UNIVERSITY OF CALGARY. Nontrivial Decay of Aftershock Density With Distance in Southern California. Javad Moradpour Taleshi A THESIS

Transcription

1 UNIVERSITY OF CALGARY Nontrivial Decay of Aftershock Density With Distance in Southern California by Javad Moradpour Taleshi A THESIS SUBMITTED TO THE FACULTY OF GRADUATE STUDIES IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE GRADUATE PROGRAM IN PHYSICS AND ASTRONOMY CALGARY, ALBERTA July, 2014 c Javad Moradpour Taleshi 2014

2 Abstract The decay of the aftershock density with distance plays an important role in the discussion of the dominant underlying cause of earthquake triggering. Here, we provide evidence that its form is more complicated than typically assumed and that in particular a transition in the power law decay occurs at length scales comparable to the thickness of the crust. This is supported by an analysis of a high-resolution catalogue for Southern California and surrogate catalogues generated by the Epidemic-Type Aftershock Sequence (ETAS) model, which take into account short-term aftershock incompleteness, anisotropic triggering and variations in the observational magnitude threshold. Our findings indicate specifically that the asymptotic decay in the aftershock density with distance is characterized by an exponent larger than 2.0, which is much bigger than the observed exponent of approximately 1.35 observed for shorter distances. This has also important consequences for time-dependent seismic hazard assessment based on the ETAS model. i

3 Acknowledgements The completion of this thesis would not be possible without supports of my supervisor, Jörn Davidsen. I thank him for the suggestion of this project to me, for helping me to understand different parts of the problem and for his feedback on my work. I also thank my colleagues in the Complexity Science Group, specially Aicko Yves Schumann for providing the core of the numerical code for the ETAS simulation and Chad Gu for his helps in the early stages of the project. ii

4 Table of Contents Abstract i Acknowledgements ii Table of Contents iii List of Figures iv 1 Introduction Moment magnitude Spatiotemporal clustering of seismicity Triggering and definition of aftershocks Inference problem Motivation Outline Aftershock Identification Gutenberg Richter law Space-time-magnitude distance Data Density plot of the nearest neighbour distances Definition of aftershocks and main shocks Spatial distribution of aftershocks in Southern California Rupture length Functional form of the spatial distribution of aftershocks in SC The ETAS model Definition of the ETAS model Temporal distribution of aftershocks for the ETAS model Spatial distribution of aftershocks for the ETAS model Different version of ETAS model Comparing different versions of the ETAS model to the SC Inference of triggered events Spatial distribution Omori Utsu law GR relation for aftershocks and background Aftershock productivity Dependence on the observational magnitude threshold Discussion Bibliography iii

5 List of Figures and Illustrations 1.1 Focus (hypocenter) and epicenter of an earthquake The number of earthquake with magnitude bigger than 2.5 per month in Southern California from 1981 to The position of earthquakes (blue dots) which occurred within 10 days after Landers Earthquake which are recorded by SCSN. The Landers earthquake is shown by red star in this figure. This figure is taken from [Gao, 2006] Density plot of the set {n j} represented in log τ log l space for SC (time is measured in seconds and distances are measured in meters). The black line (log n = 7.0) separates two populations corresponding to triggered events (below) and background events (above) Panel (a) shows a time oriented tree which has been created by connecting all the events of a toy catalogue with 22 earthquakes to their nearest neighbours based on the Eq. (2.2). The root of this tree is the first event in the catalogue. Panel (b) shows a forest which has been created by removing all the insignificant links of the tree. Each tree in this forest corresponds to an earthquake cluster. Roots of these trees are background events, but all other events are triggered by other earthquakes Spatial distribution of aftershocks of SC for different ranges of the main shock magnitude, using epicenter distances Rescaled spatial distribution of aftershocks using epicenter distances, for different ranges of the main shock magnitude for SC. All the distances are rescaled by a factor m such that λ = r m, where m is the average magnitude for the considered magnitude range Spatial distribution of aftershocks using epicenter distances for main shock magnitudes in the range 3.0 < m < 4.0 for SC considering all aftershocks (squares) and considering only aftershocks that occurred within one hour of their main shocks (circles). Three separate power-law regimes can be identified in both cases. For comparison, the spatial distribution given by Eq. (4.4) with exponents γ = 0.6, q = 0.35 and d = 1.2 (dashed line) and the spatial distribution obtained by simulating the anisotropic case (see text for details) using Eq. (4.4) with exponents γ = 0.0, q = 0.35 and d = 1.2 for the epicenterto-source distance (dotted line) are shown as well. Both agree very well with the time-restricted data Spatial distribution of aftershocks using epicenter distances for main shock magnitudes in the range 2.5 < m < 3.0 (circles) and the range 4.0 < m < 5.0 (squares) for SC considering only aftershocks that occurred within one hour of their main shocks. For comparison, the associated spatial distributions given by Eq. (4.4) with exponents γ = 0.6, q = 0.35 and d = 1.2 (dashed lines) are shown as well. As this figure shows the functional forms agree well with the time restricted spatial distribution in these magnitude ranges iv

6 4.5 Definition of spatial distances in the anisotropic case. H 1 and H 2 are the epicenters of the main shock and aftershock, respectively. L R is the rupture length of the main shock and S is the source point. r is the epicenter-tosource distance drawn from the given P m (r ) and r is the epicenter distance used to estimate P m (r) Density function of epicenter-to-fault distances, epicenter-to-source distances and epicenter-to-epicenter distances for numerically simulated and anisotropically distributed aftershocks of a main shock with a rupture length of 350m (see main text for details). For the simulation, Eq. (4.4) from the main text with γ = 0, q = 0.35 and d = 1.2 was used as the functional form for the distribution of epicenter-to-source distances Density plot of the set {n j}, same as Fig. 3.1, but for model I. There is a striking resemblance between the two, both indicating a clear separation between triggered events and background events Density plot of the set {n j} represented in log τ log l space for model II. The black line is the threshold that we chose and it represents log n = 7.0. We consider earthquakes, which are below this line, as triggered earthquakes. Time is measured in seconds and distances are measured in meters Density plots of the set {n j} represented in log τ log l space for model IV. The black line is the threshold that we chose and it represents log n = 7.0. We consider earthquakes, which are below this line, as triggered earthquakes. Time is measured in seconds and distances are measured in meters Density plots of the set {n j} represented in log τ log l space for model V. The black line is the threshold that we chose and it represents log n = 7.0. We consider earthquakes, which are below this line, as triggered earthquakes. Time is measured in seconds and distances are measured in meters Density plots of the set {n j} represented in log τ log l space for model V I. The black line is the threshold that we chose and it represents log n = 7.0. We consider earthquakes, which are below this line, as triggered earthquakes. Time is measured in seconds and distances are measured in meters Density plots of the set {n j} represented in log τ log l space for model V II. The black line is the threshold that we chose and it represents log n = 6.0. We consider earthquakes, which are below this line, as triggered earthquakes. Time is measured in seconds and distances are measured in meters Same as Figs. 3.1 and 6.1 but for model III. There is no clear separation into two populations in this case Rescaled spatial distributions of aftershocks, same as Fig. 4.2, but for model I. There is a clear resemblance between the two. Inset: Original spatial distribution of aftershocks before rescaling Rescaled spatial distributions of aftershocks, same as Fig. 4.2, but for model II. There is a clear resemblance between the two. Inset: Original spatial distribution of aftershocks before rescaling v

7 6.10 Rescaled spatial distributions of aftershocks, same as Fig. 4.2, but for model V. There is a clear resemblance between the two. Inset: Original spatial distribution of aftershocks before rescaling Rescaled spatial distributions of aftershocks, same as Fig. 4.2, but for model V I. There is a clear resemblance between the two. Inset: Original spatial distribution of aftershocks before rescaling Rescaled spatial distributions of aftershocks, same as Fig. 4.2, but for model V II. There is a clear resemblance between the two. Inset: Original spatial distribution of aftershocks before rescaling Same as Figs. 4.2 and 6.8 but for model IV. A full collapse of the distribution over the full range occurs in this case. Inset: Original spatial distribution of aftershocks before rescaling Spatial distribution of aftershocks using epicenter distances for main shock magnitudes in the range 3.0 < m < 4.0 for SC, model I and model IV, considering only aftershocks that occurred within one hour of their main shocks. For comparison, the spatial distribution given by Eq. (4.4) with exponents γ = 0.6, q = 0.35 and d = 1.2 is also shown in this figure Spatial distribution of aftershocks which happened in less than one hour after their main shocks for main shocks magnitude range 3.0 < m < 4.0 for model I, model II model V and model V I Rescaled aftershock rate r(t) at time t after the main shock averaged over different ranges of main shock magnitudes for SC (solid line) and model I (dashed line) Rescaled average aftershock rate for different ranges of main shocks magnitude for mpdel I (solid line), and the model II (dash line). In this figure r(t) is the average aftershock rate at time t after the main shocks Rescaled aftershock rate r(t) at time t after the main shock averaged over different ranges of main shock magnitudes for SC (solid line) and model V (dashed line) Rescaled aftershock rate r(t) at time t after the main shock averaged over different ranges of main shock magnitudes for model I (dashed line) and model V I (dotted line) Rescaled aftershock rate r(t) at time t after the main shock averaged over different ranges of main shock magnitudes for SC (solid line) and model V II (dashed line) Histogram of earthquake magnitudes for triggered and background events for SC. The maximum likelihood estimates are indicated as solid lines Histogram of earthquake magnitudes for triggered and background events for model I. The maximum likelihood estimates are indicated as solid lines Histogram of earthquake magnitudes for triggered and background events for model II. The maximum likelihood estimates are indicated as solid lines Histogram of earthquake magnitudes for triggered and background events for model IV. The maximum likelihood estimates are indicated as solid lines Histogram of earthquake magnitudes for triggered and background events for model V II. The maximum likelihood estimates are indicated as solid lines. 71 vi

