Physics of Aftershocks in the South Iceland Seismic Zone—Insights Into the Earthquake Process from Statistics and Numerical Mo

Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology 592

Physics of Aftershocks in the South Iceland Seismic Zone

Insights into the earthquake process from statistics and numerical modelling of aftershock sequences

MATTIAS LINDMAN

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List of Papers

This thesis is based on the following papers, which are referred to in the text by their Roman numerals.

I Jónsdottir, K., Lindman, M., Roberts, R., Lund, B., Bödvarsson, R. (2006). Modelling fundamental waiting time distributions for earthquake sequences. Tectonophysics, 424(3-4): 195-208. doi: 10.1016/j.tecto.2006.03.036. Copyright (2006) Elsevier B.V. II Lindman, M., Jónsdottir, K., Roberts, R., Lund, B., Bödvarsson, R. (2005). Earthquakes descaled: on waiting time distributions and scaling laws. Phys. Rev. Lett., 94(10): 108501. doi: 10.1103/Phys- RevLett.94.108501. Copyright (2005) by the American Physical Society. III Lindman, M., Jónsdottir, K., Roberts, R., Lund, B., Bödvarsson, R. (2006). Comment on Earthquakes descaled: On waiting time distributions and scaling laws - Lindman et al. reply. Phys. Rev. Lett., 96(10): 109802. doi: 10.1103/PhysRevLett.96.109802. Copyright (2006) by the American Physical Society. IV Lindman, M., Lund, B., Roberts, R., Jónsdottir, K. (2006). Physics of the Omori law: Inferences from interevent time distributions and pore pressure diffusion modeling. Tectonophysics, 424(3-4): 209-222. doi: 10.1016/j.tecto.2006.03.045. Copyright (2006) Elsevier B.V. V Lindman, M., Lund, B., Roberts, R. (2008). Spatiotemporal characteristics of aftershock sequences in the south Iceland seismic zone: interpretation in terms of pore pressure diffusion and poroelasticity. Manuscript.

Reprints were made with permission from Elsevier B.V. (paper I and IV) and the American Physical Society (paper II and III). The Scriptures on the dedication page are taken from the Good News Bible published by The Bible Societies/Collins, copyright American Bible Societies.

Contents

1 Introduction ...... 9 2 Statistical seismology ...... 13 2.1 Typical earthquake distributions ...... 13 2.1.1 Magnitude distribution of earthquakes ...... 13 2.1.2 Temporal distribution of aftershocks ...... 16 2.1.3 Spatial distribution of faults and earthquakes ...... 20 2.2 Earthquake interevent time distributions ...... 21 2.2.1 Mainshocks - the Poisson interevent time distribution . . . . 21 2.2.2 Aftershocks - the Omori law interevent time distribution . 22 2.3 Earthquakes and self-organised criticality ...... 38 2.3.1 Self-organised criticality, the Gutenberg-Richter law and earthquake models ...... 39 2.3.2 Self-organised criticality and the Omori law ...... 40 2.3.3 Self-organised criticality and aftershock interevent time distributions ...... 43 2.3.4 Summarising comment ...... 46 3 Statistical methods ...... 49 3.1 Simulation of aftershock sequences obeying the Omori law . . . 49 3.2 Maximum likelihood estimation of Omori law parameters . . . . 53 3.3 Comparison of two datasets ...... 56 4 The physics of aftershocks ...... 59 4.1 Physical models for the occurrence of aftershocks ...... 59 4.1.1 Rate and state friction ...... 59 4.1.2 Subcritical crack growth ...... 60 4.1.3 Transient deformation processes ...... 61 4.2 Summarising comment ...... 64 5 Pore pressure diffusion ...... 65 5.1 Constitutive poroelastic equations ...... 65 5.2 Earthquake induced stresses and pore pressures ...... 66 5.3 Poroelastic diffusion equation ...... 67 5.4 Pore pressure diffusion equation ...... 70 5.5 Pore pressure diffusion modelling ...... 71 5.6 Modelling of pore pressure induced seismicity ...... 72 6 Tectonics of Iceland and the South Iceland Seismic Zone ...... 75 6.1 Tectonic setting of Iceland ...... 75 6.2 The South Iceland Seismic Zone ...... 78 6.2.1 Mechanics of faulting ...... 78 6.2.2 Historical seismicity ...... 79 6.3 The SIL network ...... 81 6.4 The June 2000 earthquakes in the SISZ ...... 82 6.5 Studied aftershock sequences ...... 83 6.6 Data completeness in studied aftershock sequences ...... 85 6.7 Assessment of systematic detection problems ...... 87 7 Summary of papers ...... 89 7.1 Paper I ...... 89 7.2 Paper II ...... 93 7.3 Paper III ...... 95 7.4 Paper IV ...... 96 7.5 Paper V ...... 99 8 Discussion and conclusions ...... 105 9 Summary in Swedish ...... 109 10 Acknowledgements ...... 113 A Errata ...... 115 Bibliography ...... 117 1. Introduction

The many casualties of the recent magnitude 7.9 Sichuan earthquake in China on May 12, 2008, are a reminder of the need to improve our understanding of the earthquake process. This process can be considered as a cycle where there is a vast difference in the time scales of the constituent phases. The duration of the different phases is in the order of seconds to minutes in the coseismic phase (the actual rupture process of an earthquake), days to months and years in the pre- and postseismic phases and several decades in the interseismic phase (the slow build-up of stresses within the crust in between earthquakes). The pre- and postseismic phases are associated with processes that, due to the closeness in time of the earthquake, differ from the interseismic process. Since a very long time period may be required in order to observe a complete earthquake cycle, it is difﬁcult to build an understanding of the earthquake process from observations at a particular location only. We can, however, assemble our knowledge and understanding by piecing together observations of different parts of the earthquake cycle from different locations. In my thesis, I look into one piece of this puzzle, namely the postseismic phase and processes in operation following large earthquakes. I do this by studying aftershock sequences taking place in Iceland, within the south Iceland seismic zone. By improving our understanding of the aftershock process, we can gain important insight into the physics of earthquakes in general. Understanding the physics of aftershocks is also of great practical importance as large aftershocks may result in additional damage and casualties through the collapse of structures weakened by the main shock. Earthquakes are complex phenomena as they are an expression of the forces driving tectonic plate motions, the state of temperature, stress and pore pressure within the Earth, structural heterogenities, material properties of rocks, existing zones of weakness (faults) as well as complex interactions with other earthquakes. From a large scale point of view, the majority of all earthquakes occur along boundaries of lithospheric plates. Stresses are built up and released in earthquakes as these plates move slowly relative to each other, de- forming the rocks in the plate boundary zones. In this sense, the physics of earthquakes is easy to understand. Elastic strain energy is accumulated in the plate boundary zones until the rock fails and the accumulated strain energy is released as seismic waves and heat. Despite this deceptively simple driving process earthquake prediction has proved difﬁcult, as information regarding the state of stress within the crust is largely unavailable for many locations.

9 In earthquake prediction the desired goal is to be able to accurately predict the location, timing and magnitude of future earthquakes (Wallace et al., 1984). To date a variety of phenomena, such as seismic quiescense, foreshock activity, seismic gaps, anomalous crustal deformation and hydrological and geochemical anomalies have been observed and interpreted as possible earthquake precursors (Scholz, 2002). Ultimately, the prediction of individual earthquakes in a certain area must be based on observing and detecting the process leading to, and associated with, the initiation of rupture in the crust. Utilising precursory phenomena is therefore valuable for earthquake prediction, provided that we can understand the physical connection between the rupture initiation process and the precursor. However, the nature of an observed precursor, such as a radon anomaly, may only reveal the general area of higher earthquake probability and not the exact location, magnitude or occurrence time of the possible earthquake. A detailed understanding of the earthquake process is therefore important for attempts of earthquake prediction. In my thesis, a reason for my focus on the postseismic period is that a large amount of aftershock data is typically available within a relatively short time (∼3 months - 1 year or more) following a major earthquake. Studies of other phases in the seismic cycle may be limited by much less data, implying that long observation time may be required (perhaps decades). Another reason is the hypothesis of self-organised criticality (SOC) (e.g. Bak and Tang, 1989; Ito and Matsuzaki, 1990). In seismology, the SOC-hypothesis is highly debated as it essentially implies that earthquakes are unpredictable. Based on observations of power law properties in earthquake statistics (including aftershock temporal behaviour) the SOC-hypothesis proposes a model for how the earthquake process works. The behaviour of seismicity in the postseismic phase has thus been used to draw conclusions regarding the underlying mechanism behind the earthquake process in general. It is possible that understanding one of the phases in the seismic cycle may shed light on the entire earthquake process. However, aftershock temporal behaviour shall, as I will point out in my thesis, be used with caution as a support for the SOC-hypothesis. In order to further our understanding of the underlying physics of the earthquake process one can either utilise statistical methods to study relevant distributions of large earthquake datasets or model the physical processes. Statistics has the advantage that a large number of different cases can be investigated while modelling has the advantage that it allows an understanding of what may and may not explain statistical properties of the data. Statistical studies and modelling are therefore complimentary and both are essential when studying the earthquake process. In my thesis I look at the earthquake process from both a statistical (Pa- pers I, II and III) and a physical point of view (Papers IV and V). I therefore start this thesis by discussing statistical seismology and the hypothesis of self- organised criticality. I also devote one chapter to describing statistical methods

10 that I apply in the thesis and one chapter to discussing physical models for the occurrence of aftershocks. In the subsequent chapter I discuss pore pressure diffusion equations and modelling of main shock initiated pore pressure diffusion processes. Following this I provide a description of my study area, the South Iceland Seismic Zone, and the data used for the thesis. I ﬁnish with a summary of the papers included in the thesis and wrap up with a ﬁnal discussion and some conclusions regarding the physics of the earthquake process.

