( ) where expq(x) is the q-exponential function defined in Eq According to Eq.5.2, for small values of T, p(t) scales as a power-law, p( T) 1 T, and for large values of T, p(t) decays asymptotically as another power-law, 1 1 q p( T) T (Queiros, 2005). In the limit of q 1 the q-generalized gamma distribution (Eq.5.2) recovers the ordinary gamma distribution (Eq.5.1). In Fig.5.9 the probability densities p(t) of the rescaled inter-event times T for the HUSN and the CRLN datasets are plotted and the corresponding fits, according to the gamma (Eq.5.1) and q-generalized gamma (Eq.5.2) distributions. The two probability densities p(t) present similar scaling behavior and fall approximately onto a unique curve (Fig.5.9). The gamma distribution (Eq.5.1) provides a good fit at short and intermediate T for the values of C = 0.4, β = 1.65 and γ = 0.38, but it deviates at longer T, where the data decay as another power-law, rather than the exponential decay incorporated in the second term of the gamma distribution. This type of scaling behavior is described by the q-generalized gamma distribution that provides an excellent fit to the entire range of T, for the values of C = 0.4, β = 1.65, γ = 0.38 and q = 1.23 (Fig.5.9). According to this scaling behavior, for short T the probability density p(t) for the various datasets decays as a power law with an exponent of about 0.6 up to 149

167 Chapter 5 Temporal Properties of Seismicity and the Inter-event Time Distribution T 1, where the q-exponential scaling emerges. For longer T, p(t) decays asymptotically as another power-law with an exponent of about 2.7. Figure 5.9: Probability density p(t) of the rescaled inter-event times T for the HUSN (squares) and CRLN (circles) datasets. The solid line represents the corresponding fit according to the q-generalized gamma distribution (Eq.5.2), for the values of C = 0.4, β = 1.65, γ = 0.38 and q = The dashed line represents the corresponding fit according to the gamma distribution (Eq.5.1) for the values of C = 0.4, β = 1.65 and γ = In Michas et al. (2013), I performed the analysis on the inter-event time series of the earthquake activity in the West Corinth Rift according to the CRLN dataset and I used the q-generalized gamma distribution to describe the observed scaling behavior. The analysis indicated an excellent agreement between the data and the proposed distribution and for similar scaling parameters values as those found here, by fitting Eq.5.2 to the earthquake activity in the West Corinth Rift (CRLN dataset) and to the earthquake activity in the Corinth Rift (HUSN dataset). This result implies self-similarity in the temporal properties of seismicity in the Rift for 150

168 Chapter 5 Temporal Properties of Seismicity and the Inter-event Time Distribution various time periods and spatial scales. To further study this result, the probability density of the rescaled inter-event times according to the three datasets and for various threshold magnitudes (M0) are plotted in Fig.5.10, along with the probability density for the earthquake activity in the Rift (M0 = 3.8) according to the NOA catalogue, which was used in Chapter 4 to study the earthquake size distribution in the Figure 5.10: Probability density p(t) of the rescaled inter-event times T for the HUSN, CRLN and NOA datasets and for various time periods and threshold magnitudes. The solid line represents the corresponding fit according to the q-generalized gamma distribution (Eq.5.2) for the values of C = 0.4, β = 1.65, γ = 0.38 and q = The dashed line represents the corresponding fit according to the gamma distribution (Eq.5.1) for the values of C = 0.4, β = 1.65 and γ = Rift. The graph shows an almost perfect collapse of p(t) into the unique curve for the various datasets, which is described by the q-generalized gamma distribution (Fig.5.10) for the values of C = 0.4, β = 1.65, γ = 0.38 and q = The scattering in short T can be attributed to the small number of events (< 1000) that are included in the datasets for higher M0 and in the NOA dataset. This result indicates the self-similar temporal 151

169 Chapter 5 Temporal Properties of Seismicity and the Inter-event Time Distribution structure of seismicity in the Rift and implies that by just knowing the average seismic rate R in the region, the probability density of inter-event times p(τ) can be obtained as p( ) Rf ( T) (Corral, 2004), where f(t) can well be approximated by the q- generalized gamma distribution (Eq.5.2) for the parameter values found previously. Figure 5.11: Probability density p(t) of the rescaled inter-event times T for various time periods of stationary seismicity in the Rift. The solid line is the gamma distribution (Eq.5.1) for the values of C = 0.5, β = 1.7 and γ = The scaling behavior of p(t) is further studied for three periods of stationary seismicity in the Rift. These periods are recognized by the absence of significant increases in the seismicity rates and the constant increasing of the cumulative number of events, marked with the double arrows in Fig.5.2 and Fig.5.5. These include the earthquake activity in the Rift, shown in Fig.5.2 and the and earthquake activity in the West Corinth Rift, shown in Fig.5.5. The probability densities p(t) of the rescaled inter-event times T for those periods are shown in Fig The probability densities p(t) fall onto a unique curve that is best described by the gamma distribution (Eq.5.1) for the values of C = 0.5, β = 1.7 and γ = 0.56 (Fig.5.11). Thus, 152

170 Chapter 5 Temporal Properties of Seismicity and the Inter-event Time Distribution during periods of stationary seismicity in the Rift p(t) decays slowly as a power-law with an exponent of about 0.45, for short T and up to T 1 and for longer T it presents a fast exponential decay. The observed scaling is in agreement with the results of previous studies, which indicate this type of scaling for stationary seismicity (Corral, 2004; 2006; Hainzl et al., 2006). In addition, the probability densities p(t) of the rescaled inter-event times T for the declustered datasets are studied, in order to investigate the effect of aftershocks in the scaling behavior of the distributions. In this case the aftershocks, which present a shortterm clustering effect, have been removed from the HUSN and CRLN datasets. The observed p(t) for the two datasets are shown in Fig.B.4 (Appendix B) and approximately fall onto a unique curve that is well described by the q-generalized gamma distribution (Eq.5.2) for the values of C = 0.9, β = 0.8, γ = 0.75 and q = Thus, we observe a similar scaling behavior as for the entire datasets regarding the two power-law regimes at short and long inter-event times respectively. The depletion of aftershocks from the datasets results in a weaker short-term clustering effect, where the power-law exponent decreases ( p( T) 1 T, with γ = 0.38 for the entire datasets and γ = 0.75 for the declustered ones), while the scaling behavior for long-inter-event times that characterize the background activity remains the same (similar q-values) A dynamic mechanism for the inter-event time distribution Queiros (2005) developed a stochastic dynamic mechanism with memory effects, whose stationary solution is the q-generalized gamma distribution (Eq.5.2) that was introduced in the previous section. The dynamic scenario is based on the local fluctuations of the observable under study that produces the observed scaling. The q- generalized gamma distribution is similar to the F-distribution known from statistics (Marchand, 2003) and was empirically proposed to describe stock traded volume distributions in financial markets (Osorio et al., 2004). Financial time series present similar characteristics to earthquake time series, such as high fluctuations, power-law scaling and multifractal behavior (Ghashghaie et al., 1996; Mantegna and Stanley, 1999). The dynamic scenario of Queiros (2005) is here proposed as a possible dynamic 153

171 Chapter 5 Temporal Properties of Seismicity and the Inter-event Time Distribution mechanism for the observed scaling behavior of the earthquake inter-event time series and is further described below. We may consider that the evolution of earthquake activity is controlled by two parts. A first deterministic one that represents the aim of keeping the seismic rate stable to the typical value of 1 and similar to a restoring force γ. The second stochastic part represents the memory effects in the earthquake activity. Mathematically, this can be expressed by the following Feller process (Feller, 1951; Queiros, 2005): t dt T dt TdW, (5.3) where T is the inter-event time series normalized to the mean value T at some time t and Wt is a Wiener process following a Gaussian distribution with zero mean and unitary variance. Wt represents the stochastic process that mimics the microscopic effects in the evolution of T and due to its random sign leads to an increase (Wt > 0) or decrease (Wt < 0) of T. The term φ can be expressed as a function of the mean interevent time and the restoring force γ as, where ζ is a characteristic constant 2 of the system (Queiros, 2005). The corresponding Fokker-Planck equation for Eq.5.3 can be written as: 2 f T, t T f T, t T f 2 T, t. (5.4) t T T Its stationary solution is the gamma distribution (Gradshteyn and Ryzhik, 1965): T f ( T) exp T. (5.5) 1 Let s now consider local fluctuations in the seismic rate associated with nonstationarities in the evolution of the earthquake activity. In this case the mean interevent time fluctuates and we assume that it follows the stationary gamma distribution: 154

172 Chapter 5 Temporal Properties of Seismicity and the Inter-event Time Distribution ( 1) P( ) exp. (5.6) In this case Eq.5.5 provides the conditional probability of T given and the joint probability for obtaining certain values of T and is P( T, ) p( T ) P( ). The marginal probability of T, independent of, is now given by: P( T) P( T, ) d p( T ) P( ) d. (5.7) 0 0 By performing the integration in Eq.5.7 and from Eq.5.5 and Eq.5.6 we get 1 P( T) ( T) (1 T ). (5.8) By carrying out the change in the variables q 1 1,, 1 q 1 (5.9) and considering the q-exponential distribution given in Eq.2.43, Eq.5.8 can be written as (Queiros, 2005): 1 1 q 1 q 1 T T PT ( ) expq. (5.10) q 1 Eq.5.10 has the exact form of the q-generalized gamma distribution given in Eq.5.2. Dynamically, the gamma distribution is recovered if does not fluctuate. P( ) then equals the Dirac delta function centered in some value (Queiros, 2005). The previous stochastic dynamic mechanism is verified in the Corinth Rift. The nonstationary earthquake time series, where the mean inter-event time fluctuates, present two power-law regions at short and long inter-event times and scale according 155

173 Chapter 5 Temporal Properties of Seismicity and the Inter-event Time Distribution to the q-generalized gamma distribution. For stationary periods, where approximately constant, the scaling behavior approximates the gamma distribution. is 5.4 Multifractality in the earthquake time series In the previous section the histograms and the probability density of inter-event times were studied for various time periods and spatial scales in the Rift. The analysis indicated scaling and temporal clustering of seismicity at both short and long interevent times, in the case where the non-stationary earthquake time series were considered. The previous analysis represents only a first order approximation of the temporal structure of seismicity. In order to further study the structure and the clustering variability of the earthquake time series in the Corinth Rift, multifractal analysis is performed on the inter-event time series of the HUSN and CRLN datasets. The standard technique for applying the multifractal analysis was described in 2.2 and is based on the probability distribution of the variable and the calculation of its moments of order q. This technique, although quite useful for stationary or normalized series (Feder, 1988; Barabási and Vicsek, 1991), can produce erroneous results in the case of non-stationary series that present trends that cannot be normalized (Kantelhardt, 2009). To overcome this drawback, the wavelet transform modulus maxima (WTMM) method was introduced in the early 1990s (Muzy et al., 1991; Arneodo et al., 1995) and later on the multifractal detrended fluctuation analysis (MF-DFA) (Kantelhardt et al., 2002). Both WTMM and MF-DFA methods can eliminate polynomial trends from nonstationary series and reliably estimate the multifractal structure. However, MF-DFA is in many cases preferred due to its simple implementation and computational effort. In addition, MF-DFA seems to perform slightly better than WTMM for short series (Kantelhardt et al., 2002) and it is recommended in the majority of situations in which the fractal structure of the series is unknown a priori (Oświecimka et al., 2006). In general, two types of multifractality can be distinguished in the series (Kantelhardt et al., 2002). The first type is related to the range of fractal dimensions that are present due to different correlations in small and large fluctuations of the series. The second type is engendered due to a broad probability distribution. The simplest way to 156

174 Chapter 5 Temporal Properties of Seismicity and the Inter-event Time Distribution distinguish between the two types of multifractality is to perform the analysis on the randomly shuffled series (Kantelhardt et al., 2002). By randomly shuffling the series, any correlations due to the order of the successive events are destroyed, while the probability density remains unchanged. Hence, the shuffled series will exhibit nonmultifractal structure and random behavior if the observed multifractality is due to different correlations of small and large fluctuations. On the other hand, if the multifractal structure appears due to a broad probability distribution, it will not be affected by the shuffling procedure. In the case where both types of multifractality are present in the series, then the shuffled series will exhibit a weaker multifractality than the original. In the following, the MF-DFA method is described and then applied to the inter-event time series of the HUSN and CRLN datasets. The analysis is also performed on randomly shuffled copies of the original series, in order to distinguish between the different types of multifractality. The results for different threshold magnitudes and time periods are also compared Multifractal Detrended Fluctuation Analysis The implementation of the multifractal detrended fluctuation analysis (MF-DFA) to a fluctuating series u(i) of total length N (i=1,,n) consists of the following steps (Kantelhardt et al., 2002). The average <u> is first subtracted from the series u(i) and the resulting series is then integrated: k y( k) [ u( i) u ], (5.11) i1 with k = 1,2,,N. Then the integrated series y(k) is divided into non-overlapping segments of equal size n resulting in Nn=(N/n) segments. Since the length N of the series is not always a multiple of the segments size n, a short part at the end of y(k) is not included in the analysis. In order not to disregard this part, the procedure is repeated starting from the end of y(k), resulting in 2Nn total segments (Kantelhardt et al., 2001). In each of the 2Nn segments, y(ν) (ν = 1,2, 2Nn) is fitted to a polynomial function yn(ν) that represents the local trend in the series. The polynomial function yn(ν) can be of 157

175 Chapter 5 Temporal Properties of Seismicity and the Inter-event Time Distribution different orders l, thus removing different trends from the series. The polynomial trend is linear when l = 1, quadratic when l = 2 and cubic when l = 3. Then y(ν) is detrended by subtracting the local trend yn(ν) in each segment ν and the root mean-square fluctuation F(n, ν) is calculated as: 2N 1 n F( n, ) y( ) y ( ) 2 2 n. (5.12) Nn 1 The previous steps correspond to the standard Detrended Fluctuation Analysis (DFA) (Peng et al., 1994), a technique that is widely used in the estimation of long-range correlations in non-stationary fluctuating series (e.g., Bashan et al., 2008). Its main advantage over other scaling techniques, such as the autocorrelation function, spectral analysis or rescaled range analysis (e.g., Kantelhardt, 2009), is that it can reliably estimate the correlations in non-stationary series embedded in polynomial trends, thus avoiding any artifacts that are produced due to non-stationarities (Hu et al., 2001; Chen et al., 2002). The next step in MF-DFA is to estimate the qth order fluctuation function F q (n) by averaging over all segments n, as: 1 q 2Nn 1 2 q 2 N. (5.13) n 1 Fq ( n) F ( n, ) 2 The index variable q can take any real value and for q = 2 the standard DFA is retrieved. In the limit q 0 the exponent in Eq.5.13 diverges and instead a logarithmic averaging procedure is employed in the estimation of Fq(n) (Kantelhardt et al., 2002), 2N 1 n 2 F0 ( n) exp ln F ( n, ) 4N. (5.14) n 1 The previous steps are repeated for various segment sizes n and the scaling relation between the q-dependent fluctuation functions Fq(n) and n is estimated. For each value of q, the scaling behavior of the fluctuation functions can be determined by analyzing the plots of Fq(n) versus n. If the series u(i) are long-range power-law correlated, Fq(n) will increase as a power-law as a function of n, 158

176 Chapter 5 Temporal Properties of Seismicity and the Inter-event Time Distribution ( ) F ( ) ~ hq q n n, (5.15) with h(q) > 0.5. For h(q) < 0.5, the series are anti-correlated and for h(q) = 0.5 uncorrelated (random). For stationary series, h(2) is identical to the Hurst exponent H (Hurst, 1951; Fedder, 1988). Thus, h(q) can be called the generalized Hurst exponent or alternatively the q-order Hurst exponent. For very large segment sizes, n > N/4, the number of segments Nn in the averaging procedure of Eq.5.13 becomes very small and the estimation of Fq(n) becomes statistically unreliable (Kantelhardt et al., 2002). The previous steps are thus repeated for segment sizes up to N/4. If the series includes a range of fractal dimensions, then the series is multifractal and the exponent h(q) will depend on the various values of q. In this case the fluctuation functions scale differently. For positive values of q, the segments ν with large F 2 (n, ν) will dominate the average Fq(n) (Eq.5.3) and h(q) will describe the scaling behavior of the segments ν with large fluctuations. For negative values of q, h(q) will describe the scaling behavior of the segments ν with small fluctuations, as the segments ν with small F 2 (n, ν) will dominate the average Fq(n) (Kantelhardt et al., 2002). On the other hand, if the series is described by only one fractal dimension (monofractal series), the scaling behavior of F 2 (n, ν) is similar for all segments ν and h(q) will be independent of q. The generalized Hurst exponents h(q) are only one of the various types of scaling exponents that are used to characterize the multifractal structure of the series ( 2.2). The scaling exponents h(q) can directly be related to the q-order mass exponents τ(q) that correspond to the multifractal formalism based on the standard partition function (Kantelhardt et al., 2002), as: ( q) qh( q) 1. (5.16) For random and monofractal series, the mass exponent τ(q) is linearly dependent on q. If the structure of the series is multifractal, then this linearity breaks. The mass exponent τ(q) is related to the generalized fractal dimension Dq (Eq.2.4) as ( q) ( q 1) Dq (Kantelhardt, 2009). 159