8 6.26 Histogram of earthquake magnitudes for triggered and background events for model V. The maximum likelihood estimates are indicated as solid lines Average number of triggered events as a function of average main shock magnitude estimated for SC (black circles), for model I (red squares) and model II (blue diamonds). The solid, dashed and dotted line correspond to the best fit, respectively Spatial distribution of aftershocks using epicenter distances for main shock magnitudes in the range 3.5 < m < 4.5 and considering only aftershocks that occurred within one hour of their main shocks. Only events with magnitude 3.5 and higher are considered for estimating triggering relations and identifying aftershocks. For comparison, the anisotropic spatial distribution with exponents γ = 0.0, q = 0.35 and d = 1.2 describing the expected behavior for model I is also shown in this figure Histogram of earthquake magnitudes for triggered and background events for SC after changing the magnitude threshold to 3.5. The maximum likelihood estimates are indicated as solid lines Histogram of earthquake magnitudes for triggered and background events for model I after changing the magnitude threshold. The maximum likelihood estimates are indicated as solid lines Average number of triggered events as a function of average main shock magnitude estimated for SC (black circles) and for model I (red squares) after changing magnitude threshold to 3.5. The solid, dashed and dotted line correspond to the best fit, respectively vii

9 Chapter 1 Introduction Complexity science is one of the most interdisciplinary areas of research in our time. This research area is so diverse which makes it difficult to have a unique definition of it. Complexity science is often used as a broad term for research, addressing problems in different areas including anthropology, artificial intelligence, artificial life, chemistry, computer science, economics, evolutionary computation, meteorology, molecular biology, neuroscience, physics, psychology, sociology and geophysics. This science is trying to investigate and understand complex systems. A complex system is usually characterized by a large number of interacting components, showing some properties which emerge from the collective interaction based on a small number of rules. These properties are not obvious from knowing all these rules. Because of the large number of components, we often can not identify and explore these systems only by using analytical tools; we use simulation and other kinds of narratives and methods for this purpose. However, different complex systems share common features which suggests that understanding them in one context might help to proceed in others. One example of a complex system is the crust of Earth. Chaos and fractals are two important concepts which are widely used in this research area [Baas, 2002]. Chaos theory studies the behaviour of dynamical systems that are highly sensitive to initial conditions. Chaos is applicable for different geophysical phenomena such as mantle convection which is the slow creeping motion of Earth s solid silicate mantle, caused by convection currents carrying heat from the interior of Earth to the surface. On the other hand, a fractal is a mathematical set that typically displays self-similar patterns which means that there is no 1

10 preferred length scale in this pattern. Many natural forms, such as coastlines and faults look alike on many scales and then the concept of a fractal is applicable for investigating them. Scaling equations connect self-similarity to power laws. For a self-similar observable A(x), which is a function of a variable x, a scaling relationship holds by definition [Baas, 2002]: A(λx) = λ s A(x), (1.1) where λ is a constant factor and s is the scaling exponent. The power law (A(x) = x n, where n is a constant) is the only function which obeys this scaling relationship [Bak & Tang, 1989] A(λx) = (λx) n = λ n x n = λ n A(x). (1.2) The power law distribution as a signature of fractals is ubiquitous in Nature. The frequency of observing power laws in Nature can potentially be explained by self-organized criticality [Turcotte, 1999]. The most cited power law in geology is the Gutenberg Richter relationship [Gutenberg & Richter, 1949, Bak & Tang, 1989]: the expected number of earthquakes N with magnitude bigger than m in a given (large) area over a given (large) time interval is given by N = a 10 bm, (1.3) where the constant b is known as the b-value. The constant a is a measure of the intensity of the regional seismicity. If we write the magnitude m as a function of rupture area of the earthquake A, as will be discussed in detail in Section 1.1, and express Eq. (1.3) in terms of this parameter, it becomes a power law [Bak & Tang, 1989]: N A D, (1.4) where D is determined by b. Therefore the Gutenberg Richter relationship is an example of a power law in geology and we can use the framework of studying fractal behaviour for investigating certain aspects of earthquakes. 2

11 1.1 Moment magnitude Now we want to be more precise and define precisely the terms earthquake, rupture area and magnitude which we used in the previous section. An earthquake is a sudden release of stored strain energy in the Earth s crust. In most cases the underlying cause of an earthquake is the motion of tectonic plates. Tectonic plates are pieces of the Earth s crust which move slowly relative to each other. A fault is a planar fracture or discontinuity in a volume of rock which makes this motion possible. If there are no irregularities along a fault surface, the sides of the fault would move past each other smoothly. But all natural fault surfaces have irregularities and these irregularities prevent them from sliding on each other. Therefore stress and stored strain energy in the volume around the fault surfaces increase. This process continues until the stress increases sufficiently to break through the irregularity. This break suddenly allows faults to slide over each other and release some of the stored energy. The area which moves during this process is called rupture area. The position where the strain energy is first released and the rupture is initiated is called the hypocenter or focus and the point at ground level directly above the hypocenter is also called the epicenter. These concepts are illustrated in Fig Figure 1.1: Focus (hypocenter) and epicenter of an earthquake. 3

12 The moment magnitude is the most common scale on which earthquake sizes are reported. It is used to measure the size of earthquakes in terms of the energy released. The magnitude is based on the seismic moment of the earthquake which is a measure of the total amount of energy that is released during an earthquake. Mathematically speaking the moment magnitude m is a dimensionless number defined by m = 2 3 log 10(M 0 ) 6.0, (1.5) where M 0 is the seismic moment [Ben-Zion, 2008, Kanamori & Brodsky, 2004]. The scalar seismic moment M 0 is defined by the equation M 0 = µ A D, (1.6) where µ is the shear modulus of the rocks involved in the earthquake, A is the rupture area and D is the average displacement of this area. Earthquakes with magnitude 3.0 or less are almost imperceptible and earthquakes with magnitude 7.0 or more can cause serious damage over larger areas. This damage is dependent on the depth of earthquakes and shallower ones cause more damage to structures. 4

13 1.2 Spatiotemporal clustering of seismicity After talking about a single earthquake, now we want to see how these earthquakes are distributed in space and time. Fig. 1.2 shows the number of earthquakes that occurred in a month from 1981 to 2011 at Southern California. As we can see in this figure the rate of earthquakes occurrence is not homogeneous and it drastically changes during some periods of time. We can see three major peaks in this figure at 1992, 1999 and These are the years when three biggest earthquakes with magnitude bigger than 7.0 occurred in Southern California: Landers Earthquake (m = 7.3), Hector Mine Earthquake (m = 7.1) and El MayorCucapah Earthquake (m = 7.2) respectively. It indicates that seismic activity increases significantly after a big earthquake Figure 1.2: The number of earthquake with magnitude bigger than 2.5 per month in Southern California from 1981 to

14 Now we want to see how this increased seismic activity after a big earthquake is distributed in space. As an example Fig. 1.3 shows seismic activity close in time to the Landers Earthquake. Blue dots in this figure indicate the positions of earthquakes occurred withen 10 days after Landers Earthquake. The red star also shows the Landers earthquake which occurred on 1992/06/27. As we can see in this figure, most of earthquakes which occurred right after the Landers Earthquake are close to this event in space. It suggests that a big earthquake triggers other earthquakes locally such that it increases the earthquakes rate of the region significantly. 6

15 Figure 1.3: The position of earthquakes (blue dots) which occurred within 10 days after Landers Earthquake which are recorded by SCSN. The Landers earthquake is shown by red star in this figure. This figure is taken from [Gao, 2006]. 7

16 1.3 Triggering and definition of aftershocks We consider all the events which are not triggered by other earthquakes as background seismicity. The term trigger is used as a reference to the initiation of a process which can cause an earthquake. These triggered earthquakes occur after a time delay that can range from seconds to years after the causative event [Freed, 2005, Belardinelli et al., 2003, Shelly et al., 2011]. The physics of the earthquake triggering is complex. It is a process by which an earthquake redistributes the stress in the surrounding region and induces seismic activity in that region or triggers other earthquakes at longer distances. Both observations and theoretical models support the idea that earthquakes perturb the state of stress of neighbouring faults [Freed, 2005]. To be more precise these perturbations modify the mechanical condition of active faults. The stress in certain regions is actually increased by the occurrence of a fault slip which can relieve built-up elastic stress in the crust. Understanding earthquake triggering can help us to gain insight into the physics of earthquake occurrence which, as we will discuss in Chapter 7, may increase our ability to assess seismic hazards by pointing to where the next events may occur in an earthquake sequence. The induced change in the stress field consists of transient (dynamic) and permanent (static) stress changes. Both of them can trigger other earthquakes. The first type of triggering is due to static stress changes in the critical state. As we mentioned in Section 1.1, an earthquake is the result of a sudden release of stored strain energy between tectonic plates. In other word, an earthquake occurs when the shear stress is large enough to overcome the normal stress which, in combination with friction, prevents a locked fault from slipping. Stress changes caused by an earthquake can change both the shear and normal stress on neighbouring faults. A fault can be brought closer to failure if the shear stress is increased or the effective normal stress is decreased. Therefore stress changes caused by an earthquake can increase shear loads on regional faults or decrease the normal force, make faults slip 8