2. Statistical seismology

Observations of the processes driving the occurrence of earthquake within the Earth’s crust are (apart from boreholes, mines and laboratories) limited to the surface of the Earth. The use of statistics and statistical methods is thus an essential ingredient in a science such as seismology, where earthquake processes are not fully understood. Statistical seismology involves both the study of distributions of earthquake related information and the application of statistical methods to the problem at hand. One role of distribution studies is to condense the available information into a manageable form for interpretation of the underlying physical process, primarily by estimating parameters describing the distributions of the studied quantity. In seismology, distributions of the basic earthquake quantities magnitude, occurrence time and location have been among the most commonly studied in order to interpret the underlying physical processes. Recently, with a ris- ing interest in understanding the spatial and temporal complexities of earthquake occurrence and interaction, distributions of derived quantities, such as interevent time (also referred to as waiting time), have also been studied. In this chapter I will discuss the magnitude, temporal and spatial earthquake distributions in section 2.1 and interevent time distributions in section 2.2. In section 2.3 I will then provide a discussion on how the characteristics of these distributions have contributed to forming a view of earthquakes as being an expression of a self-organised critical system, and to what extent this concept appears to be viable.

2.1 Typical earthquake distributions 2.1.1 Magnitude distribution of earthquakes A general observation in seismology is that small earthquakes occur more fre- quently than large earthquakes. The Gutenberg-Richter law (G-R law) is, since its formulation by Gutenberg and Richter (1944), a well established empirical law in seismology. In a given area, the G-R law describes the frequency- magnitude distribution of earthquakes above a certain magnitude as: (≥ )= − ,( ≥ ) log10 N m a bm m Mc (2.1)

13 where N(≥ m) is the number of earthquakes with magnitudes equal to or greater than m, a and b are constants and Mc is the magnitude of completeness, above which the earthquake record is considered to be complete. The G-R law has been found to hold in a variety of tectonic regimes, with b-values generally in the range 0.8-1.2 (Frohlich and Davis, 1993) and close to one in many cases (Kanamori and Brodsky, 2004). In the many papers studying b-values that have been published since the formulation of the G-R law, the b-value has either been estimated by using least squares regression to fit a straight line (≥ ) to the linear regime in a plot of log10 N m versus magnitude (e.g. Kaly- oncuoglu, 2007) or by using maximum-likelihood methods (e.g. Aki, 1965; Utsu, 1965). Here I will not discuss the assumptions, limitations or implications of various ways to estimate the value of b as it is beyond the scope of my thesis. For a discussion on b-value estimation I refer to papers by Bender (1983); Tinti and Mulargia (1985); Kijko (1988); Rhoades (1996); Marzocchi and Sandri (2003) and Sandri and Marzocchi (2007). Given an earthquake dataset with magnitudes m1,....,mi,...mN a maximum likelihood estimate of the b-value is (Aki, 1965; Utsu, 1965): 1 b = (2.2) ln10 · (m − Mc) where m is the mean-value of the magnitudes in the dataset and Mc is the magnitude of completeness. As for the b-value, the magnitude of completeness has also been estimated in a variety of ways, and I refer to a discussion by Woessner and Wiemer (2005) for details on this topic. The maximum likelihood estimate of the a-value in the Gutenberg-Richter law is: = (≥ )+ · a log10 N Mc b Mc (2.3) where N(≥ Mc) is the number of earthquakes with magnitudes greater than or equal to the magnitude of completeness, Mc. In figure 2.1 I illustrate some aspects of the distribution of earthquake magnitudes and the application of the G-R law by using a set of earthquakes regis- tered in the south Iceland seismic zone and the Hengill region in Iceland (see figure 6.2 in chapter 6 for the geodynamical setting) between January 1992 and February 2004. Figure 2.1 shows some characteristic features of earthquake magnitude distributions. First, the G-R law does not hold below the magnitude of completeness, Mc. With increased detectability of seismological networks Mc has become smaller and smaller, extending the range where the G-R law holds to smaller and smaller magnitudes. The unanswered question is whether this behaviour continues down to molecular level or whether other physical limitations restricts the applicability of the Gutenberg-Richter law before this level is reached. Secondly, figure 2.1 also shows that the G-R law holds well in the magnitude range m = 0.65 to m ≈ 2.8 and approximately in the magnitude range m ≈ 2.8tom ≈ 4.5, where the G-R law overestimates the

14 8 10 Data G−R law 7 10

6 10

M =0.65 5 10 c

4 10

3 10

No. of eqs with magnitude > m 10 M =4 u

1 10

0 10 −2 −1 0 1 2 3 4 5 6 7 Earthquake magnitude, m Figure 2.1: Illustrating the G-R law using an earthquake dataset from south Iceland covering the period January 1992 - February 2004. Black solid curve: Gutenberg- Richter frequency-magnitude distribution for the earthquake dataset, black dashed curve: Predicted distribution according to the G-R law, with parameters a = 5.11 and b = 0.92. a and b represent the maximum-likelihood estimates using magnitudes in the range Mc <= m < Mu. The magnitude of completeness, Mc = 0.65, is a maximum likelihood estimate found by assuming that magnitudes below Mc can be described by a truncated normal distribution (see e.g. Woessner and Wiemer, 2005). Mu = 4is selected as the uppermost magnitude limit where the G-R law approximately holds for this dataset.

number of earthquakes. Although approximately agreeing with the G-R law, the distribution of earthquake magnitudes may, as we can see, also exhibit sig- niﬁcant deviations. Understanding both the origin of the G-R law and of such deviations is of great interest for an improved understanding of the physics of the earthquake process. A third characteristic feature seen in ﬁgure 2.1 is that the occurrence of large earthquakes changes the slope of the frequency- magnitude curve and limits the range where the G-R law holds. In this dataset, it is the occurrence of two magnitude 6.5 earthquakes in June 2000 that is the cause. This implies that an extrapolation of the G-R law to larger magnitudes may underestimate the magnitude of future large earthquakes. In seismology, the b-value in the G-R law is an important parameter. Math- ematically, it represents the relative proportion of large and small magnitude earthquakes. Physically, the b-value has been interpreted as an inverse measure of the stress state in a region (e.g. Scholz, 1968; Bufe, 1970; Gibowicz, 1973; Wyss, 1973). According to this interpretation, a low b-value implies a larger magnitude of the differential state of stress than a high value, with proportionally more large earthquakes occurring. The interpretation of b act-

15 ing as a stress meter is supported by an observation that b-values are largest for normal faulting earthquakes (low mean stress), intermediate for strike- slip earthquakes (intermediate mean stress) and smallest for thrust faulting earthquakes (high mean stress) (Schorlemmer et al., 2005). For earthquake forecasting purposes, studies of temporal and spatial distributions of b-values have therefore been undertaken in order to map stress variations as a function of time and/or space. In, for example, the Parkfield section along the San An- dreas Fault, Schorlemmer and Wiemer (2005) find that seismicity in the two decades prior to the M6.0 Parkfield earthquake on September 28, 2004, exhibits unusually low b-values in areas where most of the slip occurred in the subsequent earthquake. Schorlemmer and Wiemer (2005) thus interpret these areas as highly stressed patches of the fault. In Iceland, low b-values have also been found in the hypocentral regions of the two M6.5 June 2000 earthquakes in the south Iceland seismic zone (Wyss and Stefánsson, 2006). As an example of temporal variations in b-values, Nuannin et al. (2005) associate distinct drops in b-values in the Andaman-Sumatra region with two large earthquakes during the fall of 2002 as well as the major Sumatra earthquake by the end of 2004. In this section I have given a brief introduction to the G-R law for the frequency-magnitude distribution of earthquakes and discussed the importance of the parameter b. A detailed analysis of the distribution of earthquake magnitudes and b-values is beyond the scope of my thesis. However, I will discuss the G-R law in relation to the theory of self-organised criticality in section 2.3 of this chapter. In Paper IV of the thesis I address the issue of incompletely detected aftershocks utilising the G-R law and interevent time distributions.

2.1.2 Temporal distribution of aftershocks The Omori law for aftershocks is another well established empirical law in seismology, describing the rate of decay of aftershocks with time following a main shock as (Omori, 1894; Utsu, 1961; Utsu et al., 1995): dn K = = λ(t) (2.4) dt (c +t)p where n is the number of events following the main shock, K, c and p are empirical constants, and t is the time since the main shock. Of the Omori law parameters, we can immediately see that K may be related to aftershock productivity, c to a period of time after the main shock during which the aftershock rate is roughly constant and p to the power law decay in the rate of aftershocks for t c. Later, it will be necessary to calculate the number of aftershocks in a given time interval from the Omori law. Integration of equation

16 2.4 yields the number of aftershocks in the interval [t1, t2] for p = 1 as:

N = K · [ln(c +t2) − ln(c +t1)] (2.5) and for p = 1: = − · 1 − 1 N K p−1 p−1 (2.6) (p − 1) · (c +t2) (p − 1) · (c +t1)

Figure 2.2 demonstrates the Omori law (equation 2.4) for a case when K = 50, p = 1.2 and c takes the values c = 0 and c = 1500 seconds, respectively. This example illustrates both the power law decay of the Omori law and the constant rate for t c when c = 0. Later, I will use the same Omori law parameters to illustrate the Omori law interevent time distribution (see section 2.2.2).