177 Chapter 5 Temporal Properties of Seismicity and the Inter-event Time Distribution The other way to characterize a multifractal series is the singularity spectrum f(a). The singularity spectrum can directly be estimated from the mass exponents τ(q). By using the Legendre transform (Fedder, 1988): d, (5.17) dq the singularity spectrum f(a) is obtained as: f ( ) q ( q), (5.18) where a is the singularity strength or Hölder exponent and f(a) indicates the dimension of the subset of the series that has the same singularity strength a (see also 2.2). In monofractal series, the singularity strength is similar over the entire range of values and f(a) collapses into a single point Multifractal Analysis of the inter-event time series To characterize the structure of the earthquake time series in the Corinth Rift, the MF- DFA method is applied to the inter-event time series of the HUSN and CRLN datasets, as those were described in and respectively. The inter-event time series τ (for τ 60 sec) were initially divided into 50 non-overlapping and logarithmically spaced time segments n, of minimum size n = 10 events and up to the maximum size of n = N/4. The analysis was performed for q 5,5 and step size of 0.2. The Matlab script for performing the MF-DFA is provided in Appendix C. In Fig.5.12 the logarithm of the fluctuation functions Fq(n) that resulted from the analysis are shown as a function of the logarithm of the segment sizes n for the two datasets. The analysis was performed for various orders l of the detrending polynomial function yn(ν) with similar results for l 2, indicating that a second order (l = 2) polynomial function is sufficient to remove any trends from the series. In general, Fq(n) grows as a power-law with n for both datasets (Fig.5.12), indicating the presence of scaling. For the HUSN dataset and for q = -5, Fq(n) scales with n as a power-law with exponent h(q) = 1 ±0.05 and for q = 5 with exponent h(q) = 0.64 ±0.03, indicating that large (for q > 0) and small fluctuations (for q < 0) scale differently (Fig.5.12a). For 160

178 Chapter 5 Temporal Properties of Seismicity and the Inter-event Time Distribution segment sizes greater than 289 events (logn = 2.46) a crossover (dotted line in Fig.5.12a) appears in the scaling behavior of Fq(n). The power-law exponents h(q) increase for n > 289 events and take the values of h(q) = 0.8 ±0.1, for q = 5, and h(q) = 2 ±0.34, for q = -5 (Fig.5.12a). For the CRLN dataset a similar crossover in the scaling behavior of Fq(n) appears at n = 361 events (logn = 2.56, marked with the dotted line in Fig.5.12b), indicating that the fluctuation functions Fq(n) scale differently for small and large segment sizes n. The segment size of n = 361 events in the CRLN dataset usually corresponds to time-scales of a month or more. For segment sizes n < 361 the fluctuation functions scale as a power-law with n with h(q) = 0.56 ±0.02, for q = 5 and h(q) = 0.59 ±0.04, for q = -5. The similar exponents h(q) for the various orders of q indicate that for small segment sizes, the inter-event time series in the West Corinth Rift behaves as a single fractal (monofractal), forming single temporal clusters. For larger segment sizes (n > 361), small and large fluctuations scale differently for the various orders of q, with h(q) = 0.84 ±0.05, for q = 5 and h(q) = 2.82 ±0.22, for q = -5 (Fig.5.12b). The entire range of generalized Hurst exponents h(q) for the various orders of q is shown in Fig.5.13a, for the HUSN dataset and in Fig.5.13b, for the CRLN dataset. The exponents h(q) are estimated for the various orders of q, for n < 289 and for n > 289 for the HUSN dataset, and for n > 361 for the CRLN dataset. For both datasets, small and large fluctuations scale differently and the generalized Hurst exponents h(q) decrease monotonically with increasing q (Fig.5.13), behavior that is typical for multifractal series. The values of h(q) for long segment sizes n are larger for both datasets (Fig.5.13) and the decrease of h(q) with increasing q is more pronounced, indicating a more persistent behavior at long time-scales. 161

179 Chapter 5 Temporal Properties of Seismicity and the Inter-event Time Distribution a) b) Figure 5.12: The logarithm of the fluctuation function Fq(n) versus the logarithm of the segment sizes n for a) the HUSN dataset and b) the CRLN dataset. The fluctuation functions are estimated for various values of q 5,5 and step size 0.2, increasing from bottom to top. A second order polynomial (l = 2) was used for detrending the inter-event time series. 162

180 Chapter 5 Temporal Properties of Seismicity and the Inter-event Time Distribution a) b) Figure 5.13: The range of generalized Hurst exponents h(q) for various values of q [-5, 5] and step size 0.2, for a) the HUSN dataset and b) the CRLN dataset. The corresponding confidence intervals are plotted as error bars. The mean h(q) that resulted from 10 randomly shuffled copies of the original inter-event time series is also plotted. The q-order mass exponents τ(q) for the two datasets are estimated from Eq.5.16 and are plotted in Fig.5.14 as a function of q. For both series, the mass exponents τ(q) grow 163

181 Chapter 5 Temporal Properties of Seismicity and the Inter-event Time Distribution non-linearly as a function of q, exhibiting different scaling behavior for q < 0 and q > 0, as it is expected for multifractal series. a) b) Figure 5.14: The mass exponent τ(q) for the various orders of q, for the HUSN dataset and b) the CRLN dataset. The mean τ(q) that resulted from 10 randomly shuffled copies of the original inter-event time series is also shown. Furthermore, by using the Legendre transform (Eq.5.17), the singularity spectrum f(a) is estimated from Eq.5.18 for the various Hölder exponents a. The estimated f(a) for 164

182 Chapter 5 Temporal Properties of Seismicity and the Inter-event Time Distribution the two datasets is shown in Fig In both cases the spectrum is wide, indicating multifractality in the earthquake time series. The spectrum f(a) for the HUSN dataset is wider for long segment sizes than for short ones, indicating a wider range of fractal dimensions (Fig.5.15a). The maximum frequency (f(a) = 1) is observed for Hölder exponents a in the range of for the HUSN dataset and for short segment sizes (Fig.5.15a), while for long segment sizes the maximum frequency is observed for a in the range of for both datasets (Fig.5.15). In the latter case, the singularity spectrums f(a) are broader to the right for both datasets, indicating that small fluctuations due to the negative orders of q are more inhomogeneous than the large fluctuations (for q > 0). In addition, the width of the singularity spectrum f(a), defined as W = amax amin, indicates the range of fractal exponents that are present in the series and it can then be considered as a measure of the degree of multifractality (Ashkenazy et al., 2003; Telesca and Lapenna, 2006). For the HUSN dataset and for short segment sizes W 0.64, whereas for the longer ones W 1.6. For the CRLN dataset, W 2.5, indicating a wider spectrum and a richer structure in fractal exponents in the earthquake activity at the West Corinth Rift. To distinguish between the types of multifractality that are present in the inter-event time series, the previous analysis is performed for randomly shuffled copies of the original series. In this way, the correlations due to the order of the successive earthquakes are destroyed. For each of the two datasets, ten randomly shuffled copies were generated from the original series and MF-DFA was performed. The resulting values were averaged and the mean hshuf(q), τshuf(q), αshuf and fshuf(α) are shown in Fig.5.13, Fig.5.14 and Fig.5.15 respectively, along with the results of the analysis for the original series. For the HUSN dataset, the mean hshuf(q) varies between hshuf(-5) = 0.58 ±0.03 and hshuf(5) = 0.46 ±0.03 and for the CRLN dataset between hshuf(-5) = 0.56 ±0.02 and hshuf(5) = 0.46 ±0.03. For both datasets, hshuf(2) that corresponds to the scaling exponent of the standard DFA analysis is hshuf(2) 0.5, indicating random behavior of the shuffled series. The variations of the mean hshuf(q) around the value of 0.5 indicate the loss of correlations in the shuffled series. However a weak dependence of hshuf(q) on q still remains for both datasets (Fig.5.13). The loss of multifractality in the shuffled series can also be observed in Fig.5.14, where the mean τshuf(q) is almost linear with q and in Fig.5.15, where the range of Hölder exponents αshuf has substantially decreased. These results indicate that the observed multifractality in the 165

183 Chapter 5 Temporal Properties of Seismicity and the Inter-event Time Distribution original inter-event time series is due to different long-range correlations of small and large fluctuations. However, a small contribution from the broad probability densities, found for the two datasets in 5.3.1, to the observed multifractality cannot be excluded. a) b) Figure 5.15: The singularity spectrum f(a) as a function of the Hölder exponent a, for a) the HUSN dataset and b) the CRLN dataset. The mean singularity spectrum f(a), which resulted from ten randomly shuffled copies of the original inter-event time series, is also plotted for the two datasets respectively. 166

184 Chapter 5 Temporal Properties of Seismicity and the Inter-event Time Distribution In summary, the results of the analysis indicated that for short segment sizes, the interevent time series has a multifractal structure for the entire area of the Rift (HUSN dataset), while for the west part of the Rift (CRLN dataset) the series is more homogeneous and approaches a monofractal structure. For long segment sizes the structure of the series is multifractal for both datasets and richer in structure, with a wider range of fractal exponents present. The crossover between the two scaling behaviors in the fluctuation functions occur at similar segment sizes for the two datasets (n 300 events), which usually correspond to time-scales of a month or more. The latter suggests that for short time-scales the time dynamics in the west part of the Rift are more homogeneous with earthquakes belonging to single temporal clusters. For the entire area of the Rift this effect is not observed, as the activation of multiple seismogenic sources along the Rift favors heterogeneity in the temporal structure of seismicity and multifractality. The latter effect and the multifractal temporal structure of seismicity become more pronounced for longer time-scales, where multiple temporal clusters are active. Multifractal analysis was also performed on the inter-event time series of the declustered HUSN and CRLN datasets and the results are shown in Appendix B. In this case, the depletion of aftershocks results in weaker multifractality in both datasets (Fig.B.5 and Fig.B.6). A crossover in the scaling behavior of Fq(n) for the HUSN dataset appears for large Fq(n) (q > 0) (Fig.B.5a), while this crossover vanishes for small Fq(n) (q < 0). The range of generalized Hurst exponents h(q) varies between h(- 5) = 0.74 ±0.06 and h(5) = 0.36 ±0.03 for the HUSN dataset (n < 80) (Fig.B.5b) and between h(-5) = 1.05 ±0.05 and h(5) = 0.76 ±0.04 for the CRLN dataset (Fig.B.6b) The effect of the threshold magnitude The effect of the threshold magnitude (Mth) in the multifractal structure of the interevent time series for the two datasets is further studied by applying MF-DFA for various Mth. In this case the analysis is performed on the inter-event time series, for the events with M Mth, with Mth varying between for the HUSN catalogue and for the CRLN catalogue. The results are presented in terms of the singularity spectrum s 167

185 Chapter 5 Temporal Properties of Seismicity and the Inter-event Time Distribution width W as a function of Mth, which shows how multifractality degree changes with Mth. The results of the analysis for the two datasets are shown in Fig The multifractal structure of the inter-event time series is evident for the entire range of threshold magnitudes Mth. The degree of multifractality, as expressed through the singularity spectrum s width W, is quite stable for Mth in the range of and for the CRLN (n > 361) and HUSN (n < 289) datasets, respectively. For greater Mth, W increases for both datasets, indicating a wider range of Hölder exponents a, while for long segment sizes (n > 289) in the HUSN dataset W remains quite stable for greater Mth (Fig.5.16). Figure 5.16: Singularity spectrum s width W as a function of the threshold magnitude Mth, for the CRLN (circles) and the HUSN (squares) datasets Temporal variations Multifractal analysis on the inter-event time series provides insights into the temporal structure of earthquake activity. However, it does not provide any information on the 168

186 Chapter 5 Temporal Properties of Seismicity and the Inter-event Time Distribution dynamic changes that may occur in the evolution of earthquake activity during the considered period. In order to perform such an analysis, MF-DFA is applied to the interevent time series at different time intervals by using a sliding temporal window F, which is defined by the width w and the sliding factor Δ (Gamero et al., 1997). The set of inter-event times τj in each temporal window is defined as: F, j 1 m,..., w m, m 0,1,2,..., N, (5.19) m j m where m controls the time displacement of the sliding window. For instance, if w = 100 and Δ = 50, the set of inter-event times is W,,...,, W,,..., etc. The width of w = 10 3 and the sliding factor of Δ = 100 were used for the analysis, resulting in 90% overlap between the successive time windows. The particular selection of w and Δ assures a sufficient number of events in each temporal window and a good smoothing and resolution between the estimated values and time. MF-DFA is performed on the inter-event time series in each temporal window and the width W of the singularity spectrum is estimated as an indication of the multifractality degree during this period. The estimated W is then associated with the time of the last earthquake in each temporal window F and W is plotted as function of time. The results of the analysis are shown in Fig For both datasets, the degree of multifractality exhibits strong variations with time that are associated with the range of Hölder exponents, a, that is present in each time period. In particular for the HUSN dataset, these variations are more intense during , where the seismicity is non-stationary and the rate exhibits strong variations (Fig.5.2). The most abrupt change occurs after the 2010 Efpalion earthquake (Fig.5.17a). After this earthquake, the triggered aftershocks dominate the earthquake activity in the Rift. For small segment sizes (n < 289) the loss of multifractality and the tendency towards monofractality is observed (Fig.5.17a), indicating more homogeneous time dynamics. Telesca and Lapenna (2006) observed similar behavior in central Italy, where a loss of multifractality in the inter-event time series occurred during the aftershock sequence that followed a strong event (MD = 5.8) on September 26, For larger segment sizes (n > 289) the multifractality degree increases after the 2010 Efpalion earthquake 169

187 Chapter 5 Temporal Properties of Seismicity and the Inter-event Time Distribution a) b) Figure 5.17: Singularity spectrum s width W, as a measure of the degree of multifractality over time, for sliding temporal windows of width w = 10 3 and sliding factor Δ = 100, for a) the HUSN dataset and b) the CRLN dataset (n > 361). indicating a heterogeneous clustering degree that is enhanced by mixing the background activity with aftershocks. During , where the seismicity rate is almost constant in the Rift (Fig.5.2), the degree of multifractality is nearly constant (Fig.5.17a). During periods where both high and low seismicity rates are incorporated in the dataset, the singularity spectrum is richer in structure, exhibiting a wider range 170

188 Chapter 5 Temporal Properties of Seismicity and the Inter-event Time Distribution of Hölder exponents a. This effect can be observed in the second half of 2010 and during , where the background earthquake activity mixes first with the Efpalion aftershock sequence and then with two earthquake swarms that occurred in the Rift ( 5.2.1). Larger variations in the degree of multifractality appear in the inter-event time series for the West Corinth Rift (CRLN dataset) (Fig.5.17b). However, this dataset shows similar characteristics to the HUSN dataset. Lower values of multifractality and the tendency towards monofractality are observed after the 2010 Efaplion earthquake and in the beginning of 2007, where the Efpalion aftershock sequence and the 2007 earthquake swarm sequence (Bourouis and Cornet, 2009) dominate the activity in the West Corinth Rift ( 5.2.2). Other low values of multifractality are observed during , the end of 2005 and the beginning of 2006 and during 2012 and are associated with periods of stationary activity (Fig.5.5). However, during the stationary periods of , mid-2006, 2008 and 2013 (Fig.5.5), abrupt increases in the degree of multifractality are also observed (Fig.5.17b), indicating the highly variable degree of clustering in the temporal evolution of earthquake activity in the Rift. 5.5 Discussion In the present chapter the temporal properties of earthquake activity in the Corinth Rift were studied in terms of the inter-event time series, during the period of , for the entire area of the Rift and during for the West Corinth Rift. The earthquake activity during the considered periods exhibit non-stationary character, with periods of low to moderate activity interspersed with periods of high seismicity rates, which are associated with frequent earthquake swarms and the occurrence of the two 2010 Efpalion strong events that were followed by numerous aftershocks. The probability densities of the inter-event times for the two datasets (Fig.5.7 and Fig.5.8) exhibit a crossover behavior between slow power-law decay for short interevent times and faster power-law decay for long inter-event times. This property suggests clustering effects at all timescales and memory in the evolution of the earthquake activity in the Rift. Short and long-term clustering in the time intervals between successive earthquakes has been found in previous studies by Kagan and 171