17 and cause an earthquake [Freed, 2005]. The term static or elastic triggering is used as only elastic stress changes are considered for this process. Earthquake triggering is also observed at distances much larger than the regional fault lengths. Since at these distances static stress changes are small, such distant triggering challenges that the only process for the earthquake triggering is static stress changes associated with earthquake slip. An alternative way which has been proposed to explain remotely triggered seismicity over such distances is dynamic stress changes associated with the passage of seismic waves that can transmit large stress to great distances with great speed [Freed, 2005]. This can either immediately trigger earthquakes or initiate secondary mechanical processes that eventually lead to triggering. Because of these secondary mechanical processes, dynamically triggered events can occur at any time after the seismic waves have passed and even with a considerable delay. Delayed earthquake triggering can be explained by a variety of time-dependent stress transfer mechanisms, such as viscous relaxation, poroelastic rebound and afterslip, or by reductions in fault friction [Freed, 2005]. One obvious problem of explaining triggering with seismic waves is that the dynamic stress is transient and leaves no permanent changes to the stress state once the waves have passed. In order to detect the dynamic triggering, it is necessary to observe that the rate of earthquakes after the passage of seismic waves is larger than the rate of events before the transient perturbation. Since for an affected fault it is irrelevant how far the triggering dynamic stresses travelled to reach the fault, it is likely that dynamic stresses associated with passing seismic waves influence local seismic activity as well. The effects of the dynamic stress perturbations and static stress changes are similar and it is extremely difficult to distinguish which of these two processes triggered a specific local event. Independent of triggering process, all events which have been triggered by an earth- 9

18 quake through any of these two mechanism are called aftershocks of that earthquake. But some aftershocks classifications not only include those earthquakes that are directly triggered by the main shock (the first generation of aftershocks), they also include earthquakes that are indirectly triggered, e.g., earthquakes directly triggered by first generation aftershocks. 1.4 Inference problem Distinguishing triggered events from background seismicity is the key step in the study of the earthquake triggering, but in the field it is currently extremely difficult to measure the stress field. Thus it is typically impossible to find out how exactly an earthquake changes the stress in the surrounding area and which events have been triggered by that specific event. Therefore we can only use some statistical methods for aftershock identification. The main idea of all of these methods is to find out how an earthquake changes the seismic activity in the region, in comparison to the background seismicity in that specific region, and then events which are not expected to happen in the area during the time interval are considered as aftershocks. As we saw in Fig. 1.2 and Fig. 1.3, after big earthquakes seismic activity locally increases significantly. The spatiotemporal clustering of these activities is very different from uniform background activity. Sometimes the largest earthquake in such a cluster is denoted as the main shock, then all the following events would be denoted as aftershocks and preceding ones as foreshocks (see, for example Ref. [Felzer & Brodsky, 2006]). But it is known that earthquakes independent of their size can trigger other smaller or bigger earthquakes and there is no physical distinction in the relaxation mechanism between main shocks, foreshocks, and aftershocks [Houghs & Jones, 1997, Helmstetter & Sornette, 2003]. Then the classification of earthquakes into main shocks, foreshocks, and aftershocks can only rely on identifying trig- 10

19 gering relationships between them and there is no physical difference between these classes of events. Several declustering algorithms have been proposed for identifying aftershocks and main shocks (see, for example [Gardner & Knopoff, 1974, Keilis-Borok et al., 1980, Reasenberg, 1985, Molchan & Dmitrieva, 1992]). Some of these methods have arbitrary rules and they are also rich in parameters. Because of that the objectivity of these methods is in question. For example a common way for aftershock identification is to determine a sharp space-time aftershock window following a large earthquake and all the earthquakes in this window would be considered as aftershocks of that large event. The method can easily suffer from the loss of long range triggering and also the inappropriate selection of the window shape can be a problem (see, for example [Richards-Dinger et al., 2010]). A number of methods have been proposed to overcome these problems. The method which we used for the aftershock identification in this work is one of them. To identify aftershocks, we follow the method first proposed in [Baiesi & Paczuski, 2004]. In this method a space-time-magnitude distance is defined between earthquakes based on phenomenological laws of seismicity related to the Gutenberg-Richter law. As shown in [Zaliapin et al., 2008], the distribution of all nearest-neighbour distances can allow us to identify potential triggering relationships and distinguish between populations of events which are triggered by other events and background activity. Based on these statistical methods we can not exactly identify all the triggering relationships between earthquakes and these methods only identify the most probable candidate for triggering aftershocks. As we mentioned before if we want to be sure about the triggering relationship, we should know all the stresses including the maximum yield stress of the in- 11

20 volved faults, which is currently impossible. This lack of knowledge about the interactions of faults makes it difficult to understand the triggering process. But using statistical methods for aftershock identification and investigating different properties of aftershocks can help us to get some insight about the physics of triggering. 1.5 Motivation As we mentioned in Section 1.4, we can use statistical methods for distinguishing aftershocks from background activity and then investigate different features of both groups of events. As we saw in Fig. 1.2, the rate of local activity increases after a big earthquake because of aftershocks of that event. The rate of these aftershocks is described by the Omori-Utsu law [Utsu et al., 1995]. The Omori-Utsu law is an empirical observation which states that the rate of aftershocks decreases with the time t passed after the main shock as: r(t) = χ (t + C) p, (1.7) where χ is a constant and determines the intensity of the rate. We know C > 0 which guarantees that the function does not diverges at t = 0 and p > 1 which guarantees a finite total number of aftershocks [Gu et al., 2013]. In contrast to the temporal evolution of aftershock sequences, much less is known about the spatial distribution of aftershocks. As we saw in Fig. 1.3 aftershocks are localized around the main shock, but is is not clear how the distribution of aftershocks decreases with distance. Recent studies have investigated this question and produced conflicting results: In [Felzer & Brodsky, 2006], it was found that the aftershock density with distance decays as a single power-law with exponent 1.35 over the observable range with no dependence on the size of the main shock. The results were later challenged in [Richards-Dinger et al., 2010] and attributed to the specific aftershock selection scheme. Other studies provided evidences that 12

21 the size of the main shock does have an influence on the aftershock density for length scales less than the rupture length [Baiesi & Paczuski, 2004, Marsan & Lengliné, 2008, Lippiello et al., 2009, Gu et al., 2013]. Some studies also provided evidences that the power-law decay in the aftershock density for larger distances is characterized by exponents of about 2.0 [Lippiello et al., 2009, Marsan & Lengliné, 2010, Shearer, 2012]. A very recent study [Gu et al., 2013] even indicated that the decay of aftershock density for large distances might not even follow a simple power law. Different triggering mechanisms have been proposed based on these different forms of the spatial distribution of aftershocks. More investigation of this distribution can be helpful for understanding the physics of triggering. Here, we challenge the often made assumption that the decay of aftershock density for large distances follows a simple power law. We test this hypothesis further by investigating aftershock sequences in a very recent high-resolution catalogue from Southern California and also by comparing our findings with surrogate catalogues generated by the simulation of seismic activity in this region. We also investigate other features of aftershocks which may have effects on our results. Most of these results has been published in [Moradpour et al., 2014] 1.6 Outline In this thesis, I will address the issues posed in Section 1.5 following the outline we give below. In Chapter 2 I will introduce a method for aftershock identification. By using this method we are able to distinguish between aftershocks and background events. In Chapter 3 I will introduce one of the catalogues for Southern California and illustrate our aftershock identification method as applied to this catalogue. In Chapter 4 I will investigate the spatial distribution of aftershocks in Southern Califor- 13

22 nia. In that chapter first I will introduce proposed functional forms for the spatial distribution of aftershocks and at the end I will suggest a new functional form for this distribution which is matched with real data in different distance ranges. In Chapter 5 first I will introduce the Epidemic-Type Aftershock Sequence (ETAS) model which is used to generate seismic surrogate catalogues, then I will introduce different versions of this model based on different assumptions about the spatial or temporal distribution of aftershocks. In Chapter 6 I will compare different statistical features of catalogues generated by these models to the associated features in Southern California catalogue to see which of our assumptions are suitable and necessary for the modelling of the real world data. At the end in Chapter 7 I will summarize and discuss different results presented in this thesis. 14