The Omori law 2 10

1 10 c=0 0 10

−1 10

−2 10 c ≠ 0 −3

(t) 10 λ

−4 10

−5 10 c

−6 10

−7 10

−8 10 1 2 3 4 5 6 7 8 10 10 10 10 10 10 10 10 Occurrence time t [s] Figure 2.2: Illustrating the Omori law with parameters K = 50 and p = 1.2. The dashed curve represents λ(t) (equation 2.4) with c = 0 while the thick solid curve represents λ(t) with c = 1500. The thin vertical line shows the location of c = 1500.

Since the formulation of Omori’s law in 1894, aftershock sequences of both shallow and deep main shocks have been studied in tectonic environments such as oceanic spreading centers and transform faults (e.g. Bohnenstiehl et al., 2002), continental transform fault zones (e.g. Kisslinger and Jones, 1991), continental collision zones (e.g. Pavlis and Hamburger, 1991; Nyf- fenegger and Frohlich, 2000) and subduction zone environments (e.g. Wiens and Gilbert, 1996; Wiens and McGuire, 2000; Nyffenegger and Frohlich, 2000). Aftershock sequences associated with nuclear explosions have also been studied (e.g. Gross, 1996). Utsu et al. (1995) review statistical aspects

17 of the Omori law and summarise a large number of studies of aftershock sequences. Kisslinger (1996) discusses the relationship between fault zone properties and the Omori law and Davis and Frohlich (1991) investigate regional variations in aftershock sequence decay laws. A result of these studies is the observation that both aftershock productivity in terms of the number of aftershocks as well as the Omori law decay parameter p varies significantly between different tectonic regimes. A high p-value means that the rate of aftershocks decays faster with time than for a lower value of p. Davis and Frohlich (1991) find that earthquakes occurring at depths greater than 70 km have the fewest and the smallest aftershock sequences. For shallow earthquakes they find shallow subduction zone earthquakes to produce more aftershocks than shallow ridge-transform zone earthquakes. Although rare, Pavlis and Hamburger (1991) find that intermediate depth earthquakes in a collisional plate boundary setting, the Pamir-Hindu Kush region, are capable of producing aftershock sequences. Studies of aftershock sequences in subduction zones have revealed that earthquakes as deep as ∼650 km may be very productive in terms of the number of aftershocks (Wiens and Gilbert, 1996). The 564 km deep Mw 7.6 Tonga earthquake (Wiens et al., 1994; Wiens and Gilbert, 1996; Nyffenegger and Frohlich, 2000; Wiens and McGuire, 2000) and the 631 km deep Mw 8.3 Bolivia earthquake (Wiens and Gilbert, 1996; Nyffenegger and Frohlich, 2000), both taking place in 1994, are examples of such earthquakes. For the 1994 Tonga event, Wiens and McGuire (2000) find that its aftershock productivity is at the lower end of the normal range found for shallow earthquakes of similar seismic moment. On the other hand, deep earthquakes in, for example, Japan and Indonesia have shown very low or no aftershock productivity (Wiens and Gilbert, 1996). The Omori law decay parameter p has received most attention in studies of aftershock sequences. Utsu et al. (1995) report that published values of p vary between 0.6 and 2.5, with a median of 1.1. For shallow aftershock sequences in California, Kisslinger and Jones (1991) report p-values to be in the range 0.7 to 1.8 with a mean of 1.1. In an oceanic environment, Bohnen- stiehl et al. (2002) present p-values of 1.74 and 2.37 for two aftershock sequences associated with the spreading of the mid-Atlantic ridge, and p-values in the range 0.94-1.29 for three aftershock sequences associated with transform faults offsetting the ridge. In the study of intermediate depth aftershock sequences in the Pamir-Hindu Kush region by Pavlis and Hamburger (1991), a p-value close to 1 is indicated for three sequences where the main shocks occur at ∼200 km depth. Nyffenegger and Frohlich (2000), however, find p- values of 0.53 and 0.83 for the two most significant aftershock sequences studied by Pavlis and Hamburger (1991). For the aftershock sequences following the deep 1994 Tonga and Bolivia earthquakes, p-values similar to those for shallow sequences were observed, 1.19 for the Bolivia event (Nyffenegger and Frohlich, 2000) and 1.006 for Tonga event (Wiens and McGuire, 2000).

18 A general observation regarding the value of p is that it shows no correlation with either the magnitude of the main shock or the cutoff magnitude (the magnitude limit below which earthquakes are not used) (e.g. Utsu, 1961; Kisslinger and Jones, 1991; Utsu et al., 1995). The parameter p may therefore be interpreted to reflect properties of the fault system and the surrounding rock material (e.g. Mogi, 1967; Kisslinger, 1996; Nanjo et al., 1998). Moreover, a correlation between high values of p and high surface heat flow has been observed and discussed by several authors (e.g. Mogi, 1967; Kisslinger and Jones, 1991; Creamer and Kisslinger, 1993; Kisslinger, 1996; Rabinowitz and Steinberg, 1998). Bohnenstiehl et al. (2002) conclude that higher p-values in ridge settings may be consistent with the higher temperature associated with spreading centers, but that continued studies are needed to determine if high p-values are a characteristic of spreading environments. Such an investigation is interesting for Iceland, where plate spreading takes place on land. That temperature is one of the important factors controlling aftershock behaviour is also suggested by Wiens and Gilbert (1996), who show that deep earthquakes that produce aftershocks are, in general, associated with the subduction of cold slabs. However, Utsu et al. (1995) conclude that the significance of factors such as structural heterogenity, stress and temperature in controlling the value of p is not clear. Contrary to the behaviour of the decay parameter p, the remaining Omori law parameters K and c exhibit significant variation with the cutoff magnitude used. Using data from the 1993 Hokkaido-Nansei-Oki aftershock sequence, Utsu et al. (1995) show that increasing the cutoff magnitude results in low- ering the values of both K and c. An implication of this is that there is a possibility that K and c are not independent of each other, and that they in some way reflect the same physical process. Due to the intense activity at short times after the mainshock, the parameter c in Omori’s law is commonly considered as a time offset accounting for incomplete detection of aftershocks (e.g. Utsu et al., 1995; Kisslinger, 1996; Woessner et al., 2004). Utsu et al. (1995), however, also point out that positive values of c have been observed for adequately recorded aftershock sequences. It is also a fact that physical models of aftershock behaviour can provide explanations for Omori’s law (see section 4.1), including a non-zero value of c. However, how to interpret c is controversial and some authors argue, based on statistical investigations of earthquake catalogues, that c solely reflects incomplete detection of aftershocks (e.g. Kagan, 2004; Kagan and Houston, 2005; Lolli and Gasperini, 2006). In my thesis I will discuss a non-zero value of c as a reflection of a postseismic pore pressure diffusion process (Paper IV and Paper V). I will also discuss (sections 2.3.2-2.3.4) the implications of a non- zero c for the view of earthquakes as an expression of a self-organised critical state of the Earth’s crust.

19 2.1.3 Spatial distribution of faults and earthquakes Analysis of fault networks (e.g. Okubo and Aki, 1987; Hirata, 1989a) as well as regional and local seismicity (e.g. Robertson et al., 1995; Lei and Kusunose, 1999) has shown that earthquakes exhibit characteristics of fractal distributions and power law behaviour, indicating self-similarity in fault patterns and the spatial distribution of earthquake locations over certain spatial scales. Under the assumption that earthquakes are point sources described by their hypocentral or epicentral locations, the spatial fractal dimension D (also called the capacity or box dimension) characterises the degree with which an area (in 2-D) or a volume (in 3-D) is filled with earthquakes. In a volume (3- D) D=3 implies that the entire volume is filled with earthquakes while D=2 implies that earthquake hypocentres fill entire planes within the volume. Sim- ilarly, in an area (2-D) D=2 implies that the entire area is filled while D=1 implies that earthquake epicentres align along linear segments (Turcotte, 1997). In the three-dimensional study of Robertson et al. (1995), they interpret fractal dimensions D ≤ 2.0 of background seismicity and aftershock sequences in California as representing earthquakes occurring only where finite deformation is located and accommodated within the fault network. The spatial fractal dimension D for earthquakes is typically estimated by using box-counting techniques, where volumes (3-D) or areas (2-D) are di- vided into equally sized cubes or squares, respectively. The number of cubes containing earthquake hypocentres (3-D) or squares containing earthquake epicentres (2-D) can then be counted for varying cube and square sizes, respectively. In a log-log plot of the counted number of cubes/squares versus cube/square size, the value of the fractal dimension D can then be estimated from the slope of the linear part of the resulting curve. Such an analysis also reveals the spatial range where power law behaviour and self-similarity in the spatial distribution hold. As an example, large scale fault patterns along the San Andreas fault exhibit fractal behaviour over 1 - 15 km while local fault patterns with greater detail exhibit an upper limit for the fractal behaviour varying from ∼ 400 m to ∼ 1 km (Okubo and Aki, 1987). In the analysis of Robertson et al. (1995), the spatial distribution of earthquake hypocentres exhibits both lower and upper limits of the fractal behaviour. These observations imply that self-similarity in fault patterns and distributions of earthquake epicentres and hypocentres cannot be said to be generally valid for all spatial scales. Based on theoretical considerations and certain assumptions regarding the relationship between earthquake magnitude and seismic moment, the Gutenberg-Richter law implies that rupture lengths of earthquakes are power law (fractally) distributed with a power law exponent (fractal dimension) equal to twice the b-value (Aki, 1981). It is important to note, however, that this exponent is not the same as the spatial fractal dimension D of earthquakes or faults, estimated using box-counting techniques. Empirical investigations have shown that both positive (e.g. Guo and Ogata, 1995) and negative (e.g.

20 Henderson et al., 1994; Hirata, 1989b) correlations between b and D exist, implying a complex and non-general relationship between the distribution of earthquake locations and the distribution of earthquake magnitudes. Barton et al. (1999) suggest that a correlation between a large value of D and a small value of b implies a mature fault zone where diffuse and large earthquakes may take place, while the opposite implies an immature fault zone where relatively small earthquakes occur in tighter clusters.