189 Chapter 5 Temporal Properties of Seismicity and the Inter-event Time Distribution Jackson (1991), for various earthquake catalogues and by Corral (2003) for Southern California. This property can be viewed in terms of probabilities of subsequent earthquakes (e.g., Sornette and Knopoff, 1997). The probability densities in the Corinth Rift suggest that the probability of a subsequent earthquake is high immediately after the occurrence of the previous one and decreases slowly up to a characteristic time, where a crossover to faster decaying probabilities is observed. Short and long-term clustering effects further suggest that short and long inter-event times are more likely to be followed by short and long ones respectively, similar to the results of Corral (2003; 2004), Livina et al. (2005) and Lennartz et al. (2008), which suggested longterm correlations and memory in the seismogenic process. After rescaling inter-event times with the mean seismic rates during the considered periods, the probability densities fall approximately onto a unique curve that is well described by the q-generalized gamma distribution (Eq.5.2), which presents two powerlaw regimes at both short and long inter-event times. This type of scaling suggests that the rescaled inter-event times scale as 1 T, for short time intervals and as T 1 1 q, for long time intervals. In Michas et al. (2013), we suggested this type of scaling behavior for the earthquake activity in the West Corinth Rift and here it is verified for various time periods, spatial scales and threshold magnitudes in the Rift (Fig.5.10). This result indicates the self-similarity of earthquake time series in the Rift and suggests that the probability density of inter-event times p(τ) can be drawn from p() Rf T, where R is the mean seismicity rate and f(t) the scaling function that can well be described by the q-generalized gamma distribution. However, if periods of stationary seismicity in the Rift are considered, the rescaled inter-event times exhibit probability densities that scale according to the gamma distribution (Eq.5.1) (Fig.5.11), in accordance with the results of Corral (2004). This type of scaling and the exponential decay of long inter-event times for stationary periods suggest a fraction of uncorrelated activity in the Rift. According to Molchan (2005) and Hainzl et al. (2006), the fraction of uncorrelated background activity can be estimated as 1, where β is the scaling parameter incorporated in the gamma distribution (Eq.5.1). Following Molchan (2005) and Hainzl et al. (2006), for β = 1.7, this fraction corresponds to almost 60% of uncorrelated background seismicity, a value that is almost similar to the fraction 172

190 Chapter 5 Temporal Properties of Seismicity and the Inter-event Time Distribution resulted from the declustering analysis, which indicated 50% and 65% of background activity in the HUSN and CRLN datasets respectively. Furthermore, the stochastic dynamic mechanism with memory effects, which was here proposed as one that reproduces the observed scaling behavior, is consistent with the obtained results. During non-stationary periods, where the seismic rate fluctuates, the solution of this mechanism is the q-generalized gamma distribution, which was found to describe the scaling behavior of inter-event times during periods of non-stationary earthquake activity in the Rift (Fig.5.9 and Fig.5.10). This mechanism recovers the standard gamma distribution during periods of stationary activity, similar to the results found for the probability densities of the rescaled inter-event times during periods of stationary earthquake activity in the Rift (Fig.5.11). In addition, the clustering variability in the temporal structure of seismicity was further studied by applying multifractal detrended fluctuation analysis (MF-DFA) to the interevent time series. The multifractal analysis showed that small and large fluctuations scale differently, indicating a multifractal structure in the inter-event time series and a heterogeneous degree of clustering. A crossover behavior between short and long segment sizes appears for both datasets. For short segment sizes, the inter-event time series has a multifractal structure for the entire area of the Rift (HUSN dataset), while for the west part of the Rift (CRLN dataset) the series approaches a monofractal structure. For long segment sizes the structure of the series is multifractal for both datasets. The range of generalized Hurst exponents further indicated long-term correlations and non-poissonian temporal behavior. Analysis of the randomly shuffled series indicated that the observed multifractality is mainly due to different long-range correlations in the series. Analysis of the singularity spectrum s width, as a measure of the degree of multifractality with time, showed large fluctuations for the different time periods (Fig.5.17). The large fluctuations are related to periods of high and low earthquake activity in the Rift and indicate the clustering variability in the temporal evolution of seismicity. 173

191 Chapter 6 Earthquake Diffusion 6.1 Introduction Earthquake diffusion refers to the expansion or migration of the spatial zone of earthquakes with time. Such phenomena are frequently observed in cases of triggered seismicity, such as in aftershock sequences or in earthquake swarms (Tajima and Kanamori, 1985; Noir et al., 1997; Jacques et al., 1999; Antonioli et al., 2005). The simplest mechanism for aftershock triggering is coseismic stress transfer and increasing Coulomb stress due to the occurrence of a strong earthquake (King et al., 1994; Stein, 1999). The conditions for which a fault fails are expressed mathematically by the empirical Coulomb failure criterion: c s s n p, (6.1) where τc is the Coulomb stress, μs the friction coefficient, σs and σn the shear and normal stresses to a specified fault plane and p the pore-pressure in the fault (King et al., 1994; Stein, 1999). Aftershock diffusion can thus be related to a stress diffusion process, which corresponds to the propagation of a stress pulse away from the initial strong event at time scales much larger than those involved in the propagation of seismic waves (Marsan et al., 2000; Helmstetter et al., 2003). Another plausible mechanism is

192 Chapter 6 Earthquake Diffusion the existence of a pore-pressure diffusion process at depth, induced by the regional stress redistribution caused by the occurrence of strong events (Bosl and Nur, 2002). In a fluid saturated and critically stressed crust, even small pore-pressure perturbations are known to decrease the effective strength along pre-existing faults (Eq.6.1) and trigger earthquakes (Nur and Booker, 1972; Talwani and Acree, 1984; Costain and Bollinger, 2010). Other mechanisms that have been proposed to induce aftershock diffusion is the rate and state friction model (Dieterich, 1994; Marsan et al., 2000), a subcritical growth mechanism (Huc and Main, 2003) or a cascade triggering of events that leads to the expansion of the aftershock zone (Helmstetter and Sornette, 2002a; Helmstetter et al., 2003). Fluid diffusion as a likely mechanism for triggering earthquake swarms has been reported in numerous studies (e.g., Kisslinger, 1975; Yamashita, 1999; Parotidis et al., 2005). Such a process is often assumed to be caused by an intrusion of fluids into a fluid saturated seismogenic zone (Hainzl and Fischer, 2002). The diffusion of the porepressure through the fractures and faults that act as conduits can decrease their effective strength (Eq.6.1) and trigger earthquake swarm sequences, which are characterized by the strong clustering of events in time and space and can neither be described by a dominant earthquake nor by the Omori scaling relation (Eq.2.12) known in aftershock sequences (Hainzl, 2003). Characteristic cases can be drawn from volcanic earthquake swarms, where the spatio-temporal evolution of seismicity is consistent with the intrusion and movement of magma (e.g., Hough et al., 2000; Yukutake et al., 2011). Other cases can be found in geothermal regions (Parotidis et al., 2005; Kato et al., 2010), fluid injection experiments (Shapiro et al., 2002) and other seismogenic regions (Bourouis and Cornet, 2009; Bisrat et al., 2012). In the present chapter, the earthquake diffusion properties in the Corinth Rift are studied for two earthquake sequences. The first one refers to the 2010 Efpalion aftershock sequence that has been associated with coseismic stress transfer due to the occurrence of the two strong Efpalion earthquakes (Karakostas et al., 2012; Ganas et al., 2013). The second case is the 2001 Agios Ioannis earthquake swarm sequence that has been associated with fluid diffusion at depth (Pacchiani and Lyon-Caen, 2010). The spatiotemporal scaling properties of the two sequences are studied in terms of nonextensive statistical mechanics and the Continuous Time Random Walk (CTRW) 175

193 Chapter 6 Earthquake Diffusion theory that represents one of the main physical theories for the occurrence of anomalous non-gaussian diffusion in highly heterogeneous media (Bouchaud and Georges, 1990; Berkowitz and Scher, 1998). In the following sections, an overview to the CTRW approach to earthquake diffusion is initially provided, followed by the analysis and discussion of the diffusion properties of the two earthquake sequences. 6.2 Continuous Time Random Walk approach to earthquake diffusion The Continuous Time Random Walk (CTRW) model was introduced by Montroll and Weiss (1965) as a generalization of the simple random walk model, where the walker jumps by ± 1 spatial step on a discrete lattice at each time step. The CTRW model considers a random walker, which starts on the origin (x0 = 0) at time t0 = 0. The walker stays fixed to this position until time t1, where he makes a jump of length r1 to the position x1. He then stays at this position until time t2 (t2 > t1), when he jumps to a new location x2 of length r2 from the previous one and the process is renewed. In the CTRW model the spatial step or jump of the walker r, as well as the time step or the waiting time between the successive jumps τ, are drawn from a joint probability density function (pdf) ψ(r, τ), which is usually called the jump pdf. In the present overview of the CTRW theory, the jump lengths and waiting times are considered as independent random variables that correspond to the decoupled form of the jump pdf r, r. r x x From ψ(r, τ), the jump length pdf λ(r) and the waiting time pdf (τ) can be deduced as (Metzler and Klafter, 2000): ( r) d ( r, ) 0 (6.2) and ( ) dr ( r, ). (6.3) 176

194 Chapter 6 Earthquake Diffusion According to the previous, λ(r)dr produces the probability for a jump length in the interval (r, r+dr) and (τ)dτ the probability for a waiting time in the interval (τ, τ +dτ). Different types of CTRW processes can now be distinguished according to the characteristic waiting time T: T d() 0 (6.4) and the jump length variance Σ 2 : 2 2 dr() r r. (6.5) If both are finite, over a long time limit the CTRW corresponds to Brownian motion (Metzler and Klafter, 2000). Let s now assume that λ(r) and (τ) have a power-law asymptotic behavior: (6.6) and. (6.7) The cumulative distributions of λ(r) and (τ) are given respectively by: ( r) ( r) dr r 0 (6.8) ( ) ( )d 0. (6.9) For μ 2 and β 1, the characteristic waiting time and the jump length variance are finite and the standard diffusion case (Brownian diffusion) is recovered. If β 1 and μ 2 2 < 2, the variance r is infinite. This regime of long jumps corresponds to the socalled Lévy flights and super-diffusion. If μ 2 and 0 < β < 1, the characteristic waiting time is infinite and the waiting time pdf exhibits a broad distribution with asymptotic 177

195 Chapter 6 Earthquake Diffusion power-law behavior. This regime corresponds to sub-diffusion (Bouchaud and Georges, 1990). One of the objectives in CTRW theory is to estimate p(x,t), the probability distribution of the walker being at position x at time t in arbitrary dimensions (1-D, 2-D or 3-D). In the case of normal diffusion, the pdf p(x,t) is governed by the diffusion equation 20 (1- D case): 2 p D p( x, t), (6.10) 2 t x where D is the diffusion coefficient. The diffusion equation (6.10) is a direct outcome of the central limit theorem and its solution at long timescales for the initial condition,0 x p x, where δ(x) is the Dirac s delta function, is the Gaussian distribution (e.g., Bouchaud and Georges, 1990). In this case, the mean squared displacement (or variance) depends linearly on time: x 2 () t Dt. (6.11) Instead, the hallmark of anomalous diffusion is the non-linear growth of the mean squared displacement with time. Many functions of x 2 () t exist (e.g., Hughes, 1995), but a very frequent one is the function:, (6.12) which is found to describe a wide class of complex systems (e.g., Bouchaud and Georges, 1990; Shlesinger et al., 1993) and earthquake diffusion (Marsan et al., 2000, Huc and Main, 2003; Helmstetter et al., 2003) 21. The latter equation manifests anomalous diffusion and is connected with the breakdown of the central limit theorem, caused by broad probability distributions and long-range correlations (Metzler and Klafter, 2000). Thus, these properties lead to non-gaussian propagation of the walker 20 The diffusion equation (6.10) corresponds to the so-called heat equation and is connected to diffusion and Brownian motion through the Fokker-Planck equation (e.g., Hughes, 1995). 21 Following these studies, the anomalous diffusion exponent is here denoted as H. 178

196 Chapter 6 Earthquake Diffusion with time. The anomalous diffusion exponent H in Eq.6.12 characterizes the different domains of anomalous diffusion. For H > 0.5 the transport is super-diffusive, for 0 < H < 0.5 it is sub-diffusive and for H = 0.5 normal diffusion is recovered. In the case of seismicity, the diffusion exponent H provides a quantitative measure of the rate of triggered earthquake diffusion (McKernon and Main, 2005). The average properties of the percolation of stress outward from a triggering event can partly control the diffusion of triggered seismicity and as such H can also be thought of as an indicator of a possibly stress-related diffusive process (Marsan et al., 2000; McKernon and Main, 2005). In the anomalous diffusion case, the pdf p(x,t), also called the propagator, has been approximated by various models, including CTRW models and fractional equations (Metzler and Klafter, 2000; 2004) or non-extensive statistical mechanics (Tsallis and Bukman, 1996; Borland, 1998; Tsallis 2009a). These approximations are out of the scope of the present thesis and the following analysis is focused on the spatio-temporal scaling properties of seismicity and the mean squared displacement (Eq.6.12) in order to describe the characteristics of earthquake diffusion in the Corinth Rift. 6.3 The 2010 Efpalion earthquakes On January 18, 2010 a strong earthquake (Mw = 5.1, NOA), followed four days later by another strong event (Mw = 5.1, NOA) and numerous aftershocks, occurred in the West Corinth Rift near the city of Efpalion (Fig.6.1). These two events were the strongest earthquakes in the area since the 1995 Aigion earthquake (Ms = 6.2). Focal mechanism solutions for the two strong events indicated the activation of two normal fault planes of an approximately E-W direction (Sokos et al., 2012; Karakostas et al., 2012). Slip vector directions for the two strong events and the aftershocks have a NNW-SSE to NNE-SSW orientation that is almost parallel to the direction of extension in the Corinth Rift (Karakostas et al., 2012). 179

197 Chapter 6 Earthquake Diffusion Figure 6.1: Spatial distribution of the 2010 Efpalion earthquake sequence (earthquake catalogue from Karakostas et al., 2012). The two stars indicate the location of the two strong events. Red and white colors indicate the first and second strong events and their aftershocks, respectively. The faults are after Fig The spatial distribution of the Efpalion earthquake sequence, for the period January 18, 2010 March 9, 2010, is shown in Fig.6.1. The earthquake dataset is from the work of Karakostas et al. (2012) and has 587 events for the 50 days period after the first strong event, with earthquake magnitudes down to M = 1.5. Karakostas et al. (2012) used the P- and S-wave recordings from the HUSN network to relocate the events and to compile the earthquake catalogue that is shown in Fig.6.1. The mean horizontal error of the relocated events is equal to 1.42 km and the mean vertical error 1.51 km. 180

198 Chapter 6 Earthquake Diffusion a) b) Figure 6.2: Coulomb stress changes due to the coseismic slip a) of the first main shock and b) of the second strong event. White and black solid lines represent the fault traces, associated with the first and second strong events respectively in a) and vice versa in b). The second strong event and most of the aftershocks are located inside the positive Coulomb stress changes (~ 10 bars) (red areas) (from Karakostas et al., 2012). 181

199 Chapter 6 Earthquake Diffusion The spatial distribution of the relocated events indicates an E-W striking active zone of about 15 km total length in the NW part of the Corinth Rift (Fig.6.1) (Karakostas et al., 2012). After the occurrence of the first strong event on January 18 th the earthquake activity propagates to the eastern part of the active zone, in the epicentral area of the second strong event (Fig.6.1), which occurred 4 days later on January 22 nd. After the occurrence of the second strong event, the aftershock activity continues to be higher in the eastern part of the active zone and after several days a cluster of events appear to the west of the first strong event, defining the westernmost part of the active aftershock zone (Fig.6.1). Fault plane solutions for all events and the spatial distribution of aftershocks indicate the activation of multiple fault segments at depths of 7 11 km (Karakostas et al., 2012; Ganas et al., 2013). The static stress transfer due to the occurrence of the first strong event was calculated by Karakostas et al. (2012), Sokos et al. (2012) and Ganas et al. (2013) by using the change in the Coulomb failure function (Eq.6.1). In Fig.6.2 the Coulomb stress changes due to the coseismic slip of the two strong events are shown after Karakostas et al. (2012). The analysis indicated that the second strong event and most of the aftershocks that followed the first strong event occurred in the zone of positive Coulomb changes, which take values up to ~ 10 bars (Fig.6.2a). The latter implies that the static stress transfer due to the occurrence of the first strong event could have triggered the second strong event and most of the aftershocks, in accordance with the conclusions of Sokos et al. (2012) and Ganas et al. (2013). After the occurrence of the second strong event, aftershocks in the western part of the active zone occurred in the positive Coulomb changes zone ( 0.1 bar) (Fig.6.2b). The correlation between the Coulomb stress changes and the spatial distribution of the aftershocks supports the cascade triggering of events after the occurrence of the first strong event (Karakostas et al., 2012) Spatiotemporal scaling properties In the case of seismicity, the jump lengths r and the waiting times τ of the random walker correspond to the inter-event distances r and times τ between successive earthquakes. The inter-event times τ were defined in 5.2, whereas the inter-event distances r are defined as the 3-D distances between the successive events r xi 1 xi 182