23 Chapter 2 Aftershock Identification For investigating different features of aftershocks, first we need a method for aftershock identification. Traditionally space-time window techniques (see, for example [Felzer & Brodsky, 2006]) or declustering methods [Gardner & Knopoff, 1974, Keilis-Borok et al., 1980, Molchan & Dmitrieva, 1992] have been used for this purpose. These methods are very simple to implement, but have arbitrary rules and are rich in parameters. Because of that their objectivity is in question as discussed, for example, in [Richards-Dinger et al., 2010]. Specifically they may suffer from the loss of long-range triggering and/or the inappropriate selection of the window shape. We follow a recently established methodology to identify aftershocks, which was first proposed by Baiesi and Paczuski [Baiesi & Paczuski, 2004] and later refined by Zaliapin et al. [Zaliapin et al., 2008], to avoid these short-comings. This methodology has been shown to be robust and quite effective [Zaliapin et al., 2008, Gu et al., 2013, Zaliapin & Ben-Zion, 2013a, Zaliapin & Ben-Zion, 2013b]. As a first step, this aftershock identification method defines a space-time-magnitude distance between each pair of earthquakes in order to find the nearest neighbour of each event. Then, as we will show later, the density plot of all of these nearest neighbour distances can be used for identifying potential triggering relationships [Zaliapin et al., 2008]. The main idea of this method is to sparsify the weighted network of space-time-magnitude distances between all earthquakes and keep only the backbone, composed by the set of significant relations. 15

24 2.1 Gutenberg Richter law The mentioned aftershock identification method is based on the Gutenberg-Richter (GR) relation [Gutenberg & Richter, 1949]. This relation is an empirical observation that the expected number of earthquakes N with magnitude bigger than m is given by N = a 10 bm. (2.1) In this equation, the value of the exponent b is close to one and it seems to be independent of the specific geographic area as long as we consider sufficiently large areas over sufficiently long time intervals [Kanamori & Brodsky, 2004, Naylor et al., 2009, Gulia & Wiemer, 2010]. In contrast, the prefactor a is a function of the size of the area and the length of considered time interval. If we assume homogeneity in time, the number of earthquakes should increase linearly with the length of the time interval, then the prefactor a should also increase linearly with time. We know seismic activity is not homogeneous in time and it increases after a big earthquake, but if we consider a long enough time interval, this assumption of homogeneity will be a reasonable approximation. We are also aware of the fractal appearance of earthquake epicentres [Turcotte, 1997] [Kagan, 1994]. If we assume earthquakes occur randomly in this fractal space, then the number of earthquakes should be proportional to the volume of this space and the perfactor a should increase as a power law with a fractal dimension d f with the linear size of the area. This assumption is also valid if we consider a large enough area. We can conclude from this argument that the prefactor a, on average, increases linearly with the length of the time interval and roughly as a power law with a fractal dimension, d f, with the linear size of the area [Keilis-Borok et al., 1989]. It is important to realize that the spatial distribution of epicenters is typically not a simple fractal but instead shows signs of multifractality [Davidsen & Goltz, 2004, Molchan & Kronrod, 2005]. It means there is uncertainty in choosing d f [Davidsen & Goltz, 2004], but different features of aftershocks which we will look at in this work are not sensitive to changing d f and this uncertainty in choosing d f does not increase the uncertainty of our results [Gu et al., 2013]. 16

25 In the GR relation (Eq. (2.1)), the number of earthquakes is an exponential function of the minimum magnitude of these events. We know the derivative of an exponential function is also an exponential function with the same exponent. Thus we can conclude that the magnitude distribution of earthquakes, which is the derivative of the number of the earthquakes with respect to the minimum magnitude, is also an exponential function of the magnitude of events. Mathematically speaking, f(m) = d dm N(m), then f(m) = d dm a 10 bm = a b log (10)10 bm. As we mentioned in Section 1.1, the magnitude m is a logarithmic measure of the energy of an earthquake [Turcotte, 1997]. Thus the energy of earthquakes follows a power law which is a sign of scale invariance [Gu et al., 2013]. 2.2 Space-time-magnitude distance If we assume there is no correlation between earthquakes and they do not trigger each other, we would expect earthquakes to occur randomly in space and time with a rate given by the GR relation. The main idea of the aftershock identification method, which we used here, is to find violations of this null hypothesis, indicating that the opposite is true and then identify likely triggering relations between earthquakes [Baiesi & Paczuski, 2004, Baiesi & Paczuski, 2005, Zaliapin et al., 2008, Zaliapin & Ben-Zion, 2013a, Zaliapin & Ben-Zion, 2013b]. This method consists of two steps. In the first step, for each earthquake j, which occurred in position r j, at time t j with a magnitude of m j, we identify its most likely trigger. For this purpose, we consider all earthquakes i, which occurred before the earthquake j, 17

26 and calculate the respective expected number of earthquakes n ij in the space-time window spanned by i and j with magnitude larger or equal to m i. Based on the argument we had about the prefactor a in the GR relation (Eq. (2.1)) in Section 2.1, if we define the linear extent of this spatial area as r ij = r j r i and the time interval as t ij = t j t i, the expected number is given by n ij = c t ij r ij d f 10 bm i. (2.2) Here, c is a constant for the specific region under consideration. Event i that minimizes the expected number in Eq. (2.2) for fixed j corresponds to the strongest observed violation of the null hypothesis of uncorrelated events. Because the number of earthquakes, which are expected to occur in the space-time window spanned by i and j, is the smallest one, i and j are the most unlikely pair of events to occur randomly. Thus the earthquake i is the most likely candidate of being the trigger of j. Event i is often called the nearest neighbour of j and we define n j n i j. If we connect all these nearest neighbours, we would have a time oriented tree for which the first event in the catalogue will be its root. In the second step of this aftershock identification method, we identify which of the observed values of n j are statistically significant. As first discussed in [Zaliapin et al., 2008], we identify a threshold value n such that it separates two statistically distinct populations of earthquakes. One of them corresponds to randomly occurring background events which are characterized by a Poisson process. These events can be inhomogeneously distributed in space. The other corresponds to triggered events which are much closer to each other in time and space than is expected for the previous group of events. Thus only those events j with n j n are considered being triggered or aftershocks. Note that based on this definition of aftershocks, the magnitude of aftershocks is not necessarily smaller than its trigger or main shock. An event also can be an aftershock and a main shock at the same time based on this definition. Events with n j n are considered background events, which are not 18

27 triggered by any other event in the catalogue at hand. We will discuss the second step of the methodology in more detail in Chapter 3 and illustrate it for a specific example. 19

28 Chapter 3 Data In this work we use a very recent relocated high-resolution Southern California (SC) catalogue. This catalogue contains earthquakes from 1981 to 2011 which occurred in the region extending from Baja California in the south to Coalinga and Owens Valley in the north [Hauksson et al., 2012]. It consists of earthquakes and three of them have magnitude bigger than 7.0: Hector Mine earthquake (m = 7.1), El Mayor Cucapah earthquake (m = 7.2) and Landers earthquake (m = 7.3). We assume this catalogue contains all of the events above a lower magnitude threshold m th = 2.5 [Schorlemmer & Woessner, 2008]. For finding this magnitude threshold, detection probability for each station, as a function of magnitude and hypocentral distance, has been determined and then the detection probabilities of different stations were combined to determine the probability that a hypothetical earthquake with a given size and location could escape the detection [Schorlemmer & Woessner, 2008]. The number of earthquakes in this catalogue, which their magnitude are above this lower magnitude threshold, is While this magnitude of completeness is generally a good estimate, the situation is typically very different directly after a big earthquake, because during that period of time the seismic noise is high and detectors often miss small earthquakes. Specifically, it has been observed that the magnitude of completeness temporarily increases after a big earthquake [Kagan, 2004, Helmstetter et al., 2006]. This will play an important role later in our study of surrogate catalogues. Now we need to estimate b as defined in Eq. (2.1) and d f as defined in Eq. (2.2) in order 20

29 to identify triggering relations between earthquakes and define aftershocks using Eq. (2.2). We used the maximum likelihood method discussed in [Naylor et al., 2009] to estimate the b value. We obtained b = 1.01 ± 0.01 for SC using this method. The estimation of d f depends on whether we consider epicenters or hypocenters. It has been found that the asymptotic correlation dimension of hypocenters for shallow-crust seismic activity in Southern California, which takes place at depths less than 30 km, is about 2.2 if we consider hypocenters [Kagan, 2007]. In the same study it was also found that strong finite size effects and depth dependence are present. As mentioned in Chapter 2, if we consider epicenters, the spatial distribution would be rather a multifractal such that there is no unique d f [Davidsen & Goltz, 2004, Molchan & Kronrod, 2005]. We will focus nevertheless on epicenters in the following. This is because one of the goals here is to directly compare the statistical properties of aftershocks in SC with those in surrogate catalogues generated by the ETAS model and the ETAS model has only been formulated for epicenters. Since the depth of earthquakes is not estimated as accurately as their epicenter [Hauksson et al., 2012], using epicenters and not hypocenters has the additional benefit of lower uncertainties in the relative locations. The estimation of d f also may slightly depend on seismic region. We chose the estimated box-counting dimension for SC d f = 1.6 throughout [Corral, 2003]. Yet, we tested d f = 1.2 and d f = 2.0 in addition to d f = 1.6 and found that all of our results are quite independent of the specific choice of d f, which indicates that the influence of multifractality is negligible for our analysis. A similar robustness has also been established for hypocenters [Gu et al., 2013]. The primary wave or P-wave is a type of elastic waves and, compared to the other waves, it has the highest velocity in propagating in the Earth s crust. Therefore if there is a causality relationship between two earthquakes, there should be enough time separation between them for P-wave to travel the distance between these two earthquakes. Because of that, to avoid nonphysical cases in our aftershock identification scheme, we also set a condition on 21