2.2 Earthquake interevent time distributions In this section, I will discuss interevent time distributions for earthquakes distributed in time according to a Poisson distribution as well as for aftershocks obeying Omori’s law. The term interevent time refers to the time interval between two successive events in a sequence. In my thesis I use the term interevent time for this property in Paper IV but waiting time in Papers I, II and III. Although only used in one of the papers, I ﬁnd interevent time to be a better description than waiting time for what this property measures. Except in the summaries of Papers I-III in chapter 7, I have therefore chosen to use the term interevent time in my thesis summary.

2.2.1 Mainshocks - the Poisson interevent time distribution In seismology, a homogenous Poisson process is often considered as a model for the temporal distribution of main shocks or as a null hypothesis in test- ing some statistical model against purely random behaviour (e.g. Shlien and Toksoz, 1970; Gardner and Knopoff, 1974; Gasperini and Mulargia, 1989; Console and Murru, 2001). A homogeneous Poisson process is characterised by statistically independent events with a constant underlying rate, μ, and is deﬁned by the probability of k events occurring in a time interval of duration t according to: e−μ·t (μ ·t)k P[(N(t) − N(0)) = k]= ,k = 0,1,2,.... (2.7) k! where N(t) and N(0) are the number of events that have occurred before time t and time 0, respectively. In such a process the interevent time distribution is deﬁned by the probability that the interevent time Tk+1 −Tk is greater than Δτ, i.e. that no events occur during a time interval [Tk, Tk + Δτ]:

P[Tk+1 − Tk > Δτ]=P[(N(Tk + Δτ) − N(Tk)) = 0]= −μ·( +Δτ− ) Tk Tk (μ · ( + Δτ − ))0 = e Tk Tk = 0! −μ·Δτ = e (2.8)

21 As equation 2.8 is valid for all values of k, this implies that all interevent times in a homogeneous Poisson process share the same exponential distribution and that the expected interevent time equals 1/μ. The exponential distribution also implies memorylessness, i.e. that interevent times are independent of earlier events. The cumulative probability distribution of interevent times is given by: −μ·Δτ F(Δτ)=P(Tk+1 − Tk < Δτ)=1 − e ,k = 0,1,2,.... (2.9) where Δτ is the interevent time, i.e. the time between any two successive events in the sequence. The interevent time probability density function is then given by: dF −μΔτ f (Δt)= = μ · e (2.10) dΔτ In order to determine the expected proportion of interevent times within a certain range Δτ1 < Δτ < Δτ2, equation 2.10 can be integrated between Δτ1 and Δτ , giving: 2 −μΔτ1 −μ(Δτ2−Δτ1) P(Δτ1 < Δτ < Δτ2)=e · 1 − e (2.11)

For a real earthquake sequence, we can make a histogram of the associated interevent times by counting the number of interevent times into histogram bins. An empirical interevent time probability density function for the given sequence is then obtained by dividing the number of interevent times in each bin with the total number of interevent times (i.e. the number of events - 1). Using equation 2.11, empirical interevent time distributions for real earthquake sequences can be compared with the expected distribution for a Poisson process. Paper I discusses properties of interevent time distributions for homogeneous Poisson processes.

2.2.2 Aftershocks - the Omori law interevent time distribution In this section, I will use two different approaches to derive an equation for the interevent time distribution of an aftershock sequence obeying the Omori law (equation 2.4). The ﬁrst approach assumes that the Omori law deﬁnes continuously changing rates of homogeneous Poisson processes that are only valid momentarily, and the corresponding theoretical equation for the interevent time probability density functions is utilised in Papers I, II, III and IV of this thesis. Based on considering the aftershock sequence as an inhomogeneous Poisson process, however, Shcherbakov et al. (2005) have presented another formulation for the Omori law interevent time distribution. I will show that the two different formulations exhibit the same scaling behaviour, implying that the formulation of Shcherbakov et al. (2005) does not change the conclusions of this thesis.

22 2.2.2.1 Approach 1 In the ﬁrst approach, we assume that the aftershock process at a particular instant in time can be described by a homogeneous Poisson process that is only valid momentarily, with a rate given by the Omori law. Thus, at a time t after a main shock, the interevent time probability density function is (as shown in equation 2.10): −λ( )·Δτ f (Δτ|t)=λ(t) · e t (2.12)

During a short time interval dt centered around t, we expect λ(t) · dt events. The probability density function for the particular time interval then becomes: λ(t)2 · dt · e−λ(t)·Δτ f (Δτ|t,dt)= (2.13) N(T) − 1 where N(T) is the expected number of aftershocks occurring until time T (the time period the aftershock sequence is observed). By letting dt tend to zero when integrating equation 2.13 over time, we obtain an equation for the interevent time probability density function of an aftershock sequence that is observed for a time period T: TorT−Δτ 1 −λ( )·Δτ f (Δτ|T)= · λ(t)2 · e t · dt (2.14) N(T) − 1 0

In equation 2.14, the aim of the division with N(T) − 1 is a normalisation so that an integration of the probability density function over Δτ, from 0 to Δτmax, will become equal to one. N(T) can be calculated using equation 2.5 (when p = 1) or equation 2.6 (when p = 1). When integrating equation 2.13 in order to derive an equation for the Omori law interevent time probability density function, we have two choices for the range of integration as shown above: [0, T] or [0, T − Δτ]. If the aftershock sequence is observed for a time period T, integrating to T − Δτ results in a probability density function for interevent times that can be observed in the sequence. If we look at a particular time t, for example, we cannot observe interevent times larger than T −t originating from the activity at this time. Al- though this activity may produce aftershocks with interevent times larger than T − t, these cannot be observed as at least one of the corresponding events would occur after time T. For a speciﬁed observation period of an aftershock sequence T, the largest interevent time that can be observed can never be larger than T itself. A suitable choice for the upper end of the range of interevent times to be investigated when integrating to T −Δτ is therefore given by Δτmax = T. If, however, we integrate equation 2.13 to time T instead of T − Δτ,we obtain a probability density function for interevent times that we may expect from aftershock activity during the time period T. Some of these interevent

23 times can be observed while others cannot. Consider, for example, the rate of aftershocks at time T, i.e. λ(T). The interevent times associated with this rate are distributed according to a Poissonian interevent time probability density function, i.e. λ(T) · exp(−λ(T)Δτ). With a rate λ(T), the largest expected interevent time is given by the zero probability of observing interevent times larger than Δτmax, i.e. exp(−λ(T) · Δτmax)=0. If Δτmax is larger than T, this implies that such interevent times cannot be observed, although they are expected. In an aftershock sequence observed for a time period T, the choice of integration range in equation 2.14 thus governs which interevent times the probability density function is valid for. Equation 2.14 is either valid for interevent times that can be observed (integration to T − Δτ) or expected (integration to T) as a result of the aftershock activity. Further below, we will illustrate how the choice of integration range can affect the interevent time distribution. Inserting the Omori law (equation 2.4) in equation 2.14 gives the following expression for the Omori law interevent time probability density function: −Δτ 1 TorT K2 − K ·Δτ (Δτ| )= · (c+t)p · f T 2p e dt (2.15) N(T) − 1 0 (c +t)

For values of p different than one, the integral in equation 2.15 has to be integrated numerically. When p = 1, however, an analytical solution for the probability density function exists and is, depending on the choice of integration range, given by: 1 K − K ·Δτ − K ·Δτ f (Δτ|T, p = 1)= · · e c+T − e c (2.16) N(T) − 1 Δτ or: 1 K − K ·Δτ − K ·Δτ f (Δτ|T, p = 1)= · · e c+T−Δτ − e c (2.17) N(T) − 1 Δτ

Empirical interevent time probability density functions for observed aftershock datasets have been analysed (see e.g. Bak et al., 2002) by utilising histograms with bins that are evenly spaced logarithmically (i.e. exponentially increasing in size). The empirical interevent time probability density functions are obtained by dividing the number of interevent times in the different bins with the size of each respective bin. These empirical probability density functions can be directly compared with the Omori law interevent time probability density function derived in equation 2.15 by ignoring the normalisation 1/(N(T) − 1) in the equation. The comparison can also be made by integration of equation 2.15 over Δτ to determine the expected number of interevent times in the aftershock sequence for each histogram bin. Given the boundaries Δτ1 and Δτ2 of a particular histogram bin, the expected proportion of

24 interevent times in this bin is given by: (Δτ < Δτ < Δτ | )= P 1 2 T −Δτ ·Δτ ·Δτ 1 TorT K − K 1 − K 2 = · · (c+t)p − (c+t)p · p e e dt(2.18) N(T) − 1 0 (c +t)

Now, we want to investigate the characteristics and scaling behaviour of the Omori law interevent time probability density function (equation 2.15). In figure 2.3 a) we illustrate the interevent time probability density function corresponding to the Omori law example in figure 2.2 when K = 50, c = 1500, p = 1.2 and integrating to either T or T − Δτ. The figure shows that the interevent time probability density function is characterised by a constant part for short interevent times, a power law part for intermediate to long interevent times and a falloff at the largest interevent times which is more dramatic when integrating to T −Δτ than T. Interevent times separating the different parts of the probability density function are shown in the figure and labelled ΔτUL=1 (separating the constant and the power law part) and ΔτLL=1 (separating the power law part from the falloff). Note that ΔτLL=1 when integrating to T −Δτ is less than that when integrating to T. I will now show that the example in figure 2.3 a) illustrates general features of Omori law interevent time probability density functions. I am also going to explain why the boundaries between the different regimes have been labelled the way they have. In order to evaluate the behaviour of the Omori law interevent time probability density function, we consider equation 2.15 written in the form of equation 2.14, where λ(t) represents the Omori law rate of aftershocks (K/(c+t)p) in equation 2.15. We then perform a change of variables from time t to rate λ in the integral of equation 2.14: λ(0) 1 −λ·Δτ dt f (Δτ|T)=− · λ 2 · e · · dλ (2.19) N(T) − 1 λ(T) or λ(T−Δτ) dλ