200 Chapter 6 Earthquake Diffusion (i = 1, 2,...,N-1, with N the total number of events), where xi is the location of the i th event, given by its longitude, latitude and depth. Rather than calculating the distance between two points in the Earth s crust by using simple trigonometry and the Pythagorean theorem, the curvature of the Earth is taken in account and the Vincenty s formula (Vincenty, 1975), which considers the shape of the Earth as an oblate spheroid, was used to calculate the distances between successive events (specified by their longitudes and latitudes) on the surface of the spheroid. The 3-D distances were then obtained by using simple trigonometry. The Matlab script for calculating the distance between two points in the surface of a spheroid according to the Vincenty s formula is provided in Appendix C. The cumulative distributions of inter-event times P(>τ) and distances P(>r) for the Efpalion earthquake sequence are shown in Fig.6.3 and Fig.6.4 respectively. The interevent times and distances were calculated from the relocated earthquake catalogue of Karakostas et al. (2012). The cumulative distribution is here preferred to the probability distribution because for a relative small number of events, as in the case of the Efaplion earthquake sequence, it produces smoother trends in the distribution. Furthermore, the scaling properties of P(>τ) and P(>r) can directly be compared to Λ(>r) and Φ(>τ), given in Eq.6.8 and Eq.6.9 respectively, in order to characterize the diffusion regime, based on the spatio-temporal scaling properties of the sequence. P(>τ) and P(>r) are fitted to Eq.2.46 that for τ and r reads as: P( ) 1 1 q 0 2q 1q (6.13) and 2 qr 1qr r P( r) 1 1 qr r 0. (6.14) Eq.6.13 and Eq.6.14 optimize the non-additive entropy Sq (Eq.2.35) for the inter-event times and distances, as it was described in The fitting procedure follows the nonlinear least-square algorithm that was described in and Eq.6.13 provides a good fit to the majority of the observed inter-event times (up to τ = 0.6 days) for the values of qτ = 1.55 ±0.03 and τ0 = ± days (Fig.6.3). For τ > 0.6 days 183

201 Chapter 6 Earthquake Diffusion a fall-off in P(>τ) appears that can be attributed to finite size effects in the inter-event times of the sequence. In the case of P(>r), Eq.6.14 provides a good fit to the full range of the observed inter-event distances for the values of qr = 0.55 ±0.02 and r0 = 7.2 ±0.3 km (Fig.6.4). Figure 6.3: The cumulative distribution of inter-event times P(>τ) for the Efpalion earthquake sequence. The solid line represents Eq.6.13 for the values of qτ = 1.55 ±0.03 and τ0 = ± days. Scaling of P(>τ) and P(>r) according to Eq.6.13 and Eq.6.14 respectively indicates asymptotic power-law scaling according to ~ x. For the q-values of qτ = 1.55 and qr = 0.55 found previously, the latter indicates that inter-event times scale asymptotically ~ as and inter-event distances as ~ r. When compared with Eq.6.8 and Eq.6.9, the power-law exponents found for the inter-event times and distances correspond to the regime of μ 2 and 0 < β < 1, which indicates an anomalous subdiffusive process in the spatiotemporal evolution of the Efpalion earthquake sequence. 2q 1q 184

202 Chapter 6 Earthquake Diffusion Figure 6.4: The cumulative distribution of inter-event distances P(>r) for the Efpalion earthquake sequence. The solid line represents Eq.6.14 for the values of qr = 0.55 ±0.02 and r0 = 7.2 ±0.3 km Aftershock diffusion The earthquake sequence is now considered as a CTRW, which starts on the first strong event (x0 = 0) at time t0 = 0. After time equal to τ1, the walker makes a jump of length r1 to the location of the second event of the sequence (x1, t1), after time equal to τ2 he jumps to the third event of the sequence (x2, t2) and so on. For the i th event and after time ti, the walker is at distance xi from the origin. By calculating the 3-D distances and times of all the events from the origin (first strong event), the mean squared displacement 2 x over the course of time t is estimated and plotted in Fig.6.5. After the occurrence of the first strong event, the mean squared displacement remains almost constant with time and the aftershock zone does not diffuse. With respect to Eq.6.12, the diffusion exponent is H = 0. Then, the second strong event is considered at the origin (x0 = 0, t0 = 0) and the mean squared displacement 2 x of all the events that followed this event was estimated and plotted with time t (Fig.6.5). In this case, the mean squared displacement grows as a power-law with time, with a power-law 185

203 Chapter 6 Earthquake Diffusion exponent equal to ± With respect to Eq.6.12, the diffusion exponent in this case is H = The latter indicates the slow migration of the aftershock zone after the occurrence of the second strong event. For H < 0.5, this process corresponds to anomalous diffusion and the sub-diffusive regime. Figure 6.5: The mean squared displacement 2 x (in km) from the first strong event (triangles) and the second strong event (circles) of their respective subsequent seismicity over the course of time t (in days) on log-log axes Discussion The cumulative distributions of inter-event times P(>τ) and distances P(>r) of the Efpalion earthquake sequence are well described by Eq.6.13 and Eq.6.14 respectively, indicating asymptotic power-law behavior and long-range correlations in the spatiotemporal evolution of the sequence. The q-values of qτ = 1.55 and qr = 0.55 found from the analysis indicate that inter-event times scale asymptotically as ~ 0.82 inter-event distances as ~ r In terms of the CTRW approach, these power-law exponents indicate a finite jump length variance (Eq.6.5) and an infinite characteristic 186 and

204 Chapter 6 Earthquake Diffusion waiting time (Eq.6.6), which correspond to the anomalous diffusion regime and subdiffusion. Furthermore, the q-values of qτ > 1 and qr < 1 are in agreement with the qτ and qr values found for the Aigion aftershock sequence (Vallianatos et al., 2012b), for the spatio-temporal scaling properties of seismicity in California and Japan (Abe and Suzuki, 2003; 2005), for global seismicity (Vallianatos and Sammonds, 2013), for Iran (Darooneh and Dadashinia, 2008) and for the Hellenic Subduction Zone (Papadakis et al., 2013) ( and Table 2.1). The mean squared displacement of the aftershock zone remains almost constant with time (Fig.6.5), indicating a non-diffusing zone with diffusion exponent H = 0, if the first strong event is considered to be the origin. When the second event is considered to be the origin, the mean squared displacement grows as a power-law with time (Fig.6.5), with diffusion exponent H = The latter indicates that after the occurrence of the second strong event the aftershock zone migrates slowly with time, which corresponds to a slow sub-diffusive process. In previous studies, a slow sub-diffusive process has been found in earthquake diffusion in California (H < 0.1, Helmstetter et al., 2003), in global earthquake triggering (H < 0.1, Huc and Main, 2003; H 0.4, Marsan and Bean, 2003), in mining-induced seismicity (H = 0.18, Marsan et al., 1999) and various other earthquake sequences (Marsan et al., 2000; McKernon and Main, 2005; Marsan and Lengliné, 2008). The similarity between results from the Efpalion earthquake sequence and the other earthquake sequences imply that the relaxation process of the crust due to a stress perturbation depends non-linearly on the perturbation, which is in this case characterized by the two strong events. In addition, by performing numerical simulations on the ETAS model, Helmstetter and Sornette (2002a; 2002b) suggested that aftershock diffusion should only be observed for an Omori exponent of p < 1 (Eq.2.12), where p = 1 marks the existence of a cascade triggering of events, which is the mechanism at the origin of diffusion in the ETAS model ( 2.3.4). For the Efpalion earthquake sequence, the aftershock production rate n(t) decays according to the modified Omori formula (Eq.2.12) for the values of p = 0.95, K = 88.4 and c = 0.321, after the first strong event and for p = 0.84, K = and c = 0.128, after the second strong event (Fig.6.6). The parameters of the modified Omori formula were estimated by the maximum likelihood procedure (Ogata, 1983). The ZMAP software, which includes a set of scripts written in Matlab (Wiemer, 2001), 187

205 Chapter 6 Earthquake Diffusion was used to perform the analysis. In Fig.6.6, the cumulative number of events N(t) is shown as a function of time from the first strong event. The modified Omori formula for this case, where the cumulative number of events N(t) is considered, reads as (Utsu et al., 1995): t 1 p ( ) ( ) 1, for p N t n t dt K c t c p p t N( t) n( t) dt K ln t c 1, for p 1. (6.15) Cumulative number of aftershocks Time [Days after mainshock] Figure 6.6: The cumulative number of aftershocks (black circles) as a function of time (in days) after the first strong event. The red solid line corresponds to the modified Omori formula (Eq.6.15), for the values of p = 0.95, K = 88.4 and c = 0.321, after the first strong event and for p = 0.84, K = and c = 0.128, after the second strong event. The star indicates the time of occurrence of the second strong event. In the Efpalion earthquake sequence the sub-diffusion of the aftershock zone after the occurrence of the second strong event (Fig.6.5) and the correlation between the positive 188

206 Chapter 6 Earthquake Diffusion Coulomb stress changes and the spatial distribution of aftershocks (Fig.6.2b) imply that the anomalous diffusion of the aftershock zone can be related to a stress diffusion process. The initial stress perturbation due to the occurrence of the first strong event activated the fault segment to the east, where the second strong event occurred four days later. No diffusion is observed at this stage (Fig.6.5), where the Omori exponent is p 1. After the occurrence of the second strong event, the sub-diffusion of the aftershock zone can be related to the slow spatial relaxation of the stress perturbation, as aftershocks occurred at greater distances and preferantially in the positive Coulomb stress zones (Fig.6.2b), activating the fault segment further to the west. At this latter stage, the Omori exponent is p < 1, which, when compared with the ETAS simulations (Helmstetter and Sornette, 2002a; 2002b), suggests a cascade of triggering events in the evolution of the sequence. 6.4 The 2001 Agios Ioannis earthquake swarm sequence The 2001 Agios Ioannis earthquake swarm sequence occurred in the SW part of the Corinth Rift, near the city of Aigion (Fig.6.7). The sequence initiated on March 28, 2001 when a sudden increase in the seismicity rate in the area occurred and involved more than 2900 events over a period of 100 days (Pacchiani and Lyon-Caen, 2010). The largest event of the sequence was the Mw = 4.3 Agios Ioannis earthquake that occurred on April 8, 2001 (Fig.6.7). The focal mechanism of this event indicated normal faulting with a strong strike-slip component (Fig.6.7; Zahradnik et al., 2004). The spatial distribution of the sequence and the focal mechanism of the Agios Ioannis earthquake indicated the activation of a SW-NE fault plane dipping at ~40 to the northwest (Lyon-Caen et al., 2004), which coincides with the unexposed Kerinitis fault that strikes 230 N and dips at 40 NW (Pacchiani and Lyon-Caen, 2010). The Agios Ioannis earthquake sequence developed as a swarm, characterized by the absence of any dominant strong event and of an Omori regime that characterizes the decay rate of aftershock sequences (Pacchiani and Lyon-Caen, 2010). Pacchiani and Lyon-Caen (2010) relocated the earthquake sequence in order to achieve a high spatial resolution for the overall swarm. Following the relocated earthquake catalogue of Pacchiani and Lyon-Caen that has 863 events for a 60 day period, the 3-D spatial 189

207 Chapter 6 Earthquake Diffusion distribution of the Agios Ioannis sequence and its temporal evolution are shown in Fig.6.8. The spatio-temporal evolution of the sequence shown in Fig.6.8 indicates that seismicity migrated along the active fault plane towards the surface at a speed of ~20 m/day (Pacchiani and Lyon-Caen, 2010). Figure 6.7: Map of the Western Corinth Rift showing the 2001 seismicity (from Pacchiani and Lyon-Caen, 2010). The south cluster of events corresponds to the 2001 Agios Ioannis swarm sequence. The green star to the south indicates the Mw = 4.3 Agios Ioannis earthquake that occurred on April 8, The focal mechanism for this event is after Zahradnik et al. (2004). The green star to the north indicates the epicenter of the 1995 Aigion earthquake (Ms = 6.2). Yellow symbols indicate the CRLN stations. Furthermore, Pacchiani and Lyon-Caen (2010) investigated the possible involvement of fluid flow in the evolution of the sequence by comparing its spatiotemporal evolution to the pore-pressure diffusion model of Shapiro et al. (1997), which for an isotropic and homogeneous medium reads as: 190

208 Chapter 6 Earthquake Diffusion x 4 Dt, (6.16) where x is the radius of the pore-pressure triggering front, t is the time since the initiation of the process and D the hydraulic diffusivity. Pacchiani and Lyon-Caen (2010) found a good agreement between the data and the pore-pressure diffusion model (Eq.6.16), which suggests the involvement of a pore-pressure diffusion process in the evolution of the sequence. Furthermore, the analysis indicated that the possible porepressure perturbation did not initiate at the deepest point of the active fault plane but rather at some point near the early events of the sequence. Figure 6.8: The 3-D spatial distribution of the Agios Ioannis earthquake swarm sequence. The color bar indicates the temporal occurrence of the events, counting in days from 1/1/2001. The surface projection of the events is also shown at the top of the figure Spatiotemporal scaling properties 191

209 Chapter 6 Earthquake Diffusion The spatiotemporal scaling properties of the 2001 Agios Ioannis earthquake swarm sequence are studied for the inter-event times and distances, by using the relocated earthquake catalogue of Pacchiani and Lyon-Caen (2010). The cumulative distribution functions of the inter-event times P(>τ) and the inter-event distances P(>r) of the sequence are shown in Fig.6.9 and Fig.6.10 respectively. P(>τ) is fitted according to Eq.6.13 and P(>r) according to Eq.6.14, following the same procedure as previously for the Efpalion earthquake sequence. Eq.6.13 provides a good fit to the observed P(>τ), for the values of qτ = 1.52 ±0.02 and τ0 = ± hours (Fig.6.9). P(>r) is well described by Eq.6.14, for the values of qr = 0.76 ±0.01 and r0 = 1.5 ±0.13 km (Fig.6.10). Figure 6.9: Cumulative distribution function P(>τ) of the inter-event times τ for the Agios Ioannis earthquake swarm sequence and the corresponding fit according to Eq.6.13, for the values of qτ = 1.52 and τ0 = hours. As in the case of the Efpalion earthquake sequence, the cumulative distribution of the inter-event times and distances for the Agios Ioannis swarm sequence are well described by Eq.6.13 and Eq.6.14 respectively, for q-values of qτ > 1 and qr < 1. The latter indicates a correlated process in time and space that deviates from the random case and the exponential distribution. The good agreement between the observed P(>τ) 192

210 Chapter 6 Earthquake Diffusion and P(>r) and Eq.6.13 and Eq.6.14 respectively, indicate that inter-event times and distances exhibit asymptotic power-law scaling according to ~ x. For the q-values of qτ = 1.52 and qr = 0.76 found from the analysis, the latter indicates that inter-event 0.92 times scale asymptotically as and inter-event distances as ~ r. Comparing with Eq.6.8 and Eq.6.9 and in terms of the CTRW approach, the latter power-law exponents correspond to the regime of μ 2 and 0 < β < 1, which indicates an anomalous sub-diffusive process in the spatiotemporal evolution of the Agios Ionnis swarm sequence. 2q 1q ~ 5.2 Figure 6.10: Cumulative distribution function P(>r) of the inter-event distances r for the Agios Ioannis earthquake swarm sequence and the corresponding fit according to Eq.6.14, for the values of qr = 0.76 and r0 = 1.5 km Diffusion of the swarm sequence The diffusion properties of the swarm sequence are studied in terms of the CTRW approach, following the same procedure as for the Efpalion earthquake sequence. The first event of the sequence is considered to be the origin (x0 = 0, t0 = 0) and the mean squared displacement 2 x is calculated from the 3-D distances between all the events 193