30 the accepted time difference between correlated events which is t ij > r ij /v. Here v is the propagation speed of P-waves in the Earth s crust, which is on average 6 km/s [Lin et al., 2007], and r is the shortest path between two events, which we decided to approximate by the epicenter distance. 3.1 Density plot of the nearest neighbour distances Now we are in a position to evaluate Eq. (2.2) and follow through with the first step of our method to identify aftershocks. After collecting the set of n j, we need to establish a threshold n in order to distinguish significant from insignificant values. As we mentioned in Chapter 2, these insignificant values correspond to background events which occurred randomly in space and time. Our aftershock detection method should detect deviations from randomness to distinguish aftershocks from these background events. For this purpose, it is necessary to find the distribution of random events to formulate our null hypothesis. We assume a spatiotemporal marked point field N with temporal component t R, spatial component x R 2 and scalar marks m that represent the magnitude of earthquakes. The spatial component of this marked point field is a homogeneous distribution and it is different from the background distribution of Southern California catalogue, but it does not have any effects on the result of our calculation [Zaliapin et al., 2008]. We also assume that the intensity of this Poisson marked point field is µ. Magnitude marks m i are independent of fields (t j, x j ). The magnitude of an event is also independent of the magnitude of other events and they follow the GR exponential distribution (2.1) with parameter b and lower magnitude threshold m 0. We define f = b b where b is the prior parameter of the GR law used for finding nearest neighbour distances of events based on the Eq. (2.2). In other word b corresponds to an estimate, but b is the true value. Under these assumptions the nearest 22

31 neighbour distances n j have the following distribution for large time interval τ 0 and large area with linear size r 0 [Zaliapin et al., 2008]: P (n j < x) = 1 exp [ µγψ Here γ is a constant and independent of x. If we redefine function Ψ(w) in this equation we would have: ( x τ 0 r d f 0 x τ 0 r d f 0 w d f < 2, f < 1, wlnw d f = 2, f < 1, )]. (3.1) as a new variable w, for the Ψ(w) w 2/d f d f > 2, d f > 2f, w 2/d f lnw df > 2, d f = 2f, w 1/f d f < 2f, f > 1, wlnw d f < 2, f = 1, w(lnw) 2 d f = 2, f = 1. (3.2) Then the distribution of nearest neighbours n j has a Weibull distribution for b b, d 2 and d f 2f. The distribution of n j is also independent of the magnitude threshold m 0. If we define the weighted time and the weighted spatial relative distances as: τ kj = t kj 10 bm k/2, (3.3a) l kj = r ij d f 10 bm k/2, (3.3b) the distribution of nearest neighbour pairs (τ, l) for our random point field N would be concentrated along the line log 10τ +log 10l = x m, where x m is the mode (the value that appears most often) of the distribution in Eq. (3.1) [Zaliapin et al., 2008], which is a function of the 23

32 scalar marks m. Note that n kj τ kj l kj. Fig. 3.1 shows the density plot of the set {n j} in the space of the weighted time (τ) and the weighted spatial relative distance (l) for the SC catalogue. We can see two statistically distinct populations of earthquakes in this density plot and for one of them the n j s are significantly smaller than for the other. The population of events with larger n j extends around the line log τ + log l = 9.0. Thus the distribution of nearest neighbour pairs (τ, l) for these events is the same as the distribution for stationary Poisson seismicity. The line log l + log τ = 7.0 (straight solid line in Fig. 3.1) separates two populations and then earthquakes that meet the condition log n j < log n are predominantly triggered ones and all others are predominantly non-triggered ones or background events without a trigger in the catalogue [Zaliapin et al., 2008]. Note that this allows us to set c in Eq. (2.2) to one without loss of generality. There is a small overlap between the two populations and a perfect separation is not possible, but it does not have a significant effect on those statistical properties of events which we will look at in this work [Gu et al., 2013]. 24

33 Figure 3.1: Density plot of the set {n j} represented in log τ log l space for SC (time is measured in seconds and distances are measured in meters). The black line (log n = 7.0) separates two populations corresponding to triggered events (below) and background events (above). 25

34 3.2 Definition of aftershocks and main shocks The identification of triggering relationships is mathematically identical to mapping seismicity to a graph. As we mentioned before, if we connect all the events to their nearest neighbours based on Eq. (2.2), we would have a time oriented tree T. The root of this tree is the first event in the catalogue. After setting the threshold n, we remove all the insignificant links. As a result we will have a forest (set of trees) F = {T i }. Each tree T i in this forest corresponds to an earthquake cluster. By definition, n j > n for roots of all trees, making them background events, while all other events are triggered by one of the preceeding earthquakes [Gu et al., 2013]. This process is illustrated in Fig. 3.2 for a toy catalogue. The largest earthquake in each tree T i can be defined as the main shock (see see, for example [Zaliapin et al., 2008] and references therein). Then all the events in the tree T i, which occurred after the main shock, are considered as aftershocks; while all other earthquakes in that tree are foreshocks. Here, however, we denote all earthquakes that trigger other events as main shocks and the earthquakes that have been triggered as aftershocks. Thus, an event can be an aftershock of one of the preceding events and at the same time the main shock of following events. Therefore the definition of aftershocks and main shocks, which we use in this work, is independent of the magnitude of events and we do not use the notion of foreshocks in our definition. In the following, we refer to aftershocks of a given main shock as those that are directly triggered by the main shock. This makes the comparison with the ETAS model straightforward. Yet, as in [Gu et al., 2013] our main conclusions are independent of whether we include second and higher generation aftershocks. 26

35 (a) Tree (b) Forest Figure 3.2: Panel (a) shows a time oriented tree which has been created by connecting all the events of a toy catalogue with 22 earthquakes to their nearest neighbours based on the Eq. (2.2). The root of this tree is the first event in the catalogue. Panel (b) shows a forest which has been created by removing all the insignificant links of the tree. Each tree in this forest corresponds to an earthquake cluster. Roots of these trees are background events, but all other events are triggered by other earthquakes. 27

36 Chapter 4 Spatial distribution of aftershocks in Southern California By using the aftershock identification method, which is discussed in former chapters, we can investigate different statistical features of triggering cascades of aftershocks. One of these features, which is the focus of this work, is the spatial distribution of aftershocks. This distribution is a signature of the triggering process and can help us to understand the physics of triggering [Felzer & Brodsky, 2006, Richards-Dinger et al., 2010]. But there is some uncertainty about the functional form of this distribution, especially there is a debate about how this function decays for large distances [Gu et al., 2013]. Typically, a simple power law is proposed for the decay of the spatial distribution of aftershocks with distance from the associated main shock, but even the exponent of this power law is debatable. As we mentioned in Chapter 1, an earthquake can trigger other earthquakes by dynamic or static stress and based on these different power law exponents, either of them have been suggested for triggering [Felzer & Brodsky, 2006, Richards-Dinger et al., 2010, Marsan & Lengliné, 2010, Lippiello et al., 2009]. A very recent study suggested that the decay of the spatial distribution of aftershocks can not be characterized by a single power law and there is a transition in the decay of this distribution at about 10km [Gu et al., 2013]. Hypocenter distances have been used in that work. Here, we provide convincing evidence that the decay of the spatial distribution of aftershocks with distance indeed does not obey a simple power law and two different power laws with a transition point at about 10km would be the best option for characterizing the decay of this distribution if we use distance between epicenters. In particular, we pro- 28

37 vide evidence that this is independent of whether one considers distance between epicenters, epicenter-to-source or epicenter-to-fault distances. 4.1 Rupture length An important characteristic length scale in the spatial distribution of aftershocks is set by the rupture area of the main shock [Kagan, 2002, Davidsen et al., 2008, Wu et al., 2013, Gu et al., 2013]. Rupture area is the area over which an earthquake occurs and here, we simply define the characteristic rupture length scale as the square root of this area, while sometimes the width and the length of the rupture area are considered separately (see, for example, [Leonard, 2010, Wu et al., 2013]). As mentioned in [Wells & Coppersmith, 1994, Kagan, 2002, Leonard, 2010, Gu et al., 2013], this rupture length (L R ) scales with the magnitude of the earthquake (m) as: L R = l 0 10 σm, (4.1) where l 0 and σ are constants with typical values of 10 < l 0 < 20 meter and 0.4 σ 0.5. As we will show in the following paragraph, this scaling relation of the rupture length is directly observable in the spatial distribution of aftershocks of main shocks with different magnitudes. We denote this spatial distribution by P m (r) which the distance r is measured between epicenters without considering directionality. As we can see in Fig. 4.1, the maximum of the spatial distribution of aftershocks moves to larger distances by increasing the main shock magnitude. If we use Eq. (4.1) to rescale all the distances with the main shock magnitude, different distributions for different magnitude ranges would collapse around the maxima and then we can estimate l 0 and σ [Gu et al., 2013]. Indeed, we should choose these parameters such that the best collapse happens by this rescaling. Fig. 4.2 shows the rescaled spatial distribution of aftershocks of different main shock magnitude ranges for SC. Based on this figure, we estimated l 0 = 10 ± 5m and 29