The minus sign in front of the integral is necessary as we have changed the order of integration to be from the smallest rate, λ(T), to the largest, λ(0). The Omori law (equation 2.4) can be rewritten in such a way that: 1/p − λ(t) (c +t) 1 = (2.20) K1/p

dλ Taking the derivative of the Omori law, dt , and inserting equation 2.20 gives: 1+1/p dλ − p+1 p · λ = −p · K (c +t) 1 = − (2.21) dt K1/p so that: dt K1/p = − (2.22) dλ p · λ 1+1/p

25 a) Interevent time pdf, approach 1 2 10

0 Δτ 10 LL=1

−2 10 −>T−Δτ −>T

) −4 10 Δτ −6 f( 10

−8 Δτ 10 UL=1

−10 10

0 2 4 6 8 10 10 10 10 10 Interevent time, Δτ [s] b) Functions f and f , approach 1 1 2 2 10

) 10

Δτ −2 ( 10 2

), f −4 10 f

Δτ 1 (

1 −6 f −> T f 10 2 f −> T−Δτ 2 −8 10 0 2 4 6 8 10 10 10 10 10 Interevent time, Δτ [s] Figure 2.3: a) Illustrating the Omori law interevent time probability density function (equation 2.15) when K = 50, c = 1500, p = 1.2 and T = 180 days. Dashed curve: Probability density function when integrating to T. Solid curve: Probability density function when integrating to T − Δτ. Thin vertical lines: Boundaries between the regimes of the probability density function. ΔτUL=1 is the boundary between the constant part and the power law part. ΔτLL=1 is the boundary between the power law part and the falloff part. The location of ΔτLL=1 depends on whether integration is done to T or T −Δτ in the probability density function (equation 2.15). The labelling of these boundaries is explained in the text. b) A separation of the probability density function in a) into two functions f1 (dotted curve) and f2 (solid/dashed curves). f1 is responsible for the power law behaviour in the probability density function while f2 is responsible for the constant part and the falloff. Note that the falloff behaviour of f2 depends on whether integration is to T (dashed curve) or T − Δτ (solid curve) in the probability density function (equation 2.15).The thin vertical lines represent the same interevent times as in a).

26 By inserting equation 2.22 in equation 2.19, we get: 1/p λ(0) K − / −λ·Δτ f (Δτ|T)= · λ 1 1 p · e · dλ (2.23) p · (N(T) − 1) λ(T) or λ(T−Δτ)

Now, changing variables from λ to x = λ ·Δτ in the integral of equation 2.23, the probability density function can be written as a multiple of two functions f1 and f2: f (Δτ|T)= f1(Δτ) · f2(Δτ|T) (2.24) where the functions f1 and f2 are: K1/p 1 f (Δτ)= · (2.25) 1 p · (N(T) − 1) Δτ2−1/p x (Δτ) 2 1−1/p −x f2(Δτ|T)= x · e · dx (2.26) x1(Δτ, T) and the upper limit of the integral in the function f2 is: K x (Δτ)=λ(0) · Δτ = · Δτ (2.27) 2 (c)p while the lower limit is, when integrating to T in the original equation: K x (Δτ, T)=λ(T) · Δτ = · Δτ (2.28) 1 (c + T)p or, when integrating to T − Δτ in the original equation: K x (Δτ, T)=λ(T − Δτ) · Δτ = · Δτ (2.29) 1 (c + T − Δτ)p

We have shown that the Omori law interevent time probability density function can be separated into a multiple of two functions, f1 and f2. The function f1 represents the power law behaviour of the probability density function 2−1/p (∼ 1/Δτ ) and f2 represents a function that modifies the power law behaviour. Figure 2.3 b) illustrates the functions f1 and f2 for our Omori law example with K = 50, c = 1500 and p = 1.2. The figure shows the pure power law behaviour of f1 and that f2 has three distinct regimes: 1) an upward sloping part for short interevent times, 2) a constant part for intermediate interevent times and 3) a falloff at large interevent times. The falloff of f2 is more dramatic when integrating to T − Δτ rather than T in the original equation for the probability density function. Comparing with the probability density function in figure 2.3 a) we can see that the first two regimes of f2 modify f1 in such a way that the power law decay is eliminated for short interevent times and is left intact for intermediate interevent times. For large

27 interevent times, the falloff of f2 dominates over the power law decay of f1, and this leads to the falloff in the probability density function, as illustrated in ﬁgure 2.3 a). The behaviour of the function f2 (equation 2.26) is controlled by its inte- 1−1/p −x grand, x · e , and the integral limits x1 (equation 2.28 or 2.29) and x2 (equation 2.27). The interevent times where x1 = 1 and x2 = 1 are of particular interest as limits of the power law regime in the Omori law interevent time probability density function. For the case when c = 0, the upper limit p x2 (equation 2.27) increases linearly with Δτ, with a rate given by K/c . The value of x2 is equal to one when: cp Δτ = = (2.30) UL 1 K

The term ΔτUL=1 refers to the interevent time where the upper limit x2 of the integral in the function f2 is equal to one. As x2 > x1, both of the integral limits will be smaller than one for interevent times Δτ < ΔτUL=1. With interevent p 1−1/p −x times well below c /K, the f2 integrand x · e can be approximated by x1−1/p as e−x rapidly approaches the value of one when x approaches the value of zero. We also consider the interevent times to be much smaller than the time period T. In the function f2, we can therefore use x1 given by equation 2.28 as a starting point of the integration, regardless of the choice of integration range in the original equation for the probability density function. Performing the integration in f2 with these approximations yields: p 2−1/p (c) − / K 1 1 f Δτ |T = −Δτ2 1 p · · − (2.31) 2 K 2 − 1/p (c + T)2p−1 c2p−1

Equation 2.31 shows that f2 eliminates the power law decay with exponent p 2 − 1/p of the function f1 for interevent times smaller than c /K. In figure 2.3 b), the upward sloping part of the function f2 at short interevent times therefore represents a power law increase with exponent 2 − 1/p. The Omori law interevent time probability density function thus approaches, asymptotically, a constant value for short interevent times. Multiplying equation 2.31 with f1 and simplifying yields an equation for the constant short interevent time asymptote of the probability density function: (c)p K2 1 1 f Δτ |T = · − K (N(T) − 1) · (1 − 2p) (c + T)2p−1 c2p−1 (2.32) In an interval around ΔτUL=1, increasing interevent times implies a transition from x1−1/p to e−x as the dominant factor in the integrand x1−1/p · e−x of the function f2. A suitable estimate of the interevent time where the short interevent time asymptote of the probability density function ends is therefore given by ΔτUL=1. For our Omori law example, figure 2.3 a) indeed shows that this is the case. Accordingly, figure 2.3 b) shows that ΔτUL=1 represents a

28 good estimate of the end of the upward sloping regime of f2 that is responsible for eliminating the power law decay of the function f1. When we now consider interevent times larger than ΔτUL=1, we also need to consider the lower integral limit (x1) in the function f2. When the lower limit is given by λ(T) · Δτ (equation 2.28), it increases linearly with Δτ, with a rate given by K/(c + T)p. The lower limit then equals one when: (c + T)p Δτ = = (2.33) LL 1 K where ΔτLL=1 is the interevent time where x1 equals one. For values of Δτ < ΔτLL=1, the lower limit is below one and approaches zero as Δτ → 0. Note that the rate of increase of x1 is smaller than the rate of increase of the upper limit x2. Moreover, the interevent time where the lower limit becomes equal to one is larger than that of the upper limit (i.e. ΔτLL=1 > ΔτUL=1). When using the lower limit given by λ(T −Δτ)·Δτ (equation 2.29), x1 has a non-linear dependence on Δτ. In order to determine the interevent time where x1 is equal to one, the equation λ(T − Δτ) · Δτ = 1 has to be solved numerically. Analytically, however, we can ﬁnd an interval containing the interevent time where x1 goes from being less than one to being larger than one. As an upper boundary of this interval, we choose: (c + T)p ΔτU = (2.34) LL K

ΔτU ΔτU Plugging LL into equation 2.29, we can show that LL yields a value for the lower integration limit x1 that is larger than one: K (c + T)p x = · ΔτU = > 1 (2.35) 1 ( + − ΔτU )p LL ( + − ΔτU )p c T LL c T LL

ΔτU Having calculated LL, we now choose a lower boundary of the interval as: (c + T − ΔτU )p (c + T)p · (K − (c + T)p−1)p ΔτL = LL = (2.36) LL K K p+1

ΔτL It can be shown that this choice for LL yields a value for the lower integration limit x1 that is less than one: K (c + T − ΔτU )p x = · ΔτL = LL < 1 (2.37) 1 ( + − ΔτL )p LL ( + − ΔτL )p c T LL c T LL

[ΔτL , ΔτU ] Δτ Having found the interval LL LL , the interevent time LL=1 where the lower limit in equation 2.29 equals one can then be found numerically. Knowing ΔτUL=1 and ΔτLL=1, we are now concerned with the behaviour of the function f2 for interevent times ΔτUL=1 Δτ ΔτLL=1. From equation 2.27, we can see that the upper integration limit x2 becomes larger than