211 Chapter 6 Earthquake Diffusion and the origin. The mean squared displacement over the course of time is shown in Fig The mean squared displacement grows approximately as a power-law with time, with a power-law exponent equal to 0.37 ±0.043, which, when compared with Eq.6.12, indicates a diffusion exponent of H The latter value of H < 0.5 corresponds to anomalous diffusion and the sub-diffusive regime and indicates the slow migration of the swarm sequence away from the early events. Figure 6.11: The mean squared displacement 2 x (in km) of the earthquake swarm sequence over the course of time t (in days) on log-log axes, with the first event of the sequence at the origin Discussion The analysis of the spatiotemporal scaling properties of the Agios Ioannis earthquake swarm sequence and the good agreement between the observed cumulative distribution functions of the inter-event times and distances and the NESP models of Eq.6.13 and Eq.6.14 indicate asymptotic power-law behavior and spatio-temporal long-range 194

212 Chapter 6 Earthquake Diffusion correlations in the evolution of the sequence. The q-values of qτ > 1 and qr < 1 are in agreement with those of the Efpalion earthquake sequence and with those presented in Table 2.1. In terms of the CTRW approach, the power-law exponents found for the inter-event times and distances indicate an infinite characteristic waiting time (Eq.6.6) and a finite jump length variance (Eq.6.5) that correspond to the anomalous diffusion regime and sub-diffusion. The mean squared displacement 2 x of the sequence scales approximately as a powerlaw with time according to 0.37 ~ t, which corresponds to the diffusion exponent of H (Eq.6.12). The latter indicates an anomalous diffusion process and the subdiffusion of the sequence. These results are in agreement with the hypothesis of porepressure diffusion as the triggering mechanism of the sequence (Pacchiani and Lyon- Caen, 2010). An initial pore-pressure perturbation in the vicinity of the early events and a sub-diffusive pore-pressure relaxation process through the active fault plane, which can act as a conduit, could have triggered the earthquake swarm and the slow subdiffusion of the sequence towards the surface. 195

213 Chapter 7 Discussion 7.1 Summary of the Results In the present thesis, the scaling properties of fault and earthquake populations in the Corinth Rift were studied by using statistical mechanics and the generalized statistical mechanics framework, termed as non-extensive statistical mechanics (NESM). NESM presents a novel approach to the study of fracturing processes and earthquakes. The results of the thesis support the idea that NESM is an appropriate methodological tool to apply to the collective properties of fault and earthquake populations in terms of probabilities, based on the specification of the relevant microscopic states and their interactions. In the following I set the work in context. I summarize the results of the thesis and discuss the implications that arise for fault growth and the fault network evolution in the Rift in section The insights gained from the analysis of earthquake physics and the evolution of seismicity in the Rift are discussed in section Finally, the implications for earthquake hazard assessment and future paths for research, based on the insights of the thesis, are discussed in sections and 7.3, respectively. Central to the generalized statistical mechanics approach that was followed in the thesis is the non-additive entropy Sq (Tsallis, 1988), rewritten here in its integral form for reference:

214 Chapter 7 Discussion S q k 1 0 p q q 1 X dx. (7.1) The main advantage of Sq is that it considers all-length scale correlations among the elements of a system, leading to asymptotic power-law behavior and it converges to the classic Boltzmann-Gibbs-Shannon entropy SBGS in the particular case of q 1. Thus, by optimizing Sq under the appropriate constraints ( 2.6.2), a range of asymptotic power-law to exponential-like distributions are obtained for the various values of q. This approach was followed in Chapter 3, in the study of the cumulative distribution function of fault trace-lengths N(>L) in the Rift. Based on published fault maps for the area, a comprehensive dataset of the fault network was compiled (Fig.3.11 and Appendix A) and the digital elevation model (DEM) of Google Earth was used to achieve maximum precision of the fault traces on surface ( 3.3.1). The NESM based analysis indicated that the scaling properties of the complete fault dataset and the various subsets in the different strain regimes ( 3.2.3) are well described in the fullrange of values by the cumulative distribution function (Eq. 3.3) that optimizes Sq, for various q-values in the range of 1 (exponential) to In addition, an analysis of synthetic fault datasets was carried out in order to test the sensitivity of the estimated q-values on missing data ( 3.5.6). The analysis showed a remarkable stability in the scaling behavior of the subsets in the case of 90% missing data, with q varying ±0.03 from the original q-value that the data were generated from. For 99.9% of missing data, the variation of the estimated q-values increases to ±0.1 from the original q-value. In previous studies, the q-value of q = 1.16 was found by Vallianatos et al. (2011a) for N(>L) in Crete, Greece, the q-values of q = 1.75 and q = 1.10 for linked and independent faults, respectively in Valles Marineris, Mars (Vallianatos and Sammonds, 2011) and the q-values of q = and q = for thrust (compressional) and normal (extensional) faults in Mars, respectively (Vallianatos, 2013a). The results of these studies and those presented here imply that the scaling properties of fault systems are universally characterized by q-values of q 1. The various q-values found in these studies correspond well to the physical properties of faults, i.e. linked versus independent, or thrust versus normal faults. In the Corinth Rift, analysis in the different strain regimes indicated that the q-value variations are associated with brittle strain in 197

215 Chapter 7 Discussion the Earth s crust. N(>L) in the eastern low-strain zone exhibits asymptotic power-law behavior, with q = 1.22, in comparison to exponential-like scaling in N(>L) of the western and central high-strain zones, where q approaches unity. Such transition has previously been observed to be a function of increasing strain in numerical models (Spyropoulos et al., 2002; Hardacre and Cowie, 2003), laboratory experiments (Spyropoulos et al., 1999; Ackermann et al., 2001) and in the fault populations of the Afar Rift (Gupta and Scholtz, 2000a). The results found for the Corinth Rift provide further evidence for such transition in a single tectonic setting. In addition, these results show that scaling analyses of natural fault systems must be carried out with caution, as the scaling behavior of the complete system can be a mixture of distributions of distinctive subsets with different physical properties. Bimodal distributions with two power-law segments have been found by Wojtal (1994; 1996), Ackermann and Schlische (1997) and by Zygouri et al. (2008) for the active fault system in the Corinth Rift. These distributions can simply resemble fault populations with different physical properties in the same tectonic setting. The collective properties of earthquakes were studied in Chapters 4 6, for various earthquake catalogues, time periods and spatial scales. In particular, in Chapter 4 the frequency-size distribution of earthquakes in the Rift was studied for an almost 50 year period of recorded seismicity. The analysis was carried out by using the empirical Gutenberg-Richter (G-R) scaling relation (Gutenberg and Richter, 1944) and the NESM formulation of the fragment-asperity (F-A) model (Sotolongo-Costa and Posadas, 2004; Telesca, 2012). Considering the magnitude of M0 = 3.8 as the minimum magnitude for the analysis, the G-R scaling relation yields the b-value of b = 1.18 ±0.043 and the F-A model the q-value of qe = ± The analysis for various M0 showed the strong dependence of b on the initial selection of M0, in contrast to qe, which remains quite stable irrespective of M0. This is quite an important aspect in earthquake hazard assessments, as more reliable estimations of the scaling behavior of the earthquake-size distribution in a region can be made with the F-A model and the expected seismicity rates can be estimated more rigorously. By using the results of the F-A model, a recurrence period of 15.3 ±3.8 years for an M = 6 earthquake was estimated, which is in agreement to the 320 year seismicity rates in the Rift. 198

216 Chapter 7 Discussion The temporal properties of seismicity in the Rift were studied in Chapter 5 by looking into the inter-event time series, which expresses the time intervals between successive earthquakes. The analysis showed that the probability densities of inter-event times exhibit scaling behavior and a crossover between two power-law regimes at both short and long inter-event times. The latter indicates clustering effects at all timescales, similar to the previous results of Kagan and Jackson (1991), for various earthquake catalogues and of Corral (2003), for the Southern California. By rescaling inter-event times with the mean seismicity rate, the rescaled probability densities, for various time periods, threshold magnitudes and spatial scales approximately fall onto a unique curve that is well described for over eight orders of magnitude by the q-generalized gamma distribution (Eq.5.2). The q-generalized gamma distribution presents two power-law regimes and memory effects at all timescales and reduces to the ordinary gamma distribution for q 1. It was further shown ( 5.3.2) that the q-generalized gamma distribution can be derived for non-stationary earthquake time series by using the stochastic dynamic mechanism with memory effects, introduced by Queiros (2005). Furthermore, multifractal analysis on the earthquake time series indicated a multifractal temporal structure and clustering variability, which can be associated with periods of high and low earthquake activity in the Rift and the earthquake clusters that are active in each time period. In Chapter 6, earthquake diffusion phenomena were studied for two cases of triggered seismicity in the Rift. The first was the aftershock sequence that was triggered by the two 2010 Efpalion strong events (Karakostas et al., 2012) and the second the 2001 Agios Ioannis earthquake swarm sequence that has been associated with fluid diffusion at depth (Pacchiani and Lyon-Caen, 2010). The diffusion properties of the two sequences were studied in terms of NESM and the continuous time random walk (CTRW) theory. The spatiotemporal scaling properties of both sequences are well described by the q-exponential distribution that optimizes Sq (Eq.7.1), for q-values that in terms of the CTRW theory correspond to the sub-diffusion regime. The mean squared displacements of the two sequences with time indicated non-linear power-law dependence, which is the hallmark of anomalous diffusion. The estimated diffusion exponents indicate a slow sub-diffusive process in the spatiotemporal evolution of seismicity. In the case of the Efpalion earthquake sequence, such a process can be related to the slow spatial relaxation of the stress perturbation triggered by the 199

217 Chapter 7 Discussion occurrence of the second strong event. The Omori exponent of p < 1, found for the aftershock sequence that followed the second strong event, suggests a cascade of triggering events in the evolution of the sequence (Helmstetter and Sornette, 2002a). In the case of the Agios Ioannis earthquake swarm sequence, the slow migration of the earthquake zone with time is consistent with the hypothesis of a sub-diffusive porepressure relaxation process that propagates along the active fault plane. 7.2 Wider Implications Implications for fault growth and fault network evolution The results found from the analysis in the different strain zones of the Rift were summarized in Fig This is reproduced here as Fig.7.1. The observed properties indicate that with increasing strain the number of faults decreases and the average faultlength of the population increases as faults start to coalesce forming larger structures and the distribution turns to exponential (Fig.7.1). In analogy to the main stages of fault population evolution, these properties suggest fracture saturation in the central and western high-strain zones of the Rift, where faults grow by linkage rather than nucleation or growth. In the eastern low-strain zone, the greater number of small faults and the asymptotic power-law behavior of the cumulative fault-length distribution suggest that nucleation and growth may still dominate over coalescence, leading to scale-invariant fault growth. Similar properties were also observed in the case where the currently active (high strain rate setting) and inactive (low strain rate setting) Rift zones were considered ( 3.5.4), suggesting fracture saturation in the currently active zone. Furthermore, the transition from asymptotic power-law to exponential-like scaling suggest the suppression of long-range fault interactions with increasing strain (Cowie, 1998a; Ackermann et al., 2001). Such processes are predicted by the numerical model of Cowie (1998b), which seems to match well the slip rate variations and the localization of strain in the narrow offshore zone and the Perachora Peninsula during the Late Quaternary (~0.7 Myr present; Roberts et al., 2009; Leeder et al., 2012; Ford et al., 2013). In particular, the stress feedback mechanism incorporated in the model of Cowie (1998b) favors deformation on optimally oriented faults that fall into Coulomb 200

218 Chapter 7 Discussion stress increase zones (e.g., King et al., 1994). Such a mechanism can lead to the localization of strain on a few large faults that either coalesce to form larger structures or increase their displacement rates to accommodate increasing strain. As deformation progresses, these large structures span the brittle layer of the crust, their vertical growth is restricted and new or pre-existing faults start to grow (Ackermann et al., 2001; Soliva et al., 2006). Such a mechanism has been proposed by Poulimenos (2000), Cowie and Roberts (2001) and Roberts et al. (2009) as a possible mechanism for the observed slip rate variations in the Corinth Rift. Figure 7.1: Summary of the observed properties in the different strain zones of the Corinth Rift. 201

219 Chapter 7 Discussion Other mechanisms that have been proposed for the localization of strain in continental rifts are crustal rheology, heat fluxes or changes in regional strain rates (Cowie et al., 2005; Bell and Jackson, 2015). As there is no evidence that such processes took place in the Corinth Rift, the present results, summarized in Fig.7.1, rather suggest that fault growth processes control the fault network evolution in the Rift. Higher strains in the crust in the last ~2 Myr led the central and western zones to fracture saturation, the suppression of long-range fault interactions, exponential scaling and progressive localization of strain in the narrow offshore zone. This process can be more rapid in the western zone, where the seismogenic depth is confined to narrower depths (Hatzfeld et al., 2000). In the eastern low-strain zone, the asymptotic power-law scaling suggests that the fault population has not reached saturation and distribution of strain is more diffuse Implications for earthquake physics and the evolution of seismicity The good agreement between the earthquake data in the Corinth Rift and the q- exponential family of distributions that optimize the non-additive entropy Sq (Eq.7.1) suggests that the latter may act as attractors for the earthquake populations. In support of this hypothesis are also the results of the numerous studies on earthquake physics and NESM, summarized in Tables 2.1 and 2.2 ( 2.6.3). These types of distributions are attractors for a wide class of complex non-equilibrium systems, as diverse as black holes, living organisms, financial markets and optical lattices, among others (Tsallis, 2009b; 2013; 2014), indicating that earthquakes belong to the same universality class as such systems! Earthquake activity in the Corinth Rift is intermittent and evolves dynamically over time as clusters of events, characterized by a range of fractal dimensions and multifractality. The bimodality in the probability density of inter-events times and the gradual crossover between two power-law regimes for short and long inter-events times respectively, shown in Fig.5.10 and reproduced here as Fig.7.2, suggest that the earthquake activity evolves as two processes. The first one is related to clustering effects at short time scales, induced by aftershock sequences and earthquake swarms 202

220 Chapter 7 Discussion Figure 7.2: Probability density p(t) of the rescaled inter-event times T for the HUSN, CRL and NOA datasets and for various time periods and threshold magnitudes. The solid line represents the corresponding fit according to the q-generalized gamma distribution (Eq.5.2) for the values of C = 0.4, β = 1.65, γ = 0.38 and q = The dashed line represents the corresponding fit according to the gamma distribution (Eq.5.1) for the values of C = 0.4, β = 1.65 and γ = and the second one to long-term clustering effects, related to the background activity. Although clustering effects at all timescales imply memory in the seismogenic process, similar to the conclusions of Livina et al. (2005) and Lennartz et al. (2008), a fraction of uncorrelated and Poissonian distributed events in the Rift cannot be excluded. In addition, the similar temporal patterns found for various earthquake catalogues, time periods and threshold magnitudes and the almost perfect collapse of the rescaled interevent times onto an unique curve (Fig.7.2) indicate the self-similar temporal structure of seismicity in the Rift. Earthquake diffusion phenomena in the Rift can be associated with pore-pressure diffusion in the seismogenic zone or to the diffusion of stress, induced by either static transfer during seismic rupture or by a cascade of triggering events. The diffusion characteristics would depend in the first case on the local distribution of fractures that 203

221 Chapter 7 Discussion act as conduits for the crustal fluid movement, while in the second case on the local distribution of asperities that act as barriers to the diffusion of stress. In either case, a hierarchical spatio-temporal clustering of events can be generated that diffuses according to the relaxation process of the perturbation. The low diffusion exponents found for the two cases of triggered seismicity in the Rift and for other earthquake sequences (Marsan et al., 2000; Huc and Main, 2003; Helmstetter et al., 2003) suggest that the relaxation response of the highly heterogeneous crust to an initial perturbation depends non-linearly on the perturbation Implications for earthquake hazard in the Rift The Corinth Rift is one of the most seismically active areas in Europe and has experienced many strong and destructive earthquakes in the past. Such earthquakes with magnitude greater than 6 are estimated to occur every ~15 years in the area. Although this average value can be quite useful for earthquake preparedness and the estimated earthquake hazard in the area, it may only prove to be an ensemble of strong earthquakes distributed irregularly in time. A characteristic case that depicts exactly this is the 1981 Alkyonides earthquake sequence that hit the eastern part of the Rift with three strong earthquakes of magnitude greater than 6 within a few hours and days. Other implications that stem from the insights of the thesis refer to the spatio-temporal properties of seismicity in the Rift. The probability densities of inter-event times, shown in Fig.7.2, indicate that the probability of a subsequent earthquake is high immediately after the occurrence of the previous one and decreases slowly thereafter. The temporal clustering effects further suggest that short inter-event times are more likely to be followed by short ones and long inter-event times by long ones. Such type of behavior implies that the longer it had been since the last earthquake, the longer it will be until the next one, referred to as the paradox of the expected time until the next earthquake by Sornette and Knopoff (1997). Furthermore, the almost perfect collapse of the rescaled inter-event time distributions onto a unique curve (Fig.7.2) implies that in earthquake hazard assessments the probability density of inter-event times p(τ) in the Rift can be estimated as p() Rf T, where R is the mean seismic rate and f(t) the 204