38 σ = 0.44 ± 0.02 for rupture length in SC. These estimated values are within the typical range. For this estimation, we have assumed that the maximum corresponds to L R /2 as in [Gu et al., 2013]. Our estimation of parameters of rupture length is compatible with the results in [Davidsen et al., 2008]. l 0 = 12m and σ = 0.45 have been estimated in that work. It is also relatively close to the estimated parameters for rupture length in [Wells & Coppersmith, 1994] who obtained l 0 = 18m and σ = As evident from Fig. 4.2, after using these parameters for rescaling all the distances, the spatial distribution of aftershocks of SC for different main shock magnitude ranges collapse over a wide range of scales only excluding the tail of the distribution < m < < m < < m < < m < < m < < m P(r) r (m) Figure 4.1: Spatial distribution of aftershocks of SC for different ranges of the main shock magnitude, using epicenter distances. 30

39 <m>=2.73 <m>=3.44 <m>=4.50 <m>=5.47 <m>=6.39 <m>=7.22 P(λ) λ Figure 4.2: Rescaled spatial distribution of aftershocks using epicenter distances, for different ranges of the main shock magnitude for SC. All the distances are rescaled by a factor m such that λ = r m, where m is the average magnitude for the considered magnitude range. 31

40 4.2 Functional form of the spatial distribution of aftershocks in SC As we mentioned in Section 4.1, after rescaling all distances with the main shock magnitude, spatial distributions of aftershocks for different main shock magnitude ranges collapse over a wide range of scales, but this collapse is absent in the tail of the distribution (Fig. 4.2). This behaviour is incompatible with the functional form of the spatial distribution of aftershocks which is reported in some studies [Marsan & Lengliné, 2008, Lippiello et al., 2009]: P m (r) = qr. (4.2) L 2 m( r2 + 1) L 1+q/2 2 m This functional form is frequently used in statistical models of seismicity such as the ETAS model [Helmstetter et al., 2003, Peixoto et al., 2010, Gu et al., 2013, Console et al., 2003, Zhuang et al., 2004]. Note that often the corresponding areal density instead of the linear density is given. Here, q is a constant and describes how fast the function decays for distances larger than L m. We mentioned in Section 4.1 that we assume L m is half of the rupture length of the main shock and it scales with the main shock magnitude in the same way as the rupture length. Note that P m (r) does not depend on the direction and it only depends on the epicenter distances of aftershocks from their main shocks [Felzer & Brodsky, 2006]. Another deviation from the Eq. (4.2) which we can see in Fig. 4.2 is that P m (r) increases for small r instead of being constant. This behaviour can be captured by a more general form of Eq. (4.2) given by: P m (r) = L γ+1 m qr γ ( rγ+1 L γ+1 + 1) 1+ q γ+1 m, (4.3) where γ is a constant and determines how fast this function increases at small distances. As we will show later, this deviation from Eq. (4.2) at small scales can be understood as a consequence of considering distances between epicenters instead of epicenter distances to source 32

41 or fault. The same interpretation about initial increase of spatial distribution of aftershocks was mentioned in [Felzer & Brodsky, 2006, Marsan & Lengliné, 2010]. If we look at Fig. 4.1 more carefully, we can see two regimes in the decay of the spatial distribution of aftershocks with distance. The first regime extends from half of the rupture length to about 10km, while the second regime starts almost from 10km. The second regime decays with distance faster than the first one. This transition in the decay exponent of the spatial distribution of aftershocks can explain why the rescaled version of this distribution for different main shock magnitude ranges do not collapse in the tail of the distribution. Note that 10km is comparable to the width of the Earth s crust in the region and a possible explanation for the existence of these two regimes can be that the width of the Earth s crust starts playing an important role in the spatial distribution of aftershocks for distances larger than 10km. If we want the functional form of the spatial distribution of aftershocks to also capture this behaviour, we need to use a more complicated form which can describe this transition in the decay exponent: P m (r) = α β L γ+1 m L γ+1 m qr γ ( rγ+1 L γ+1 + 1) 1+ q γ+1 m dr γ ( rγ+1 L γ+1 + 1) 1+ d γ+1 m if r < 10km, if r > 10km. (4.4) In this equation α and β are normalization factors and d is a constant which describes how fast the function decays for distances larger than 10km. We can see the spatial distribution of aftershocks for the main shock magnitude range 3.0 < m < 4.0 and its associated functional form in Fig As we can see in this figure, the functional form given by Eq. (4.4) with L m = L R /2 is indeed a good description of the data from Southern California. The match would improve if we take potential biases into 33

42 account. Specifically, we know aftershock detection methods will generally be more effective for shorter times, since interference effects of background earthquakes or events which are not in the considered earthquake cluster increase with time. For a long time interval, this causes a bias in the estimation of the spatial distribution of aftershocks [Felzer & Brodsky, 2006, Gu et al., 2013]. For example, by using the ML method [Marsan & Lengliné, 2008] to identify aftershocks for the time interval [0, 15 min], an exponent of 1.76±0.35 was estimated for the main shocks in the magnitude range [3, 4] while for the time interval [12 h, 24 h] an exponent of 1.97 ± 1.11 was obtained [Marsan & Lengliné, 2010], or in [Felzer & Brodsky, 2006] for 30 minutes of post main shock data, an inverse power law with an exponent of 1.36 ± 0.07 fits the 3.0 < m < 4.0 main shocks, but for 30 minutes to 25 days of post main shock earthquakes, a combined power law/background function was required to fit the data. For the aftershock detection method which we used in this work, it has been shown that the bias is negligible if we consider only those aftershocks that occur within the first hour of their respective main shock [Gu et al., 2013]. Because of that, in Fig. 4.3, we also draw spatial distribution of aftershocks for magnitude range 3.0 < m < 4.0 with this time restriction. The changes in P m (r) are small and the spatial distribution of aftershocks can be still well described by the functional form in Eq. (4.4) with a suitable choice of γ, q and d. As it has been shown in Fig. 4.3, time restricted spatial distribution of aftershocks initially increases with an exponent close to 0.6 and after the maximum of the function, it decays with an exponent close to 1.35 for less than 10km and an exponent 2.2 for larger distances. It means we should set γ = 0.6, q = 0.35 and d = 1.2 for our functional form. As Fig. 4.3 indicates, Eq. (4.4) with these parameters matches very well with the time-restricted spatial distribution of aftershocks in the SC catalogue. Fig. 4.4 compares the time restricted spatial distribution of aftershocks for magnitude 34

43 ranges 2.5 < m < 3.0 and 4.0 < m < 5.0 to their associated functional form. As evident from this figure, functional form in the Eq. (4.4) with parameters γ = 0.6, q = 0.35 and d = 1.2 still matches fairly well with the time-restricted density function of aftershocks in these main shock magnitude ranges for the SC catalogue. It indicates that this functional form works for different main shock magnitude ranges and not only for one specific magnitude range. As we can see in Fig. 4.3, if we combine the functional form given in Eq. (4.4) with the fact that the epicenter-to-source distance, r, is more relevant for triggering than epicenter distances r [Hainzl et al., 2008], this functional form will be still a good match for the spatial distribution of aftershocks in SC. To be more clear, we assume that any point along the main shock rupture, which is characterized by the rupture length given in Eq. (4.1), can trigger other earthquakes with the isotropic spatial distribution such that P m (r ) has the same functional form as Eq. (4.4). Thus, the distance vectors between epicenters, r, will no longer be isotropically distributed and we refer to this case as the anisotropic case in the following. Fig. 4.5 illustrates how the relevant distances are defined. Note that, as we can see in Fig. 4.6, P m (r ) is somewhat similar to the distribution of epicenter-to-fault distances. If we measure the distance between the epicenters to obtain P m (r) as we did for SC, for short distances, this distribution will be significantly different from isotropic functional form, P m (r ) which we used for generating the data. Fig. 4.3 shows if we use Eq. (4.4) with γ = 0 for generating epicenter-to-source distances, but measure the distance between the epicenters to obtain the spatial distribution, we would recover γ 0.6 for this distribution which has the same value as the exponent of the initial increase for the spatial distribution of aftershocks in SC. Therefore, similar to observations in [Felzer & Brodsky, 2006, Marsan & Lengliné, 2010], for this case still the functional form remains the same but the exponent characterizing the behaviour for short distances varies. 35

44 Based on the argument which we had in the previous paragraph, we can conclude that the initial increase for the spatial distribution of aftershocks for epicenter distances is a consequence of earthquakes being spatially extended events. Since γ is related to the fractal dimension of the epicenters as d f = γ + 1 [Gu et al., 2013], this interpretation about the initial increase of the spatial distribution of aftershocks can also help us to get new intuition about the (multi-)fractal distribution of epicenters. We estimated γ 0.6 which is consistent with the directly estimated box-counting dimension of 1.6 for SC [Corral, 2003, Davidsen & Goltz, 2004]. It means the (multi-)fractal distribution of epicenters might not be truly indicative of fractal behaviour and it might be just a consequence of assuming an earthquake as a single point and not a spatially extended event, in distances which are comparable to the rupture length of these events. Our finding in this chapter about the spatial distribution of aftershocks in SC shows that this distribution, as a function of distance between epicenters, can be well described by Eq. (4.4). This functional form for the spatial distribution of aftershocks has three different power-law regimes and it can explain why the spatial distribution of aftershocks for different magnitude ranges would collapse partially after rescaling all the distances with main shock magnitude. The initial increase of this function over the first power law regime can be interpreted as a consequence of earthquakes being spatially extended events and not point events and the transition between the second and third power-law regimes can be understood as a consequence of the finite thickness of the Earth s crust. 36