29 p one for interevent times larger than ΔτUL=1 = c /K. At the uppermost end of the integration range, the integrand x1−1/p · e−x can therefore be approx- −x imated by e for interevent times Δτ ΔτUL=1. This implies that the additional contribution to the value of the integral by increasing x2 (i.e. increasing the interevent time), rapidly becomes more and more insignificant with increasing interevent time. From equation 2.28 or 2.29, we can see that the lower integration limit x1 becomes smaller than one for interevent times shorter than ΔτLL=1. At the lowermost end of the integration range, the f2 integrand x1−1/p · e−x can therefore be approximated by x1−1/p for interevent 1−1/p times Δτ ΔτLL=1. The behaviour of x is characterised by a high rate of change for very small values of x, a rapid reduction in the rate of change with increasing values of x and that x1−1/p = 1 when x = 1. This behaviour implies that the reduction in the value of the integral by increasing x1 (i.e. increasing the interevent time) is relatively small for short to intermediate interevent times. Altogether, the behaviour of the f2 integrand at the upper and lower integration limits implies that the function f2 is nearly constant for interevent times ΔτUL=1 Δτ ΔτLL=1. For interevent times satisfying these conditions, the function f2 will therefore only modify the function f1 through a multiplication with a nearly constant scaling factor. For non-zero values of c, this implies that the power law decay of the probability density function is restricted by lower and upper bounds. The power law decay exponent, 2 − 1/p, corresponds to the result of Senshu (1959). In figure 2.3 b), we can indeed see that the function f2 for our Omori law example is asymptotically constant for interevent times ΔτUL=1 Δτ ΔτLL=1. As expected, the corresponding interevent time probability density function in figure 2.3 a) decays asymptotically as a power law in this interevent time interval. Finally, we consider interevent times Δτ > ΔτLL=1. In the same way as for ΔτUL=1, there is an interval around ΔτLL=1 where there is a transition from x1−1/p to e−x as the dominant factor in the integrand x1−1/p · e−x of the function f2. In our Omori law example, figure 2.3 b) shows that this transition may take place over a large range of interevent times. The transition to e−x as the dominant factor is the reason for the falloff in the function f2 that we can see in figure 2.3 b). At interevent times larger than ΔτLL=1, both the lower and the upper integral limits x1 and x2 in the function f2 are larger than one, and the integrand can be approximated by e−x. Integrating e−x over such a range implies that the value of the function f2 will drop rapidly as the interevent time increases. In figure 2.3 b), we can see that the function f2 falls off more dramatically when the lower limit x1 is given by λ(T − Δτ) · Δτ (equation 2.29) rather than by λ(T) · Δτ (equation 2.28). The reason for this is that λ(T − Δτ) · Δτ is, for the same interevent time, always larger than λ(T) · Δτ. The starting point of the integration in f2 is therefore larger and the integration −x range thus shorter if x1 = λ(T − Δτ) · Δτ.Ase also becomes smaller with increasing values of x, the larger starting point explains why the value of f2 drops more dramatically when x1 equals λ(T −Δτ)·Δτ rather than λ(T)·Δτ.

30 In our Omori law example (figure 2.3 a)), this is reflected in the more dramatic falloff in the interevent time probability density function when the integration is made to T − Δτ in equation 2.15. This behaviour is realistic, considering that integration to T −Δτ implies a probability density function for interevent times that can be observed in a time interval T. A dramatic drop in the probability density function at the largest interevent times is therefore natural as these have very low or no probability to be observed. How fast the falloff takes place, however, will depend on the transition of dominance from x1−1/p to e−x which, in turn, depends on the values of K, c, p and T. We have derived an expression for the Omori law interevent time probability density function (equation 2.15) and analysed its scaling behaviour. We have considered two cases: 1) the interevent time distribution that can be observed during a timeperiod T and 2) the interevent time distribution that can be expected from the activity occurring until time T. To summarise, the Omori law interevent time probability density function is asymptotically constant in the range [0, (c)p/K], it decays asymptotically as a power law with expo- p nent 2 − 1/p in the range [c /K, ΔτLL=1] and falls off rapidly for interevent times larger than ΔτLL=1. With a zero value of the Omori law parameter c, the asymptotically constant regime of the probability density function van- ishes, and the power law decay extends to infinitely small interevent times. Between case 1 and case 2, the only differences lie in the value of ΔτLL=1 and the falloff above ΔτLL=1. For case 1, ΔτLL=1 is located within the in- p p−1 p p+1 p terval [ (c + T) · (K − (c + T) ) / K , (c + T) /K] while ΔτLL=1 = p (c + T) /K for case 2. For case 1, ΔτLL=1 is therefore smaller than for case 2. For interevent times larger than ΔτLL=1, the case 1 interevent time probability density function may fall off more rapidly than for case 2, depending on the values of K, c, p and T.

2.2.2.2 Approach 2 Based on an inhomogeneous Poisson process, an equation for the Omori law interevent time distribution is given by Shcherbakov et al. (2005). For clarity, I will show some more details of the derivation than given in their paper. In an inhomogeneous Poisson process, the number of events in a time interval [a, b] has a distribution defined by: k − b λ( ) · b λ( ) exp a t dt a t dt P[(N(b) − N(a)) = k]= ,k = 0,1,2,.... k! (2.38) where N(a) and N(b) are the number of events that have occurred before time a and time b, respectively, and λ(t) is the time dependent rate. The interevent time distribution for the first event (the time between the main shock and the first aftershock) is: Δτ P(T1 > Δτ)=P(N (Δτ)=0)=exp − λ(t)dt (2.39) 0

31 When λ(t) is given by the Omori law (equation 2.4), it can be shown that the corresponding probability density function is given by: Δτ f (Δτ|t = 0)=λ (Δτ) · exp − λ(t)dt (2.40) 0

The interevent time distribution between subsequent events with numbers k and k + 1 is deﬁned by: +Δτ Tk P(Tk+1 − Tk > Δτ)=exp − λ(t)dt (2.41) Tk where Tk and Tk+1 is the occurrence time of events k and k + 1, respectively. With λ(t) given by the Omori law (equation 2.4) and Tk = t, it can be shown that the corresponding probability density function is: t+Δτ f (Δτ|t)=λ (t + Δτ) · exp − λ (t)dt (2.42) t

If we consider that λ(t)dt approximates the number of events during a short time interval dt, we have a probability density function for interevent times at the speciﬁc interval given by: u =t+Δτ (Δτ| , )= 1 · λ ( ) · λ ( + Δτ) · − 2 λ ( ) · f t dt ( ) − t t exp u du dt N T 1 u1=t (2.43) where N(T) is, as in approach 1, the expected number of aftershocks occurring until time T (the length of the period of observation of the aftershock sequence). The number of aftershocks, N(T), can be calculated using equation 2.5 or 2.6. In the approach of Shcherbakov et al. (2005), they integrate equation 2.43 between time t = 0 and t = T −Δτ. As discussed in approach 1, this yields an expression for the interevent time probability density function that can be observed during a time period T. Here, we also discuss the case when integrating to T, yielding an expression for the interevent time probability density function that can be expected from the activity occurring until time T. The interevent time probability density function is: 1 f (Δτ|T)= · N(T) − 1 TorT−Δτ u2=t+Δτ · λ (t) · λ (t + Δτ) · exp − λ (u)du · dt 0 u1=t (2.44)

As for the probability density function derived using approach 1, an analytical solution to equation 2.44 exists for p = 1. By using the following relation,

32 valid for all values of p: − − / / p λ(t + Δτ)=λ(t) · 1 + Δτ · K 1 p · λ(t)1 p (2.45) it can be shown that the analytical solution of equation 2.44 for p = 1 when integrating to T is: 1 K c + T K c K f (Δτ|T, p = 1)= · · − (2.46) N(T) − 1 Δτ c + T + Δτ c + Δτ and when integrating to T − Δτ: 1 K Δτ K c K f (Δτ|T, p = 1)= · · 1 − − (2.47) N(T) − 1 Δτ c + T c + Δτ

Equation 2.44 describes the probability density function for interevent times between aftershocks and does not include the probability density function for the interevent time between the main shock and the ﬁrst aftershock. Including this probability density function (equation 2.40) yields the following equation: 1 f (Δτ|T)= · N(T) TorT−Δτ u2=t+Δτ · λ (t) · λ (t + Δτ) · exp − λ (u)du · dt + = 0 u1 t Δτ + λ (Δτ) · exp − λ(t)dt 0 (2.48)

Including the interevent time between the main shock and the first aftershock yields the same number of interevent times as aftershocks in the sequence. In the probability density function above, the scaling factor is therefore 1/N(T) rather than 1/(N(T) − 1). Equation 2.48 corresponds to equation 2 of Shcherbakov et al. (2005) when the upper limit of integration is T − Δτ. For our Omori law example with K = 50, c = 1500 and p = 1.2, figure 2.4 a) shows that the interevent time probability density function derived using approach 2 (equation 2.44) is asymptotically constant at short interevent times and decays as a power law for intermediate interevent times. Comparing with figure 2.3 a), approach 1 and 2 probability density functions behave in the same way for short and intermediate interevent times. At larger interevent times, however, the falloff is more rapid in approach 1 than in approach 2. In our example, this is especially clear for the case when the integration in the probability density function is to T in both approach 1 and approach 2. Using approach 2, there is then only a small change in the slope of the probability