222 Chapter 7 Discussion scaling function that is well approximated by the q-generalized gamma distribution shown in Fig Directions for Future Work In the introduction of the thesis, the necessity of statistical mechanics for the better understanding of the transition from the microscopic physical processes that take place inside fault zones to the large scale macroscopic description of fracturing processes and earthquakes was highlighted. The results of the thesis suggest that this transition can successfully be described in terms of probabilities using NESM. The NESM approach to fracturing processes and earthquakes is still in its infancy and future research towards such an approach may lead to significant outcomes. It may be expected that the present thesis will provide a useful guide for the implementation of NESM to earthquake physics and stimulate further research towards that field. In particular, the quite good agreement between the fragment-asperity (F-A) model and the frequency-size distribution of earthquakes, found for the Corinth Rift and other seismically active sites around the globe (e.g., Sotolongo-Costa and Posadas, 2004; Silva et al., 2006; Telesca, 2010a; Papadakis et al., 2013), suggests that the F-A model can be used to estimate hazard more efficiently and to possibly determine the stress changes in the crust. The q-value incorporated in the model may then provide useful insights in the geodynamic instability of the seismogenic zone as it evolves through seismicity. Towards such insights are the works of Telesca (2010c) and Papadakis et al. (2015) on the seismicity that preceded the 2009 L Aquilla earthquake (ML = 5.8) and the 1995 Kobe earthquake (M = 7.2), respectively. In both studies a q-value increase prior to the strong event was observed, indicating the transition of the stress release rates towards instability. Such insights would greatly benefit from rock deformation experiments in a controlled environment in the laboratory. In such experiments, the q-value variations as function of the time-varying applied stress and crack growth can be monitored and useful insights about the underlying physical process can be deduced (e.g., Sammonds et al., 1992). In addition, the earthquake activity in tectonically active areas or volcanoes is typically characterized by non-stationarities. Although such non-stationarities are produced by 205

223 Chapter 7 Discussion stochastic models of seismicity like the ETAS model (Ogata, 1988; Helmstetter and Sornette, 2002a), such models consider the background activity as an uncorrelated Poisson process and produce earthquake time series that are approximately gamma distributed (Hainzl et al., 2006; Touati et al., 2009). The present study on non-stationary earthquake time series in the Rift showed that clustering effects and power-law scaling also appear at long inter-event times, in accordance with the previous results of Kagan and Jackson (1991) and Corral (2003). Such properties indicate a fraction of correlated background activity that is not captured by the ETAS model. Future studies on stochastic models that incorporate such properties will greatly improve the stochastic modeling of seismicity and the efficiency of probabilistic hazard assessments. In addition, the q-generalized gamma distribution, introduced here to describe the probability densities of non-stationary earthquake time series in the Rift, reproduce well such properties and its efficiency for describing such series requires further application to other tectonic environments, volcano seismicity and laboratory acoustic emissions. The study on earthquake diffusion and the spatial properties of the Efpalion and the Agios Ioannis earthquake sequences in Chapter 6 highlights the need for accurate earthquake catalogues in the study of such phenomena. Future research on the spatial distribution of seismicity in the Rift will be greatly improved by using such catalogues. Earthquake diffusion in the area can then be estimated for longer time periods and not only for single sequences. The well-defined diffusion properties can be incorporated into probabilistic earthquake hazard assessments by estimating, for instance, the average migration of the earthquake zone with time for a given earthquake or an initial pore-pressure perturbation. The combination of NESM and the CTRW theory in the analysis of anomalous earthquake diffusion phenomena seems promising and future research should focus on whether the propagation of triggered seismicity can efficiently be modeled using such an approach. The scaling properties of fault attributes are usually studied by using power-law or exponential distributions that in many cases do not describe the full-range of the observed distributions (e.g., Davy, 1993; Cladouhos and Marrett, 1996; Vetel et al., 2005). In the thesis it was suggested that the NESM approach may provide a more rigorous way to study the scaling properties of fault systems and further studies on other deformed regions will verify if this is the case. In addition, power-law distributions are 206

224 Chapter 7 Discussion used to extrapolate scaling to small faults, which may accommodate a significant portion of total strain due to brittle faulting (Marrett and Allmendinger, 1991). Then this extrapolation is used to estimate the amount of strain accommodated by the fault system (e.g., Marrett and Allmendinger, 1992). In comparison to such approaches, the NESM approach may provide more robust results on strain calculations and this idea should be explored for the Corinth Rift and other tectonic environments in the future. 207

225 Chapter 8 Conclusions Despite the complexity of the earthquake generation process, the collective properties of fault and earthquake populations exhibit scaling properties that can well be approximated by statistical or physical models. Such results were shown in the present thesis for one of the most seismically active areas in Europe, the Corinth Rift. In terms of probabilities of the different microstates and their interactions, the large scale properties of fault and earthquakes populations in the Rift can be deduced by following the principles of statistical mechanics and the generalized framework, termed as nonextensive statistical mechanics (NESM). The NESM approach, based on the first principles of statistical mechanics, provides a unified framework that produces a range of asymptotic power-law to exponential-like distributions that are both ubiquitous in nature. The scaling properties of fault trace-lengths in the Corinth Rift are well described by the NESM approach, for various q-values in the range of 1 (exponential) to In particular, systematic variations and the transition from asymptotic power-law (q > 1) to exponential-like scaling (q = 1) appear to be function of increasing strain in the deforming zone, providing further evidence for such transition in a single tectonic setting. Exponential-like scaling in the central and western high-strain zones of the Rift, or in the currently active Rift zone, imply that coalescence may dominate over fault nucleation and growth, while the interplay between these processes lead to scaleinvariant fault growth in the eastern low-strain zone. Other factors that seem to control

226 Chapter 8 Conclusions fault growth in the Rift are fault interactions and the thickness of the brittle layer. These factors, in synergy with higher strains in the crust, may lead to fracture saturation, the suppression of fault interactions and the transition from asymptotic power-law to exponential scaling in the fault-length distribution. Regional strain, fault interactions and the boundary condition of the brittle layer may then control fault growth and the fault network evolution in the Corinth Rift. For an almost 50 year period of recorded earthquake activity in the Rift, the NESM formulation of the fragment-asperity (F-A) model yields the q-value of qe = ±0.013 for the frequency-size distribution of earthquakes, while the empirical Gutenberg-Richter (G-R) scaling relation yields the b-value of b = 1.18 ± In comparison to the G-R scaling relation, the F-A model produces more stable results irrespective of the threshold magnitude considered in the analysis, rendering the F-A model quite a useful tool for earthquake hazard assessments. In terms of expected seismicity rates, the F-A model estimates the recurrence period of 15.3 ±3.8 years for earthquakes of magnitude M = 6 in the Rift. Earthquake activity in the Rift is typically characterized by intermittent behavior, where periods of low activity are interspersed by sudden seismic bursts, which are associated with frequent earthquake swarms and with the occurrence of stronger events followed by aftershock sequences. This type of behavior is manifested in the degree of heterogeneous clustering and the multifractal structure of the earthquake time series in the Rift. The probability densities of inter-event times exhibit a bimodal character and a gradual crossover between two power-law regimes at both short and long inter-event times, indicating clustering effects at all timescales and memory in the seismogenic process. Such scaling behavior is well reproduced by the q-generalized gamma distribution that exhibits two-power law regimes and includes the ordinary gamma distribution as a particular case. The q-generalized gamma distribution describes well the probability densities of inter-event times in the Rift for over eight orders of magnitude. After rescaling the inter-event times with the mean seismicity rate, the probability densities, for various time periods, threshold magnitudes and spatial scales, approximately fall onto a unique curve, which indicates the self-similar temporal character of seismicity in the Rift. 209

227 Chapter 8 Conclusions Earthquake diffusion phenomena in the Rift may occur during periods of high earthquake activity. Such phenomena are observed during the 2010 Efpalion earthquake sequence and the 2001 Agios Ioannis earthquake swarm sequence. In the first case, the spatiotemporal properties of seismicity correspond to a correlated process that propagates non-linearly with time away from the second Efpalion strong event (Mw = 5.1). Such a process corresponds to anomalous sub-diffusion of the aftershock zone, possibly induced by the slow spatial relaxation of the stress perturbation triggered by the occurrence of the second strong event. Similar properties are observed for the spatiotemporal evolution of the Agios Ioannis earthquake swarm, which has previously been related to fluid diffusion at depth. The slow sub-diffusion of the swarm sequence may then reflect the slow pore-pressure relaxation process that propagates along the active fault plane towards the surface. To conclude, the physical models derived in the framework of generalized statistical mechanics can successfully reproduce the scaling properties of fault and earthquake populations in the Corinth Rift, providing new insights into the physics that governs fracturing processes and earthquakes in the region. Although the results of the present thesis provide a step forward to the understanding of such complex phenomena, many questions regarding the earthquake generation process remain wide open. Using the principles of generalized statistical mechanics in a unified approach with the other known laws in fracture mechanics may lead to significant discoveries and may enhance our understanding regarding the physical mechanisms that drive the evolution of seismicity in local, regional and global scale. 210

228 Appendices Appendix A: Table of Faults Name Length (km) Dip Spatial distribution References 1. Antirio Fault (AntF) 8.77 S onshore Flotté et al., 2005; Jolivet, Drosatos Fault (DrF) 4.41 S onshore Valkaniotis, Filothei Fault (FltF) 5.95 S onshore Valkaniotis, Kalithea Fault 1 (KaF/1) 5. Kalithea Fault 2 (KaF/2) 6. Marathias Fault 1 (MrtF/1) 7. Marathias Fault 2 (MrtF/2) 8. Amigdalea Fault (AmgF) 9. Kokinovrahos Fault 1 (KnF/1) 10. Kokinovrahos Fault 2 (KnF/2) 11. Panormo Fault 1 (PaF/1) 12. Panormo Fault 2 (PaF/2) 13. Galaxidi Fault (GlxF) 14. Agios Vlasios Fault (AgVF) 15. Vounichora Fault 1 (VoF/1) 16. Vounichora Fault 2 (VoF/2) 17. Tritea Fault 1 (TrF/1) 18. Tritea Fault 2 (TrF/2) 19. Tritea Fault 3 (TrF/3) 6.65 S onshore Valkaniotis, S onshore Valkaniotis, S onshore 8.31 S onshore Valkaniotis, 2009 Beckers et al., 2015 Valkaniotis, N onshore Valkaniotis, N onshore Valkaniotis, S onshore Valkaniotis, S onshore Valkaniotis, S 7.76 S 3.28 S onshore - offshore onshore - offshore onshore - offshore Valkaniotis, 2009; Beckers et al., 2015 Papanikolaou et al., 2009; Valkaniotis, 2009 Papanikolaou et al., NE onshore Papanikolaou et al., E onshore Papanikolaou et al., NE onshore Papanikolaou et al., NE onshore Papanikolaou et al., ESE onshore Papanikolaou et al., Sernikaki Fault (SrF) 4.63 ESE onshore Papanikolaou et al., Amfissa Fault 1 (AmF/1) 22. Amfissa Fault 2 (AmF/2) 5.4 SW onshore Papanikolaou et al., 2009; Valkaniotis, NE onshore Valkaniotis, 2009

229 Appendix A Table of Faults 23. Amfissa Fault 3 (AmF/3) 1.21 NE onshore Valkaniotis, Delfoi Fault (DelF) S onshore 25. Chriso Fault (ChF) 10.8 N onshore 26. Camping Apollon Fault (CAF) Neotectonic map, Leivadia sheet; Valkaniotis, 2009 Neotectonic map, Leivadia sheet; Valkaniotis, S Valkaniotis, Kira Fault (KirF) 2.48 WSW onshore Papanikolaou et al., Makrigialos Fault (MkF) 7.82 SW onshore - offshore 29. Sikies Fault (SiF) 7.71 SW onshore 30. Kourmoutsi Fault (KMRF) 31. Kourmouli Fault (KRMF) 32. Antikyra Fault 1 (AnF/1) 33. Antikyra Fault 2 (AnF/2) 34. Distomo Fault 1 (DiF/1) 35. Distomo Fault 2 (DiF/2) 36. Abelos North Fault (AbNF) 6.47 N onshore 2.6 S onshore 6.35 SE onshore Valkaniotis, 2009 Neotectonic map, Leivadia sheet; Valkaniotis, 2009 Neotectonic map, Leivadia sheet; Valkaniotis, 2009 Neotectonic map, Leivadia sheet; Valkaniotis, 2009 Neotectonic map, Leivadia sheet; Valkaniotis, NE onshore Neotectonic map, Leivadia sheet E onshore Neotectonic map, Leivadia sheet; Valkaniotis, S onshore Valkaniotis, S offshore Stefatos et al., Tarsos Fault (TarF) 3.55 W onshore Neotectonic map, Leivadia sheet 38. Kalogeriko Fault (KagF) 39. Analipsi Fault 1 (AnaF/1) 40. Analipsi Fault 2 (AnaF/2) 4.62 S onshore Valkaniotis, N onshore Neotectonic map, Leivadia sheet 6.59 N onshore Neotectonic map, Leivadia sheet 41. Kiriaki Fault (KiF) 9.76 S onshore Valkaniotis, Elikonas Fault (ElF) 14.1 S onshore Valkaniotis, Karioti Fault (KrF) 4.33 N onshore Neotectonic map, Leivadia sheet 44. Kalamiotisa Fault (KaF) 11.4 S onshore Neotectonic map, Leivadia sheet 45. Paralia Fault (PrF) 4.29 E onshore Neotectonic map, Leivadia sheet 46. Prodromos Fault (PrdF) 2.51 S onshore Valkaniotis, Thisvi Fault (ThF) 3.62 S onshore Neotectonic map, Leivadia sheet 48. Akrotiri Domvrena Fault 1 (ADF/1) 3.79 S onshore Neotectonic map, Leivadia sheet 212

230 Appendix A Table of Faults 49. Akrotiri Domvrena Fault 2 (ADF/2) 50. Akrotiri Domvrena Fault 3 (ADF/3) 2.59 N onshore Neotectonic map, Leivadia sheet 0.82 S onshore Neotectonic map, Leivadia sheet 51. Taratsa Fault (TaF) 4.38 N onshore Valkaniotis, Neochori Fault 1 (NchF/1) 53. Neochori Fault 2 (NchF/2) 54. Agios Nikolaos Fault (ANF) 55. Domvrena Gulf Fault (DGF) 56. Aliki Fault 1 (ALF/1) 57. Aliki Fault 2 (ALF/2) 58. Aliki Fault 3 (ALF/3) 9.51 S onshore Valkaniotis, S onshore Valkaniotis, SE onshore Valkaniotis, N offshore Sakelariou et al., SW onshore 1.39 SW onshore 1.46 SW onshore IGME geologic map, Kaparellion sheet IGME geologic map, Kaparellion sheet IGME geologic map, Kaparellion sheet 59. Leontari Fault (LeF) 13.2 S onshore Tsodoulos et al., Livadostra Fault (LvF) 61. West Kapareli Fault (KWF) 62. East Kapareli Fault (KEF) 16.9 SE onshore - offshore Tsodoulos et al., 2008; Sakelariou et al., S onshore Morewood and Roberts, S onshore Morewood and Roberts, Kotroni Fault (KotF) 1.21 SE onshore Morewood and Roberts, Kokinia Fault (KokF) 65. North Kitheronas Fault (KNF) 2.1 N onshore Morewood and Roberts, N onshore Morewood and Roberts, Erithres Fault (ErF) 14.8 N onshore Tsodoulos et al., Kitheronas Fault (KthF) 68. Agia Triada Fault (AtrF) 69. Agios Vasilios Fault 1 (AVF/1) 70. Agios Vasilios Fault 2 (AVF/2) 71. Agios Vasilios Fault 3 (AVF/3) 72. Agios Vasilios Fault 4 (AVF/4) 73. Porto Germeno Fault 1 (PGF/1) 74. Porto Germeno Fault 2 (PGF/2) 6 ESE onshore Morewood and Roberts, N onshore Morewood and Roberts, S onshore Morewood and Roberts, SW onshore Morewood and Roberts, S onshore Morewood and Roberts, S onshore Morewood and Roberts, W onshore Morewood and Roberts, W onshore Morewood and Roberts,