45 P(r) ~r 0.6 ~r Restricted Southern California Unrestricted Southern California Isotropic Functional Form Anisotropic Functional Form ~r r(m) Figure 4.3: Spatial distribution of aftershocks using epicenter distances for main shock magnitudes in the range 3.0 < m < 4.0 for SC considering all aftershocks (squares) and considering only aftershocks that occurred within one hour of their main shocks (circles). Three separate power-law regimes can be identified in both cases. For comparison, the spatial distribution given by Eq. (4.4) with exponents γ = 0.6, q = 0.35 and d = 1.2 (dashed line) and the spatial distribution obtained by simulating the anisotropic case (see text for details) using Eq. (4.4) with exponents γ = 0.0, q = 0.35 and d = 1.2 for the epicenter-to-source distance (dotted line) are shown as well. Both agree very well with the time-restricted data. 37

46 < m < < m < P(r) r(meter) Figure 4.4: Spatial distribution of aftershocks using epicenter distances for main shock magnitudes in the range 2.5 < m < 3.0 (circles) and the range 4.0 < m < 5.0 (squares) for SC considering only aftershocks that occurred within one hour of their main shocks. For comparison, the associated spatial distributions given by Eq. (4.4) with exponents γ = 0.6, q = 0.35 and d = 1.2 (dashed lines) are shown as well. As this figure shows the functional forms agree well with the time restricted spatial distribution in these magnitude ranges. 38

47 Figure 4.5: Definition of spatial distances in the anisotropic case. H 1 and H 2 are the epicenters of the main shock and aftershock, respectively. L R is the rupture length of the main shock and S is the source point. r is the epicenter-to-source distance drawn from the given P m (r ) and r is the epicenter distance used to estimate P m (r). 39

48 10 0 probability density Epicenter to fault Epicenter to source Epicenter to epicenter distance(meter) Figure 4.6: Density function of epicenter-to-fault distances, epicenter-to-source distances and epicenter-to-epicenter distances for numerically simulated and anisotropically distributed aftershocks of a main shock with a rupture length of 350m (see main text for details). For the simulation, Eq. (4.4) from the main text with γ = 0, q = 0.35 and d = 1.2 was used as the functional form for the distribution of epicenter-to-source distances. 40

49 Chapter 5 The ETAS model Now we want to analyze surrogate catalogues generated by the Epidemic-Type Aftershock Sequence (ETAS) model [Kagan & Knopoff, 1987, Ogata, 1988, Helmstetter & Sornette, 2002] to support our claims about the functional form of the spatial distribution of aftershocks in SC which is given by Eq. (4.4). The ETAS model is a stochastic point-process based on empirical observations of seismic activity, which is used for modelling of this activity in the area under consideration. In this chapter first we define the ETAS model in Section and then introduce different versions of this model with distinct spatial distributions of aftershocks in Section 5.4. Investigating these different versions of ETAS model will allow us to identify which properties of the spatial distribution are essential for recovering the main statistical features of aftershocks, observed in SC by using the methodology described in Chapter 2. We also introduce some other versions of the ETAS model to test other features of the this model which may have effects on our results. 5.1 Definition of the ETAS model For the ETAS model we have a rate of background events which is not dependent on any earthquake in the catalogue. In this model each event can also trigger some earthquakes and we have a rate of events triggered by other earthquakes. Thus, the total rate of earthquakes is sum of these two terms. Mathematically speaking, the rate of the earthquake occurrence with magnitude m, at time t and in position r is defined as [Kagan & Knopoff, 1987, Ogata, 1988, Helmstetter & Sornette, 2002]: λ(t, r, m) = µ b ( r) + t i <t φ mi ( r r i, t t i ) (5.1) 41

50 First term of this equation corresponds to the background rate of events. µ b ( r) is assumed to follow a spatially non-homogeneous marked Poisson process with an average rate of µ b, which is equal to the background rate of earthquakes in the consider overall region. We want to mimic the seismic activity in SC and because of that here we use the same spatial distribution for µ b ( r) as in [Gu et al., 2013], which was estimated using the methodology described in Chapter 2. These background events can be main shocks and each of them can trigger some aftershocks with a rate given by φ mj ( r r j, t t j ), corresponding to a spatially non-homogeneous and time-varying marked Poisson process. These aftershocks also can be main shocks of other aftershocks and we can have a cascade of events. The Poisson process, which is described in the former paragraph, set the time and position of each of the background events, but we should also identify their magnitudes. We assume the magnitude distribution of these background events and their aftershocks are the same for the ETAS model and this distribution is independent of the past seismic activity. The ETAS model identifies the magnitude of each of earthquakes based on the normalized GR probability distribution: P m (m) = b ln (10) 10 b(m mth). (5.2) Here m th is the lower magnitude threshold. All the events in the surrogate catalogues have magnitude equal or larger than this magnitude threshold. The second term of the Eq. (5.1) identifies triggering processes and it corresponds to the contribution of all the events before time t to the total rate. As we can see in this term, the ETAS model assumes that triggering processes lead to a simple linear superposition in terms of the rates and the function φ mi ( r, t) sets how the individual event i triggers its aftershocks. This function quantifies the spatiotemporal distribution of aftershocks at the spatial distance r and temporal distance t t j to the main shock with magnitude m i. Typically, the functional 42

51 form is assumed to factorize into three terms: φ mi ( r, t) = ρ(m i ) ψ(t) ζ mi ( r). (5.3) In this equation ρ(m i ) is the number of aftershocks triggered by the event i. It is assumed to be determined by the Poissonian character of the process which has an average rate of ρ(m i ). This average rate is a function of the main shock magnitude m i and it is usually chosen in a way that an earthquake with a larger magnitude trigger more events than those with smaller magnitude. The empirically observed productivity law, which has this property, is typically used for this purpose [Helmstetter & Sornette, 2002, Gu et al., 2013]: ρ(m i ) = K 10 α(m i m th ). (5.4) Here K and α are constants and m th is again the lower magnitude threshold. The constant α quantifies how fast the number of aftershocks increases with the magnitude of the main shock. The productivity law is zero for earthquakes with magnitude below the magnitude threshold m th. It means events with magnitude smaller than this magnitude threshold can not trigger other earthquakes. 5.2 Temporal distribution of aftershocks for the ETAS model The term ψ(t) of the Eq. (5.3) is the temporal distribution of aftershocks at time t after the main shock. The temporal distribution of aftershocks is the normal form of the rate of aftershocks which, as we mentioned is Section 1.5, is set by the Omori Utsu law [Utsu et al., 1995]: r(t) = χ (t + C) p, see Eq. (1.7). Here χ is a constant, only sets the intensity of the rate and it is related to the productivity law. The most recent empirical evidence suggests that p is a constant with a typical value of about 1 [Gu et al., 2013], though variations in p with main shock magnitude 43

52 cannot be fully ruled out [Hainzl & Marsan, 2008]. The situation is more complicated for the parameter C. Recent results support a physical origin of C more than an instrumental origin [Peng & Zhao, 2009, Lengliné et al., 2012, Holschneider et al., 2012], though it remains unclear whether it is indeed a constant or if it varies with the main shock magnitude. The normalized form of the Omori-Utsu law is only a function of parameters C and p > 1: ψ(t) = (p 1) Cp 1 (t + C) p. (5.5) This function has two regimes: It is constant for short times less than C and it decays as a power law with the exponent p for longer times. It has been observed that the transition time when the temporal distribution of aftershocks goes from the constant regime to the power law regime changes with the main shock magnitude for earthquakes in SC. Since the parameter C in Eq. (1.7) determines this transition time, it should also vary with the main shock magnitude. In [Kagan, 2004, Lippiello et al., 2012], for SC catalogues with magnitude threshold m th = 2.5, the following behaviour has been proposed for the parameter C in the Omori Utsu law: C(m) 10 m 6.5 days. (5.6) Dependency of the transition time on the main shock magnitude might simply be a consequence of Short-Term Aftershock Incompleteness (STAI) as argued in [Kagan, 2004]. As we mentioned in Chapter 3, the seismic noise increases after a big earthquake. During this time the detectors are not able to detect all the small earthquakes and then the completeness magnitude significantly increases [Kagan, 2004]. One way to mpdel this is to use a completeness magnitude m c which is a function of main shocks magnitude m and time distance t to the main shock [Helmstetter et al., 2006, Lennartz et al., 2008]: m c (t, m) = m log t. (5.7) If we use this completeness magnitude and remove events below m c from the catalogue, the number of aftershocks which have been removed after a main shock with a bigger magnitude 44