33 density function at larger interevent times. Approach 1, however, results in a more distinct falloff from the power law behaviour. I will now show that equation 2.44 exhibits the same scaling behaviour at short and intermediate interevent times as equation 2.15 for the Omori law interevent time probability density function which is derived using approach 1. I will also discuss why the difference between the approach 1 and approach 2 probability density functions is ﬁrst noticed at larger interevent times. The Omori law probability density function derived using approach 2 can, as in approach 1, also be written as a multiple of two functions. Performing a change of variables from time t to x = λ(t) · Δτ in the integral in equation 2.44 and using equation 2.45, we eventually obtain:

f (Δτ|T)=g1(Δτ) · g2(Δτ|T) (2.49) where the functions g1 and g2 are: K1/p 1 g (Δτ)= · (2.50) 1 p · (N(T) − 1) Δτ2−1/p

g2(Δτ|T)= x (Δτ) − 2 − / − / − / / p = x1 1 p · 1 + K 1 p · Δτ1 1 p · x1 p · (Δτ, ) x1 T 1/p u K 2 − / · exp − · u 1 p · du · dx (2.51) · Δτ1−1/p p u1 where x1 is given by equation 2.28 or 2.29 and x2 by equation 2.27. The integral limits u1 and u2 in the exponential term in equation 2.51 are given in terms of the main integration variable x as: −p −1/p 1−1/p 1/p u1 = x · 1 + K · Δτ · x (2.52)

u2 = x (2.53)

By comparing the functions g1 and g2 with the functions f1 (equation 2.25) and f2 (equation 2.26) in approach 1, we can immediately see that f1 and g1 are identical while g2 is a more complex function than f2. Comparing g2 (ﬁgure 2.4 b)) with f2 (ﬁgure 2.3 b)) for our Omori law example with K = 50, c=1500 and p = 1.2, however, we can see that g2 only differs from f2 at the largest interevent times, where the respective functions falls off. In the following discussion, we shall see that this agreement between g2 and f2 at short and intermediate interevent times is a general behaviour. As in approach 1, we need to investigate the behaviour of the integrand in g2 in conjunction with the limits of the main integral. These limits are

34 a) Interevent time pdf, approach 2 2 10

0 Δτ 10 LL=1

−2 10 −>T−Δτ −>T

) −4 10 Δτ −6 f( 10

−8 Δτ 10 UL=1

−10 10

0 2 4 6 8 10 10 10 10 10 Interevent time, Δτ [s] b) Functions g and g , approach 2 1 2 2 10

0 ) 10 Δτ

( −2

2 10

), g −4 10 g

Δτ 1 (

1 g −> T −6 2 g 10 g −> T−Δτ 2 −8 10 0 2 4 6 8 10 10 10 10 10 Interevent time, Δτ [s] Figure 2.4: a) Illustrating the Omori law interevent time probability density function derived using approach 2 (equation 2.44) when K = 50, c = 1500, p = 1.2 and T = 180 days. Dashed curve: Probability density function when integrating to T. Solid curve: Probability density function when integrating to T − Δτ. Thin vertical lines: Bound- aries between the regimes of the probability density function. ΔτUL=1: Boundary between the constant part and the power law part. ΔτLL=1: Boundary between the power law part and the falloff part. The location of ΔτLL=1 depends on whether integration is done to T or T −Δτ in the probability density function (equation 2.44). The labelling of these boundaries is explained in the text. b) A separation of the probability density function in a) into two functions g1 (dotted curve) and g2 (solid/dashed curves). g1 is responsible for the power law behaviour in the probability density function while g2 is responsible for the constant part and the falloff. Note that the falloff behaviour of g2 depends on whether integration is to T (dashed curve) or to T −Δτ (solid curve) in the probability density function (equation 2.15). The thin vertical lines represent the same interevent times as in a).

35 identical with the integral limits of f2 (equation 2.26). The interevent times where the integral limits in g2 are equal to one are therefore the same as for approach 1, i.e. ΔτLL=1 (lower limit) and ΔτUL=1 (upper limit). If the lower limit of integration is x1 = λ(T) · Δτ then ΔτLL=1 equals (c + T)p/K (equation 2.33). Otherwise, if the lower limit of integration is x1 = λ(T − Δτ)· Δτ, ΔτLL=1 is then located within the interval [ (c + T)p · (K − (c + T)p−1)p / K p+1 , (c + T)p/K]. For non-zero values p of c, ΔτUL=1 equals c /K (equation 2.30). We consider a non-zero value of c and interevent times Δτ ΔτUL=1.For these interevent times, both of the integral limits x1 and x2 in the integral in − −1/p 1−1/p 1/p p g2 are below one. It can be shown that the term 1 + K · Δτ · x in the integrand of g2 approaches the value of 1 when Δτ ΔτUL=1 and the value of x approaches the value of zero. In the exponential term of the inte- 1/p K u2 −1/p grand of g , − / · u · du represents the number of events in an 2 p·Δτ1 1 p u1 interval of duration Δτ. By evaluating this expression with the integral limits u1 and u2 given by equations 2.52 and 2.53 and plugging in x = λ(t) · Δτ we get an expression for the number of events in the time interval [t, t + Δτ]. When Δτ ΔτUL=1 and the value of x approaches zero, it can be shown 1/p K u2 −1/p that exp − − / · u · du approaches exp(−0)=1 as the number p·Δτ1 1 p u1 of events then approaches zero. For interevent times Δτ ΔτUL=1, the inte- 1−1/p grand of g2 can therefore be approximated by x , yielding an expression Δτ (c)p | Δτ (c)p | for g2 K T that is identical to the expression for f2 K T (equation 2.31). This implies that the function g2 eliminates the power law decay with exponent 2 − 1/p of the function g1 for interevent times that are shorter than cp/K. An equation for the constant short interevent time asymptote of the probability density function derived using approach 2 is therefore also given by equation 2.32. In figure 2.4 b), the upward sloping part of the function g2 represents a power law increase with exponent 2 − 1/p. For interevent times ΔτUL=1 Δτ ΔτLL=1, the g2 integral limit x1 is smaller than one while the limit x2 is larger than one. This 1−1/p implies that x is a good approximation of the integrand in g2 at the start of integration (x1) but not at the end (x2). For interevent 1−1/p times in an interval around ΔτUL=1, there is a transition from x to − 1/p −1/p 1−1/p 1/p p K u2 −1/p 1 + K · Δτ · x · exp − − / · u · du as the p·Δτ1 1 p u1 dominant factors in the integrand. Both of these factors will approach the value of zero when interevent times are larger than ΔτUL=1, which implies that the value of the integrand will begin to decay to zero. The additional contribution to the value of g2 by increasing the upper limit x2 (i.e. increasing the interevent time) therefore rapidly becomes more and more insignificant with increasing interevent time. Increasing the lower integration limit x1 (by increasing the interevent time), however, does not result in a significant reduction in the value of g2 as long as x1 1. As for the function f2, the

36 reason for this is the behaviour of x1−1/p which implies that increasing x1 (i.e. increasing the interevent time) only results in a relatively small reduction in the value of the integral for short to intermediate interevent times. For interevent times in the range ΔτUL=1 Δτ ΔτLL=1, the function g2 is therefore only modifying the function g1 by a near constant scaling factor. The Omori law interevent time probability density function derived using approach 2 therefore also decays asymptotically as a power law with exponent 2 − 1/p, restricted by lower and upper bounds. Figure 2.4 b) shows that g2 for our Omori law example is near constant for interevent times between ΔτUL=1 and ΔτLL=1. Accordingly, the corresponding probability density function in figure 2.4 a) decays as a power law in this interevent time range. So far, the behaviour of g2 has not been different from the behaviour of f2. I will now discuss interevent times Δτ > ΔτLL=1 and explain why the falloff behaviour of the two functions can be different. In an interval around− Δτ 1−1/p + −1/p · Δτ1−1/p · 1/p p · LL=1, there is a transition from x to 1 K x 1/p K u2 −1/p exp − − / · u · du as the dominant factors in the integrand of p·Δτ1 1 p u1 g2. With interevent times Δτ > ΔτLL=1, the exponential factor will become the dominant of these two. As both x1 and x2 are now larger than 1, the in- 1/p K u2 −1/p tegrand in g can be approximated by exp − − / · u · du .Now, 2 p·Δτ1 1 p u1 remember that the term of the exponential represents the negative of the number of events in an interval of duration Δτ when the event rate decays with time according to the Omori law. In approach 1, the term of the exponential represents the negative of the number of events in an interval of duration Δτ when the event rate in the interval is constant and given by the Omori law rate at the start of the interval. With all things equal, i.e. the same time interval and interevent time Δτ > ΔτLL=1, approach 1 may, depending on the values of K, c and p, give a larger negative term in the exponential of f2 than approach 2 gives in the exponential of g2. For interevent times larger than ΔτLL=1, the value of f2 may therefore be lower than the value of g2. This is the case for the choice of K = 50, c = 1500 and p = 1.2 in our Omori law example and explains why the falloff in the probability density function at large interevent times is not as dramatic in approach 2 as in approach 1. This is especially clear in the case when integration is to T rather than T − Δτ (compare figure 2.4 b) with figure 2.3 b)). To summarise, we have derived and investigated an expression for the Omori law interevent time probability density function based on an inhomogeneous Poisson process as proposed by Shcherbakov et al. (2005). As in approach 1, the interevent time probability density function derived in approach 2 is asymptotically constant in the range [0, (c)p/K], decays asymptotically as a power law with exponent 2 − 1/p in the range p [c /K, ΔτLL=1] and falls off for interevent times larger than ΔτLL=1. If the in- p tegration in equation 2.44 is to T, the value of ΔτLL=1 is equal to (c + T) /K.

37 If the integration is to T − Δτ, however, the value ofΔτLL=1 is located within the interval [ (c + T)p · (K − (c + T)p−1)p / K p+1 , (c + T)p/K].