231 Appendix A Table of Faults 75. Agios Nektarios Fault 1 (ANF/1) 76. Agios Nektarios Fault 2 (ANF/2) 77. Agios Nektarios Fault 3 (ANF/3) 78. Profitis Ilias Fault (PIF) 79. South Vilia Fault 1 (SVF/1) 80. South Vilia Fault 2 (SVF/2) 81. South Vilia Fault 3 (SVF/3) 82. Krio Pigadi Fault (KPF) 83. Agia Paraskevi Fault 1 (APF/1) 84. Agia Paraskevi Fault 2 (APF/2) 85. Mytikas Fault (MytF) 86. Agios Georgios Fault 1 (AGF/1) 87. Agios Georgios Fault 2 (AGF/2) 88. Agios Georgios Fault 3 (AGF/3) 89. Agios Georgios Fault 4 (AGF/4) 90. Agios Sotiras Fault 1 (ASF/1) 91. Agios Sotiras Fault 2 (ASF/2) 92. Veniza Fault 1 (VenF/1) 93. Veniza Fault 2 (VenF/2) 2.59 WNW onshore Morewood and Roberts, WNW onshore Morewood and Roberts, W onshore Morewood and Roberts, S onshore Morewood and Roberts, S onshore Morewood and Roberts, SW onshore Morewood and Roberts, NW onshore Morewood and Roberts, SE onshore Morewood and Roberts, NNW onshore Morewood and Roberts, NW onshore Morewood and Roberts, SW onshore Morewood and Roberts, S onshore Morewood and Roberts, S onshore Morewood and Roberts, ESE onshore Morewood and Roberts, S onshore Morewood and Roberts, N onshore Morewood and Roberts, N onshore Morewood and Roberts, S onshore Morewood and Roberts, 2001; Bentham, S onshore Morewood and Roberts, Psatha Fault (PsF) 11.2 N onshore Morewood and Roberts, Alepochori Fault 1 (AlpF/1) 96. Alepochori Fault 2 (AlpF/2) 97. Alepochori Fault 3 (AlpF/3) 98. Alepochori Fault 4 (AlpF/4) 99. Megara Basin Fault 1 (MBF/1) 100. Megara Basin Fault 2 (MBF/2) 2.51 N onshore Morewood and Roberts, S onshore Morewood and Roberts, N onshore Morewood and Roberts, SE onshore Morewood and Roberts, ESE onshore Bentham, NE onshore IGME geologic map, Kaparellion sheet 214

232 Appendix A Table of Faults 101. Megara Basin Fault 3 (MBF/3) 102. Megara Basin Fault 4 (MBF/4) 103. Megara Basin Fault 5 (MBF/5) 104. Megara Basin Fault 6 (MBF/6) 105. Megara Basin Fault 7 (MBF/7) 106. Megara Basin Fault 8 (MBF/8) 107. Megara Basin Fault 9 (MBF/9) 108. Megara Basin Fault 10 (MBF/10) 109. Megara Basin Fault 11 (MBF/11) 110. Mavrolimni Fault (MavF) 111. Kakia Skala Fault (KKF) 2.42 NW onshore Bentham, SE onshore Bentham, NNE onshore Bentham, N onshore Bentham, NE onshore 1.41 NE onshore IGME geologic map, Kaparellion sheet IGME geologic map, Kaparellion sheet 8.16 NE onshore Bentham, WSW onshore Moretti et al., SW onshore Moretti et al., NE onshore 6.61 S onshore 112. Kineta Fault (KntF) 5.15 S onshore 113. Gerania Mountain Fault 1 (GMF/1) 114. Gerania Mountain Fault 2 (GMF/2) 115. Gerania Mountain Fault 3 (GMF/3) 116. Gerania Mountain Fault 4 (GMF/4) 117. Gerania Mountain Fault 5 (GMF/5) 118. Gerania Mountain Fault 6 (GMF/6) 119. Gerania Mountain Fault 7 (GMF/7) 120. Agioi Theodoroi Fault (AThF) 121. Kalithea Fault 1 (KalF/1) 122. Kalithea Fault 2 (KalF/2) 123. Alkiona Fault (AlkF) 124. Schinos Fault (SchF) 2.71 SW onshore 1.85 SW onshore 1.59 SE onshore 2.74 N onshore 12.7 SW onshore 2.4 S onshore 3.36 S onshore 2.96 S onshore 4.1 S onshore 2.61 SE onshore 1.81 N onshore IGME geologic map, Kaparellion sheet IGME geologic map, Sofikon sheet IGME geologic map, Sofikon and Kaparellion sheet IGME geologic map, Kaparellion sheet IGME geologic map, Kaparellion sheet IGME geologic map, Kaparellion sheet IGME geologic map, Kaparellion sheet IGME geologic map, Sofikon and Kaparellion sheet IGME geologic map, Sofikon sheet IGME geologic map, Sofikon sheet IGME geologic map, Sofikon sheet IGME geologic map, Sofikon sheet IGME geologic map, Sofikon sheet IGME geologic map, Sofikon sheet 13 N onshore Morewood and Roberts, Pisia Fault (PisF) 20.7 N onshore Morewood and Roberts,

233 Appendix A Table of Faults 126. Asprokambos Fault 1 (AspF/1) 127. Asprokambos Fault 2 (AspF/2) 128. Asprokambos Fault 3 (AspF/3) 129. Asprokambos Fault 4 (AspF/4) 130. Asprokambos Fault 5 (AspF/5) 131. Skalosia Fault (SklF) 132. Sterna Fault 1 (StrF/1) 133. Sterna Fault 2 (StrF/2) 134. North Vouliagmeni Fault 1 (NVF/1) 135. North Vouliagmeni Fault 2 (NVF/2) 136. Lake Vouliagmeni Fault (LVF) 137. Akrotiri Melagavi Fault 1 (AMF/1) 138. Akrotiri Melagavi Fault 2 (AMF/2) 139. Akrotiri Melagavi Fault 3 (AMF/3) 140. Akrotiri Melagavi Fault 4 (AMF/4) 141. Akrotiri Melagavi Fault 5 (AMF/5) 142. Makrigoas Fault 1 (MkgF/1) 143. Makrigoas Fault 2 (MkgF/2) 144. Makrigoas Fault 3 (MkgF/3) 145. Makrigoas Fault 4 (MkgF/4) 146. Makrigoas Fault 5 (MkgF/5) 147. Moni Agiou Ioannou Fault 1 (MAIF/1) 148. Moni Agiou Ioannou Fault 2 (MAIF/2) 149. Perachora Basin Fault 1 (PBF/1) 4.49 N onshore Roberts and Gawthorpe, N onshore Roberts and Gawthorpe, N onshore Roberts and Gawthorpe, S onshore Roberts and Gawthorpe, S onshore Morewood, N onshore IGME geologic map, Perachora sheet 0.71 S onshore Morewood, N onshore Morewood, S onshore Morewood, S onshore Morewood, S onshore Morewood, SW onshore Morewood, NE onshore Morewood, E onshore Morewood, N onshore Morewood, S onshore Morewood, N onshore Morewood, N onshore Morewood, N onshore Morewood, S onshore Morewood, S onshore Morewood, SW onshore Morewood, S onshore IGME geologic map, Perachora sheet 1.31 N onshore Morewood,

234 Appendix A Table of Faults 150. Perachora Basin Fault 2 (PBF/2) 151. Perachora Basin Fault 3 (PBF/3) 152. Perachora Basin Fault 4 (PBF/4) 153. Perachora Basin Fault 5 (PBF/5) 154. Perachora Basin Fault 6 (PBF/6) 155. Perachora Basin Fault 7 (PBF/7) 156. Osios Patapios Fault 1 (OPF/1) 157. Osios Patapios Fault 2 (OPF/2) 158. Osios Patapios Fault 3 (OPF/3) 159. Osios Patapios Fault 4 (OPF/4) 160. Loutraki Fault (LtF) 161. Isthmos Fault 1 (ISF/1) 162. Isthmos Fault 2 (ISF/2) 163. Isthmos Fault 3 (ISF/3) 164. Isthmos Fault 4 (ISF/4) 165. Examilia Fault (EXF) 166. Onia Fault 1 (ONF/1) 167. Onia Fault 2 (ONF/2) 168. Onia Fault 3 (ONF/3) 169. Kehries Fault (KHF) 170. Loutra Elenis Fault (LEF) 171. Mapsos Fault 1 (MpsF/1) 172. Mapsos Fault 2 (MpsF/2) 173. Mapsos Fault 3 (MpsF/3) 174. Mapsos Fault 4 (MAF/4) 0.83 S onshore Morewood, SW onshore Morewood, S onshore Morewood, N onshore Morewood, WSW onshore 3.77 S onshore 6.35 S onshore 2.72 S onshore 5.59 S onshore 1 W onshore 11.5 S onshore - offshore 2.95 S onshore 2.4 S onshore 3.8 S onshore 4.42 N onshore 5.35 NW onshore 2.7 N onshore 1.95 S onshore 1.87 S onshore 10.5 N onshore 2.1 S onshore 3.66 NE onshore 3.87 N onshore 2.18 S onshore 1.58 S onshore IGME geologic map, Perachora sheet IGME geologic map, Perachora sheet Neotectonic map, Korinthos sheet Neotectonic map, Korinthos sheet Neotectonic map, Korinthos sheet Neotectonic map, Korinthos sheet Charalampakis et al., 2014 Neotectonic map, Korinthos sheet Neotectonic map, Korinthos sheet Neotectonic map, Korinthos sheet Neotectonic map, Korinthos sheet Neotectonic map, Korinthos sheet Neotectonic map, Korinthos sheet Neotectonic map, Korinthos sheet Neotectonic map, Korinthos sheet Neotectonic map, Korinthos sheet Neotectonic map, Korinthos sheet Neotectonic map, Korinthos sheet Neotectonic map, Korinthos sheet Neotectonic map, Korinthos sheet Neotectonic map, Korinthos sheet 217

235 Appendix A Table of Faults 175. Mapsos Fault 5 (MpsF/5) 176. Mapsos Fault 6 (MpsF/6) 177. Mapsos Fault 7 (MpsF/7) 178. Athikia Fault 1 (AthF/1) 179. Athikia Fault 2 (AthF/2) 180. Athikia Fault 3 (AthF/3) 181. Athikia Fault 4 (AthF/4) 182. Athikia Fault 5 (AthF/5) 183. Galataki Fault (GltF) 184. Almyri Fault 1 (AlmF/1) 185. Almyri Fault 2 (AlmF/2) 186. Katakali Fault (KatF) 187. Agios Vasilios Ritos Fault zone (AVRF) 188. Pefkali Fault 1 (PfkL/1) 189. Pefkali Fault 2 (PfkF/2) 190. Pefkali Fault 3 (PfkF/3) 191. Pefkali Fault 4 (PfkF/4) 3.99 N onshore 4.1 S onshore 1.88 SE onshore 4.31 N onshore 3.77 N onshore 1.46 N onshore 2.1 N onshore 6.19 S onshore 1.64 SSE onshore 2.13 N onshore 1.97 S onshore 13 N onshore 29.3 N onshore 2.76 S onshore 3.1 N onshore 1.26 N onshore 3.12 N onshore 192. Sofiko Fault (SfkF) 6.34 SE onshore 193. Stefani Fault 1 (StF/1) 194. Stefani Fault 2 (StF/2) 195. Stefani Fault 3 (StF/3) 196. Agios Ioannis Fault 1 (AgIF/1) 197. Agios Ioannis Fault 2 (AgIF/2) 198. Agios Ioannis Fault 3 (AgIF/3) 199. Agios Ioannis Fault 4 (AgIF/4) 7.73 S onshore 3.43 N onshore 1.23 S onshore 1.91 S onshore 3.30 N onshore 5.28 N onshore 5.22 S onshore Neotectonic map, Korinthos sheet Neotectonic map, Korinthos sheet Neotectonic map, Korinthos sheet Neotectonic map, Korinthos sheet Neotectonic map, Korinthos sheet Neotectonic map, Korinthos sheet Neotectonic map, Korinthos sheet Neotectonic map, Korinthos sheet Neotectonic map, Korinthos sheet IGME geologic map, Sofikon sheet IGME geologic map, Sofikon sheet IGME geologic map, Sofikon sheet Neotectonic map, Korinthos sheet IGME geologic map, Sofikon sheet IGME geologic map, Sofikon sheet IGME geologic map, Sofikon sheet IGME geologic map, Sofikon sheet IGME geologic map, Sofikon sheet Neotectonic map, Korinthos sheet Neotectonic map, Korinthos sheet Neotectonic map, Korinthos sheet Neotectonic map, Korinthos sheet Neotectonic map, Korinthos sheet Neotectonic map, Korinthos sheet Neotectonic map, Korinthos sheet 218

236 Appendix A Table of Faults 200. Nemea Fault (NmF) 201. Stymfalia Fault (StmF) 202. Kefalari Fault (KflF) 203. Lechaio Fault (LcF) 204. Vrachati Fault (VrF) 7.1 N onshore 16.1 N onshore Neotectonic map, Korinthos sheet Neotectonic map, Korinthos sheet 8.66 N onshore Jolivet, S offshore Charalampakis et al., N offshore Charalampakis et al., Fryne Fault (FrF) 6.35 N offshore Charalampakis et al., Heraion Fault (HrF) 3.24 SE offshore Charalampakis et al., Vouliagmeni Fault 1 (VlF/1) 208. Vouliagmeni Fault 2 (VlF/2) 209. Vouliagmeni Fault 3 (VlF/3) 4.23 S offshore Charalampakis et al., S offshore Charalampakis et al., S offshore Charalampakis et al., Kiato Fault (KtF) 4.39 S offshore 211. Xylokastro Fault (XF) 212. Korfiotisa Fault (KrF) 213. Koutsos Fault (KtsF) 214. Pirgos Fault 1 (PF/1) 215. Pirgos Fault 2 (PF/2) NNE onshore NNW onshore SE onshore NW onshore SW onshore Taylor et al., 2011; Charalampakis et al., 2014 Ghizetti and Vezzani, 2005; Leeder et al., 2012 Ghizetti and Vezzani, 2005; Leeder et al., 2012 Ghizetti and Vezzani, 2005; Leeder et al., 2012 Ghizetti and Vezzani, 2005; Leeder et al., 2012; Rohais et al., 2007 Ghizetti and Vezzani, 2005; Rohais et al., Trikala Fault (TrF) 1.58 N onshore Rohais et al., Killini Fault (KilF) 8.92 N onshore Rohais et al., 2007; Leeder et al., Tarsos Fault (TarF) 5.47 NE onshore Rohais et al., Mavro Fault 1 (MF/1) 220. Mavro Fault; segment 2 (MF/s2) 221. Lagadeika Fault (LagF) 222. Lygia Fault (LyF) N onshore 5.57 N onshore Rohais et al., 2007; Leeder et al., 2012 Rohais et al., 2007; Leeder et al., E onshore Rohais et al., NW onshore 223. Aigira Fault (AgrF) 7.28 N onshore 224. Kalamias Fault (KlmF) 4.73 N onshore Ghizetti and Vezzani, 2005; Rohais et al., 2007 Ford et al., 2013; Rohais et al., 2007; Leeder et al., 2012 Ford et al., 2013; Rohais et al., 2007; 219

237 Appendix A Table of Faults 225. Akrata Delta Fault (ADF) 226. Ano Akrata Fault 1 (AAF/1) 227. Ano Akrata Fault 2 (AAF/2) 228. Paralia Platanos Fault 1 (PPF/1) 229. Paralia Platanos Fault 2 (PPF/2) 230. Prioni Fault 1 (PrF/1) 231. Prioni Fault 2 (PrF/2) 232. Prioni Fault 3 (PrF/3) 233. Voutsimos Fault (VtsF) 234. Aiges Fault 1 (AgF/1) 235. Aiges Fault 2 (AgF/2) 236. Valimi Fault (ValF) Tsivlos Fault (TsvF) 3.68 N onshore Leeder et al., 2012 Rohais et al., 2007; Leeder et al., E onshore Rohais et al., W onshore Rohais et al., W onshore Ford et al., E onshore Ford et al., NW onshore 1.1 W onshore NE onshore NW onshore Ford et al., 2013; Rohais et al., 2007; Ghizetti and Vezzani, 2005 Ford et al., 2013; Rohais et al., 2007 Ford et al., 2013; Rohais et al., 2007; Ghizetti and Vezzani, 2005 Ford et al., 2013; Rohais et al., 2007; Ghizetti and Vezzani, 2005; Leeder et al., NE onshore Rohais et al., NE onshore Rohais et al., NE onshore 9.15 N onshore 238. Vela Fault (VlF) NW onshore 239. Exochi Fault 1 (ExF/1) 240. Exochi Fault 2 (ExF/2) 241. Exochi Fault 3 (ExF/3) 242. Exochi Fault 4 (ExF/4) 243. Exochi Fault 5 (ExF/5) 244. Xerovouni Fault 1 (XeF/1) SE onshore Rohais et al., 2007; Ford et al., 2013; Ghizetti and Vezzani, 2005; Leeder et al., 2012 Ford et al., 2013; Rohais et al., 2007; Leeder et al., 2012 Ghizetti and Vezzani, 2005; Rohais et al., 2007; Leeder et al., 2012 Rohais et al., 2007; Ghizetti and Vezzani, S onshore Rohais et al., S onshore 3.73 N onshore 1.74 NW onshore 4.26 N onshore Rohais et al., 2007; Ford et al., 2013 Rohais et al., 2007; Ford et al., 2013 Ford et al., 2013; Rohais et al., 2007 Ford et al., 2013; Rohais et al., 2007; Leeder et al.,