53 will be larger than the number of removed aftershocks after a main shock with a smaller magnitude and consequently the number of removed aftershocks increases by the magnitude of the main shock. This can change the rate of aftershocks in early time after the main shock. We will see later this can explain the scaling behaviour described by Eq. (5.6), while the true parameter C is a constant. While Eq. (5.7) was originally proposed for main shocks with magnitude bigger than 6.0, it has been argued in [Hainzl, 2013] that it also can be used for main shocks with smaller magnitudes. We will show in Chapter 6 that if we extend Eq. (5.7) to all earthquakes in a given catalogue the agreement in the statistical properties of aftershocks between ETAS and SC improves. 5.3 Spatial distribution of aftershocks for the ETAS model In Eq. (5.3), the term ζ mi ( r) is the normalized spatial distribution of aftershocks with distance r from the main shock. The direction of this distance vector is typically chosen at random for the procedure we follow here [Gu et al., 2013, Peixoto et al., 2010, Helmstetter & Sornette, 2002]. The length of this distance is chosen according to the functional form of the spatial distribution of aftershocks as captured by the Eq. (4.4) with L m = L R /2. It means we have the flowing relationship between the function P m (r) and the function ζ mi ( r): P m (r) = 2π 0 ζ mi ( r)dφ. This framework allows us to consider isotropic and anisotropic spatial distributions in the epicenter distances. In the former case, the distance vector r for triggering an aftershock originates at the epicenter of the main shock, while in the anisotropic case it originates at a random point along the rupture of the main shock (the source) such that P m (r ) now determines the epicenter-to-source distance r instead [Hainzl et al., 2008]. To be more precise, to each main shock we assign a line with a length equivalent to the rupture length, centered at its epicenter, and a random orientation in space. Each point along this line is then equally likely to trigger aftershocks according to P m (r ). This gives rise to an anisotropic 45

54 distribution of aftershocks in space. Fig. 4.5 illustrates how the relevant distances are defined in the anisotropic ETAS model. Note that in our model the epicenter-to-source distance is statistically similar but not identical to the epicenter-to-fault distance (see Fig. 4.6). 5.4 Different version of ETAS model In Chapter 6 we will analyze surrogate catalogues generated by different versions of the ETAS model to test our claims about the spatial distribution of aftershocks. We list the common parameters between these models in Table 5.1 and the distinct parameters and other model distinctions in Table 5.2. We will analyze catalogues generated by model I and model II to test the effects of using isotropic and anisotropic distributions for ζ mi ( r) and investigate whether the initial increase in the spatial distribution of aftershocks for epicenter distances can be explained by an anisotropic distribution or not. Investigating model III and model IV will help us to see whether it is crucial to have two exponents for the decay in the spatial distribution of aftershocks with a break in scaling at about 10km. Finally investigating catalogues generated by model V, model V I and model V II will help us to see the effects of changing the parameter C of the Omori Utsu law with the main shock magnitude and how STAI can explain the dependency of this parameter on the main shock magnitude. We selected surrogate catalogues for different versions of the ETAS model such that they closely resemble the observed activity in SC, including the number of earthquakes and the number of events with magnitude 6 and higher. We also chose the b value in Eq. (5.2) for the different versions of the ETAS model in a way that they will be the same as the observed values for SC only after taking STAI into account. The catalogues for models I, II, III and IV are identical and only spatial aspects of them are different. For model V I we use the same catalogue as model I, but STAI was considered only for main shocks with magnitude 46

55 µ b (1/day) m th p α l 0 (m) σ T (day) t 0 (day) L (km) Table 5.1: List of the parameters which are common in all different versions of the ETAS model. We chose these parameters such that the generated catalogs closely resemble the observed activity in SC. model STAI A C (day) γ q d b K N I Yes Yes II Yes No III Yes Yes IV Yes Yes V No Yes C(m) = 10 m V I partially Yes V II No Yes Table 5.2: List of parameters that highlight differences between different versions of the ETAS model. N is the number of events in each catalog and column A indicates whether the spatial distribution of aftershocks is anisotropic. For model V I, we consider STAI only for main shocks with magnitude bigger than 6.0. bigger than 6.0 and for model V II, STAI was not considered for any of the main shocks. Finally the catalogue we chose for model V is completely different from the others, because the b value we used for generating this catalogue is different from others. We consider a time interval [0, T ] and a square spatial area of size L L with periodic boundary conditions for generating catalogues of the different versions of the model. To minimize any bias due to events which occurred before the simulation period, we remove all events in the time interval [0, t 0 ]. The values of these (common) parameters are given in Table

56 Chapter 6 Comparing different versions of the ETAS model to the SC After introducing different versions of the ETAS model in Chapter 5, in this chapter we want to compare some of the most important features of aftershocks in the SC catalogue to the same features for surrogate catalogues generated by these different versions of the ETAS model. In each section we compare one of these features for SC to the same feature for all different versions of the ETAS model and then if we see a significant difference for one of these versions, we reject that version and eliminate it from consideration. Through this process we find the most reasonable assumptions about the spatial and temporal distribution of aftershocks for the ETAS model which allow us to recover the main features of the SC catalogue. At the end of this chapter we also test the robustness of our results by changing the magnitude threshold of completeness. 6.1 Inference of triggered events At the beginning of our investigation of the different versions of the ETAS model, we want to know to which extent we can distinguish between triggered and background events, using the methodology described in Chapter 2. For using this methodology first we should see a bimodality in the density plot of the catalogue and if we can not see this bimodality, we would not be able to use this methodology for aftershock identification. As we can see in Fig. 6.1, the density plot for model I is very similar to the density plot for SC in Fig We can see two separated populations of events in the density plot for both of them. One of the populations extended around the line log l = 3 from log τ = 0 to almost log τ = 4, while 48

57 the other one extended around the line log l + log τ = 9. For both model I and SC the line log l + log τ = 7.0 can separate these two populations of events. As discussed in Chapters 2 and 3, we consider all events below this line as triggered events or aftershocks and all other events as background events. For a surrogate catalogue, which has been generated by the ETAS model. Figure 6.1: Density plot of the set {n j}, same as Fig. 3.1, but for model I. There is a striking resemblance between the two, both indicating a clear separation between triggered events and background events. The density plot of model II in Fig. 6.2 is almost identical to the density plot of model I. It indicates that choosing isotropic or anisotropic spatial distribution does not have any effect on the density plot of evens of surrogate catalogues. Triggered population for model IV in Fig. 6.3 is extended around the line log l = 2 and it is less spread out in the l direction compared to the same population for model I. Spatial distribution of aftershocks for this model decays with one steep exponent and then aftershocks are closer to their main shocks in space which can explain why the triggered population is less spread out in the l direction. 49

58 We used the magnitude dependent C in the Omori Utsu law (Eq. (5.5)) for model V instead of using STAI and for model V I we only considered STAI for main shocks with magnitude bigger than 6.0. As we can see in Fig. 6.4 and Fig. 6.5 the density plots for these cases are still the same as model I. But the density plot of model V II in Fig. 6.6 is a little different from model I or SC. For this case we used a constant C in the Omori Utsu law and did not use STAI at all. The density plot of this model is extended to small τ which is reasonable, because we did not remove aftershocks, which were close in time to their main shock, based on STAI. Even the line which separates the background population from the triggered population, changes and the line log l + log τ = 6.0 is a better separation line in this case. In all the cases listed above, triggered and background populations are still separated. Figure 6.2: Density plot of the set {n j} represented in log τ log l space for model II. The black line is the threshold that we chose and it represents log n = 7.0. We consider earthquakes, which are below this line, as triggered earthquakes. Time is measured in seconds and distances are measured in meters. As evident from Fig. 6.7, a clear separation between triggered and background populations is absent in the density plot of model III. In this density plot we can not see the 50

59 dense population around the line log l = 3, which corresponds to triggered events. We only can see a dense population of events around the line log l + log τ = 9 and some scattered events in the left side of this population. If we do not have two separated populations in the density plot, it means we are not able to set a threshold for distinguishing triggered events from background activities. We have observed the same features in all different surrogate catalogues generated by model III which we tested. So we are not able to separate aftershocks from background events in the surrogate catalogues generated by model III and then it is impossible to investigate statistical properties of aftershocks for this case. Therefore by observing the density plot of model III we can conclude that this model is not a suitable model for SC and a functional form which decays with one exponent 1.35 can not characterize important features of the spatial distribution of aftershocks in SC. The observed behaviour of model III in Fig. 6.7 can actually be understood based on the slow decay of the spatial distribution of aftershocks. Because of the slow decay, triggered events are more likely to occur much further away from the main shock. Larger distances lead to higher values in Eq. (2.2) directly explaining the absence of clear bimodality in the density plot. Since the spatial distribution of aftershocks in this case decays with exponent 1.35 which is smaller than 2.0, the spatial distance between aftershocks and their main shocks does not have a finite mean for model III. 51

60 Figure 6.3: Density plots of the set {n j} represented in log τ log l space for model IV. The black line is the threshold that we chose and it represents log n = 7.0. We consider earthquakes, which are below this line, as triggered earthquakes. Time is measured in seconds and distances are measured in meters. Figure 6.4: Density plots of the set {n j} represented in log τ log l space for model V. The black line is the threshold that we chose and it represents log n = 7.0. We consider earthquakes, which are below this line, as triggered earthquakes. Time is measured in seconds and distances are measured in meters. 52

61 Figure 6.5: Density plots of the set {n j} represented in log τ log l space for model V I. The black line is the threshold that we chose and it represents log n = 7.0. We consider earthquakes, which are below this line, as triggered earthquakes. Time is measured in seconds and distances are measured in meters. Figure 6.6: Density plots of the set {n j} represented in log τ log l space for model V II. The black line is the threshold that we chose and it represents log n = 6.0. We consider earthquakes, which are below this line, as triggered earthquakes. Time is measured in seconds and distances are measured in meters. 53