238 Appendix A Table of Faults 245. Xerovouni Fault 2 (XeF/2) 246. Kalavrita Fault (KalF) 247. Kastraki Fault (KstF) 248. Avlonas Fault (AvlF) 249. Kerpini Fault (KrpF) 3.63 SW onshore Ford et al., N onshore Ford et al., E onshore Ford et al., S onshore Ford et al., N onshore Ford et al., 2013; Ghizetti and Vezzani, Rogoi Fault (RogF) 1.59 S onshore Ford et al., Doumena Fault (DoF) N onshore 252. Korfes Fault (KorF) 5.73 S onshore 253. Kato Zachlorou Fault (KZF) 254. Vilivina Fault (VlvF/1) 255. Vilivina Fault 2 (VlvF/2) Ford et al., 2013; Ghizetti and Vezzani, 2005 Moretti et al., 2003; Ghisetti and Vezanni, S onshore Ford et al., S onshore Ford et al., S onshore Ford et al., Lofos Fault (LoF) 2.67 S onshore Ford et al., Pirgaki-Mamousia Fault zone (PMF) 258. Mamousia Vertical Fault 1 (MvF/1) 259. Mamousia Vertical Fault 2 (MvF/2) 260. Mamousia Vertical Fault 3 (MvF/3) 261. Mamousia Vertical Fault 4 (MvF/4) 262. Katafigion Fault (KtF) 263. Kastillia Fault (KslF) 264. Trapeza Fault 1 (TrpF/1) 265. Trapeza Fault 2 (TrpF/2) 266. Trapeza Fault 3 (TrpF/3) 267. Trapeza Fault 4 (TrpF/4) 268. Kerinitis Fault (KrnF) 269. East Helike Fault (EHF) N onshore Ford et al., 2013; Ghizetti and Vezzani, onshore Ford et al., onshore Ford et al., onshore Ford et al., onshore Ford et al., S onshore Ford et al., S onshore Ford et al., NE onshore Ford et al., 2013; Ghizetti and Vezzani, N onshore Ford et al., N onshore Ford et al., S onshore Ford et al., NW onshore NE onshore - offshore Ford et al., 2013; Ghizetti and Vezzani, 2005 Ford et al., 2013; Ghizetti and Vezzani, 2005; Bell et al.,

239 Appendix A Table of Faults 270. West Helike Fault (WHF) 271. Kato Fteri Fault 1 (KFF/1) 272. Kato Fteri Fault 2 (KFF/2) 273. West Kerinia Fault (WKF) 274. Kato Mavriki Fault (KMF) 275. Achladia Fault (AchF) 276. South Lakka Fault (SLF) 277. Agios Konstantinos Fault 1 (AKF/1) 278. Agios Konstantinos Fault 2 (AKF/2) 279. Agios Konstantinos Fault 3 (AKF/3) 280. Agios Konstantinos Fault 4 (AKF/4) 281. Agios Konstantinos Fault 5 (AKF/5) 282. Selinoudas Fault (SndF) NE onshore Ford et al., 2013; Ghizetti and Vezzani, NE-E onshore Ford et al., NE onshore Ford et al., NE onshore Ford et al., 2013; Ghizetti and Vezzani, S? onshore Ford et al., 2013; NW onshore Ford et al., 2013; Ghizetti and Vezzani, SE onshore Ghizetti and Vezzani, N onshore Palyvos et al., N onshore Palyvos et al., N onshore Palyvos et al., NE onshore NE onshore NE onshore Ghizetti and Vezzani, 2005 Palyvos et al., 2005 Ghizetti and Vezzani, 2005 Palyvos et al., 2005 Ford et al., 2013; Ghizetti and Vezzani, Valta Fault (VltF) 2.62 N onshore Ford et al., Famelitika Fault (FmlF) 285. Manesi Fault (MnsF) 286. Karousi Fault (KrsF) 287. Leodio Fault 1 (LdF/1) 288. Leodio Fault 2 (LdF/2) 289. Leodio Fault 3 (LdF/3) 5.52 S onshore Moretti et al., N onshore Moretti et al., N onshore Moretti et al., N onshore Jolivet, S onshore Taylor et al., S onshore Flotté et al., Lakka Fault (LakF) 7.79 NE onshore 291. Aigion Fault (AigF) 292. Fassouleika Fault (FsF) 293. Neos Erineos Fault (NEF) 294. Selianitika Fault (SeF) NE onshore - offshore NE onshore Moretti et al., 2003; Flotté et al., 2005 Moretti et al., 2003; Ghizetti and Vezzani, 2005; Palyvos et al., 2005; McNeill et al., 2005 Ghizetti and Vezzani, 2005; Palyvos et al., NE onshore Palyvos et al., NE onshore Ghizetti and Vezzani, 2005; Palyvos et al.,

240 Appendix A Table of Faults 295. Kamares Fault 1 (KmrF/1) 296. Kamares Fault 2 (KmrF/2) 297. Kamares Fault 3 (KmrF/3) 298. Kamares Fault 4 (KmrF/4) 299. Kamares Fault 5 (KmrF/5) 300. Panagopoula Fault (PngF) 301. Ano Ziria Fault (AZF) 302. Ano Rodini Fault (ARF) 0.7 NE onshore Palyvos et al., NE onshore Palyvos et al., NE onshore Palyvos et al., 2007a 1.1 NE onshore Palyvos et al., 2007a 1.3 NE onshore Palyvos et al., 2007a 2.93 NE onshore Palyvos et al., 2007a 2.12 NE onshore Palyvos et al., 2007a 4.2 NW onshore Palyvos et al., 2007a 303. Labiri Fault (LbF) 3.57 NE onshore Palyvos et al., Psathopyrgos Fault (PsthF) 305. Drepano Fault 1 (DrF/1) 306. Drepano Fault 2 (DrF/2) 10.7 N onshore Palyvos et al., 2007b Flotté et al., N onshore Palyvos et al., 2007a 1.18 NW onshore Palyvos et al., 2007a 307. Sella Fault (SelF) 9.25 NW onshore Flotté et al., Nafpaktos Basin Fault 1 (NBF/1) 309. Nafpaktos Basin Fault 2 (NBF/2) 310. Nafpaktos Basin Fault 3 (NBF/3) 311. Nafpaktos Basin Fault 4 (NBF/4) 312. Monastiraki Fault 1 (MnF/1) 313. Monastiraki Fault 2 (MnF/2) 314. Monastiraki Fault 3 (MnF/3) 315. Trizonia Fault 1 (TrzF/1) 316. Trizonia Fault 2 (TrzF/2) 317. Trizonia Fault 3 (TrzF/3) 318. Trizonia Fault 4 (TrzF/4) 319. Trizonia Fault 5 (TrzF/5) 5.48 S offshore Beckers et al., S offshore Beckers et al., SE offshore Beckers et al., N offshore Beckers et al., S offshore Beckers et al., N offshore Beckers et al., S offshore Beckers et al., S offshore Beckers et al., S onshore Beckers et al., N 2.60 N onshore - offshore onshore - offshore Beckers et al., 2015 Moretti et al., N offshore Beckers et al., Valimitika (ValF) 3.83 N offshore Stefatos et al.,

241 Appendix A Table of Faults 321. Cape Gyftisa Fault (CGF) 322. Psaromita Fault 1 (PsrF/1) 323. Psaromita Fault 2 (PsrF/2) 324. Psaromita Fault 3 (PsrF/3) 325. North Eratini Fault (NEF) 326. South Eratini Fault (SEF) 327. West Channel Fault (WCF) 3.29 N offshore 1.83 NW offshore 5.78 S offshore 2.66 N offshore N offshore S offshore McNeill et al., 2005; Stefatos et al., 2002 McNeill et al., 2005; Bell et al., 2009 McNeill et al., 2005; Stefatos et al., 2002 McNeill et al., 2005; Bell et al., 2009 Bell et al., 2008 McNeill et al., 2005 Bell et al., 2008 McNeill et al., S offshore Bell et al., Diakopto (DkF) S offshore Stefatos et al., East Channel Fault (ECF) 330. Panormo Fault 1 (PnF/1) 331. Panormo Fault 2 (PnF/2) S offshore Bell et al., S offshore Taylor et al., S offshore Taylor et al., Akrata Fault (AkrF) N offshore Bell et al., 2008; Akrata North 1 (ANF/1) 334. Akrata North 2 (ANF/2) 335. Akrata North 3 (ANF/3) 336. Anemokambi 1 (AnmF/1) 337. Anemokambi 2 (AnmF/2) 338. Anemokambi 3 (AnmF/3) 339. Anemokambi 4 (AnmF/4) 5.11 S offshore 4.32 N offshore Bell et al., 2008; 2009 Taylor et al., 2011 Bell et al., 2008; 2009 Taylor et al., S offshore Taylor et al., S offshore Bell et al., 2008; S offshore Bell et al., 2008; 2009; Taylor et al., N offshore Bell et al., 2008; S offshore Bell et al., 2008; Itea Fault 1 (ItF/1) 14.6 S offshore 341. Itea Fault 2 (ItF/2) S offshore Bell et al., 2009 Taylor et al., 2011 Bell et al., 2009; Stefatos et al., 2002; Taylor et al., Itea Fault 3 (ItF/3) 3.91 S offshore Taylor et al., Derveni (DrvF) N offshore 344. North Derveni Fault 1 (NDF/1) 345. North Derveni Fault 2 (NDF/2) 346. North Derveni Fault 3 (NDF/3) Bell et al., 2009 Taylor et al., N offshore Taylor et al., S offshore Taylor et al., N offshore Taylor et al.,

242 Appendix A Table of Faults 347. North Derveni Fault 4 (NDF/4) 348. North Derveni Fault 5 (NDF/5) 349. North Derveni Fault 6 (NDF/6) 350. North Derveni Fault 7 (NDF/7) 351. North Derveni Fault 8 (NDF/8) 352. North Derveni Fault 9 (NDF/9) 353. North Derveni Fault 10 (NDF/10) 354. North Derveni Fault 11 (NDF/11) 5.53 S offshore Taylor et al., S offshore Taylor et al., S offshore Taylor et al., S offshore Taylor et al., S offshore Taylor et al., S offshore Taylor et al., S offshore Taylor et al., S offshore Taylor et al., Likoporia (LkF) 17 N offshore 356. West Antikyra (WAF) 357. East Antikyra (EAF) 358. South Antikyra Fault 1 (SAF/1) 359. South Antikyra Fault 2 (SAF/2) 360. South Antikyra Fault 3 (SAF/3) 361. South Antikyra Fault 4 (SAF/4) 362. Pagalos Fault 1 (PgF/1) 363. Pagalos Fault 2 (PgF/2) 364. Abelos Fault 1 (AblF/1) 365. Abelos Fault 2 (AblF/2) 366. Velanidia Fault (VlF) 367. Vourlia Fault (VrlF) 368. Offshore Xylokastro Fault (OXL) 369. North Xylocastro Fault 1 (NXF/1) S offshore S offshore 12.3 S offshore Bell et al., 2009; Taylor et al., 2011 Taylor et al., 2011; Bell et al., 2009; Stefatos et al., 2002 Stefatos et al., 2002; Bell et al., 2009; Taylor et al., 2011 Taylor et al., 2011; Bell et al., S offshore Taylor et al., S offshore Taylor et al., S offshore Taylor et al., S offshore Stefatos et al., 2002 Bell et al., 2009 Taylor et al., S offshore Taylor et al., S offshore Taylor et al., N offshore Taylor et al., SW offshore Stefatos et al., S offshore N offshore Stefatos et al., 2002; Bell et al., 2009 Stefatos et al., 2002; Bell et al., 2009; Taylor et al., N offshore Bell et al.,

243 Appendix A Table of Faults 370. North Xylocastro Fault 2 (NXF/2) 371. North Xylocastro Fault 3 (NXF/3) 372. North Xylocastro Fault 4 (NXF/4) 373. North Xylocastro Fault 5 (NXF/5) 374. Perachora Fault (PchF) 8.89 N offshore 6.4 S offshore Bell et al., 2009; Taylor et al., 2011 Bell et al., 2009; Taylor et al., N offshore Bell et al., S offshore Bell et al., NW offshore 375. Strava Fault (StrF) N offshore 376. Glaronisi Fault 1 (GlF/1) S offshore 377. Glaronisi Fault 2 (GlF/2) 8.12 S offshore 378. Glaronisi Fault 3 (GlF/3) 6.14 S offshore 379. Glaronisi Fault 4 (GlF/4) NE offshore 380. Glaronisi Fault 5 (GlF/5) 2.17 S offshore 381. Glaronisi Fault 6 (GlF/6) 382. Domvrena Fault (DomF) 383. West Alkyonides (WAF) Stefatos et al., 2002; Bell et al., 2009; Taylor et al., 2011 Stefatos et al., 2002; Sakelariou et al., 2007; Leeder et al., 2002 Stefatos et al., 2002; Sakelariou et al., 2007; Leeder et al., 2002; Bell et al., 2009 Leeder et al., 2005; Bell et al., 2009; Stefatos et al., 2002 Leeder et al., 2002; Stefatos et al., 2002 Stefatos et al., 2002 Leeder et al., 2002; Bell et al., 2009; Sakelariou et al., 2007 Leeder et al., 2002; Stefatos et al., 2002; Bell et al., 2009; Sakelariou et al., N offshore Sakelariou et al., S offshore 384. East Alkyonides (EAF) 8.47 NW 385. Alkyonides Fault 1 (AlkF/1) 386. Alkyonides Fault 2 (AlkF/2) 387. Alkyonides Fault 3 (AlkF/3) N offshore offshore 2.22 N offshore 2.83 SE offshore 3.3 SE offshore Stefatos et al., 2002; Sakelariou et al., 2007; Leeder et al., 2002; Bell et al., 2009 Stefatos et al., 2002; Leeder et al., 2002; Sakelariou et al., 2007; Bell et al., 2009 Stefatos et al., 2002; Leeder et al., 2002; Sakelariou et al., 2007; Bell et al., 2009 Leeder et al., 2002; Stefatos et al., 2002; Sakelariou et al., 2007 Leeder et al., 2002; Stefatos et al., 2002 Leeder et al., 2002; Stefatos et al., 2002; Sakelariou et al., 2007; Bell et al.,

244 Appendix A Table of Faults 388. Alkyonides Fault 4 (AlkF/4) 389. Germeno Fault (GrmF) 390. North Mytikas Fault (NMF) 391. South Mytikas Fault (SMF) 4 NW offshore S onshore - offshore Leeder et al., 2002; Stefatos et al., 2002; Bell et al., 2009 Stefatos et al., 2002; Sakelariou et al., 2007; Leeder et al., 2002; Bell et al., N offshore Sakelariou et al., S offshore Sakelariou et al.,

245 Appendix B: Results for the Declustered Datasets a) b) Figure B.1: Aftershock identification intervals in a) space and b) time, as a function of the mainshock magnitude, according to the window parameter settings of Gardner and Knopoff (1974) and Uhrhammer (1986).

246 Appendix B Results for the Declustered Datasets Figure B.2: Cumulative (squares) and non-cumulative (incremental) (triangles) frequency magnitude distribution for the declustered NOA dataset. The solid line represents the G-R scaling relation for b = 1.16 and a = Figure B.3: Cumulative frequency magnitude distribution for the declustered NOA dataset and the corresponding fits according to the F-A model for various values of the minimum magnitude M0. 229

247 Appendix B Results for the Declustered Datasets Figure B.4: Probability density p(t) of the rescaled inter-event times T for the declustered HUSN (circles) and CRLN (squares) datasets. The solid line represents the corresponding fit according to the q-generalized gamma distribution (Eq.5.2), for the values of C = 0.9, β = 0.8, γ = 0.75 and q = The dashed line represents the corresponding fit according to the gamma distribution (Eq.5.1) for the values of C = 0.9, β = 1 and γ =

248 Appendix B Results for the Declustered Datasets a) b) 231