2.2.2.3 Discussion of approach 1 and 2 The Omori law interevent time probability density function of Shcherbakov et al. (2005) exhibits the same behaviour at short and intermediate interevent times as the expression we derived in approach 1 but may, depending on the values of K, c and p, have a different falloff behaviour at large interevent times. In both expressions, however, the interevent times (ΔτUL=1, ΔτLL=1) where the probability density function changes its behaviour scale in the same way with the Omori law parameters K, c, p and the time period of observation T. Regardless of which expression it is that best describes the behaviour at large interevent times, the common scaling points and behaviour at short and intermediate interevent times are important for the discussion in Paper I, II and III of this thesis. In fact, it may not be possible to determine whether it is approach 1 or 2 that correctly describes the interevent time distribution of real aftershock sequences at large interevent times. As our analysis shows, the theoretical Omori law interevent time probability density function for interevent times that can be observed during a time period T falls off dramatically for interevent times larger than ΔτLL=1. This falloff implies a very low probability of observing, in practice, interevent times in the range where it would be possible to distinguish whether it is approach 1 or approach 2 that is more correct. In my thesis, I make conclusions based on the Omori law interevent time probability density function which is derived using approach 1. Our analysis shows that these conclusions are not violated by the formulation of Shcherbakov et al. (2005), which is derived using approach 2 in my thesis.

2.3 Earthquakes and self-organised criticality In section 2.1 of this chapter, I have discussed three typical distributions associated with earthquakes. First, we have the Gutenberg-Richter law which implies a power law distribution of the seismic moment released in earthquakes. Thereafter, we have the Omori law which, with c = 0, implies a power law distribution of aftershock occurrence times. Finally, we have the spatial distributions of faults and earthquake locations whose features appear to be scale- invariant and fractal. The theory of self-organised criticality (SOC), proposed by e.g. Bak et al. (1987) and Bak et al. (1988), is an attempt to provide an explanation for power law distributions of properties associated with a wide variety of natural systems, including the Earth’s crust where earthquakes occur. A system is considered to be in a state of self-organised criticality if its evolution into a critical state is neither dependent on initial conditions nor on tuning of parameters of the system (Bak et al., 1988). In the theory of SOC, the

38 critical state is characterised by the ability of local interactions to accumulate in order to generate long-range interactions throughout the system (Kanamori and Brodsky, 2004). This ability implies that sizes of events that occur in the system will be limited only by the size of the system itself (Bak and Tang, 1989). The distributions of properties measuring sizes of events will therefore have the form of power laws at the critical state as power laws can result from long-range interactions only (Ito and Matsuzaki, 1990). As opposed to other type of distributions, the critical state is therefore characterised by a lack of a characteristic scale in the distributions of the relevant properties (Bak and Tang, 1989).

2.3.1 Self-organised criticality, the Gutenberg-Richter law and earthquake models In seismology, the widespread applicability of the Gutenberg-Richter law is considered as the primary argument supporting the hypothesis that the Earth’s crust is in a state of self-organised criticality (e.g. Bak and Tang, 1989; Ito and Matsuzaki, 1990). Most papers concerning self-organised criticality in relation to earthquakes therefore deal with earthquake models resulting in a power law distribution of earthquake sizes. For this reason, I will discuss general and common features of three typical models of earthquakes used in modelling of self-organised criticality, i.e. : slider block models (Burridge and Knopoff, 1967), percolation models (Kanamori and Brodsky, 2001) and sand-pile or cellular automata models (e.g. Bak and Tang, 1989). These models generally consist of a 1-D, 2-D or 3-D matrix of elements that is loaded at discrete time steps, with a regular rate representing the long- term geological rate of loading from plate tectonic motions. The time scale of the loading process is therefore assumed to be very large (Bak and Tang, 1989), of the order of ∼100-1000 years (Ito and Matsuzaki, 1990). At each time step, the loading typically consists of a discrete, and ﬁxed, increase in the level of stress for a randomly selected element of the system (e.g. Bak and Tang, 1989). When the level of stress for an element exceeds a failure threshold, that may be globally valid for all elements, the accumulated stresses are redistributed to the nearest neighbours according to rules that may also be globally valid for the entire system. Depending upon the state of stress of the neighbouring elements, these may also fail, possibly initiating a sequence of failures. In the real world, this sequence would represent the propagation of rupture during an earthquake. As the rupture propagation velocity is in the order of ∼ km/s, this implies that the duration of each failure sequence in these models is very short when compared to the time steps of the loading. Ito and Matsuzaki (1990) consider each sequence of failures to be an instantaneous process. Each sequence can then be referred to as an event, having a size given by the number of elements that failed. Considering each element to represent

39 a fault patch with a certain area, the number of failed elements is therefore an analogue to the rupture area and thus the seismic moment of a real earthquake. Slider block, percolation and sandpile models all give power law distributions of event sizes (Kanamori and Brodsky, 2004). According to the theory of self-organised criticality, this implies that the system, through the slow loading, has evolved into a critical state where local perturbations in the level of stress may trigger events of all sizes, limited only by the size of the system (Bak et al., 1987; Bak and Tang, 1989). The self-organised critical state is said to be statistically stationary with respect to time (e.g. Bak et al., 1987, 1988) and to exhibit power law spatial and temporal correlation functions (Bak and Tang, 1989). The stationarity implies that the system, when perturbed away from the critical state, naturally evolves back into the critical state (Turcotte, 1997). The average level of stress in the system therefore reaches a statistically stationary value (Bak and Tang, 1989). The power law spatial and temporal correlation functions represent the distribution of event sizes (i.e. the number of failed elements), and the distribution of event durations (i.e. the total time for the sequence of failures that constitutes an event) (see e.g. Bak et al., 1988). From the observation of a power law distribution of event sizes, the theory of self-organised criticality thus says that the critical state is characterised by a possibility of long-range spatial interaction from element to element during the rupture of an event. In the same way, a power law distribution of event durations says that the critical state is characterised by a possibility of a long-range temporal correlation between the first and the last elements that rupture in an event. In the spatial interaction between failuring elements, the redistribution of stresses modifies the time to failure of surrounding elements. Through the failure of the intermediate elements, the failure of the last element can therefore be said to be spatially and temporally correlated with the failure of the first element of each event. Note that the spatial correlation can extend over large distances in these models, implying that the sizes of events is only limited by the size of the system. In reality, however, earthquake magnitudes are limited by the size of the seismogenic zone. Moreover, note that the temporal correlation is only valid within the time scale of rupture (∼seconds to minutes) and not the time scale associated with aftershock sequences (∼months to years) or the geological loading rate of the models (∼thousands of years).

2.3.2 Self-organised criticality and the Omori law In my thesis, I am particularly interested in the time scale associated with aftershock sequences and the implications of the Omori law for self-organised criticality as a model for the occurrence of earthquakes. The scale-invariant power law features of the Omori law for aftershocks as well as the fractal distribution of faults and earthquake locations are considered as additional

40 support of the hypothesis that earthquakes are a result of a self-organised critical state of the Earth’s crust (see e.g. Ito and Matsuzaki, 1990; Corral, 2003). It is important to note, however, that the Omori law is not a pure power law over the entire domain of aftershock occurrence times. A non-zero value of the parameter c in the Omori law implies a power law decay in the rate of aftershocks only for occurrence times t c. A non-zero value of c does imply a characteristic scale in the distribution of aftershock occurrence times, break- ing the scale-invariance between small and large occurrence times. Later, I will discuss the implications of this for aftershock interevent time distributions (section 2.3.3). Let us first consider what the Omori law says if we interprete it in terms of self-organised criticality. Remember that a characteristic feature of earthquake models in a self-organised critical state is the initiation of a sequence of local spatial interactions that accumulate into an event. Each of the events that occur consists of a number of failures and has a duration controlled by the rupture velocity. As an analogue, we can interpret an aftershock sequence as one event, where the individual aftershocks represent the individual element failures. The number of aftershocks in the sequence would then represent the size of the event while the difference in occurrence time between the last aftershock and the main shock would represent the duration of the event. Upon failure, each aftershock will modify the time to failure of its near surroundings through the redistribution of stresses, eventually resulting in the occurrence of the next aftershock. The modification of the time to failure must be time dependent and non-linear in order for the Omori law to be obtained. The last aftershock to occur in the sequence can be said to be spatially and temporally correlated with the mainshock through the spatial interaction of the intermediate aftershocks. This is a direct analogue to what happens during an event in sandpile or slider block models of earthquakes (section 2.3.1). The interpretation of an aftershock sequence as one event implies that it is not enough to observe one aftershock sequence to draw the conclusion that the Omori law is a result of self-organised criticality. Instead, it would be necessary to infer whether the statistics of aftershock sequence durations or the total number of aftershocks from many aftershock sequences yield power law distributions. If this was the case, the statistics of aftershock sequences would not contradict the requirements in self-organised criticality of scale invariance and power law behaviour. Using empirical data for such an analysis requires, however, the observation of hundreds or thousands of aftershock sequences in order to yield meaningful statistics. In practice, this is virtually impossible as this requires access to many high quality datasets from all over the world and a very long period of observation. A possible solution for this problem is to run simulations of self-organised critical earthquake models that have been modified to yield Omori law aftershock behaviour. Ito and Matsuzaki (1990) modified a sandpile model to allow for the spatial distribution of aftershocks within main shock rupture zones and

41 show, mathematically, that their model yields a relaxation formula in agreement with the Omori law. Unfortunately, it cannot be verified that the model actually results in Omori law behaviour as timing of individual aftershocks is not included in the actual modelling. The model can, however, be used to study whether or not the total number of aftershocks in the aftershock sequences obtained with the model has a power law distribution. If this was the case, it would neither contradict nor prove that self-organised criticality may be the mechanism behind aftershocks. Hainzl et al. (2000) explicitly model aftershock occurrence times by including the contribution of transient postseismic creep in the loading of the crust following the occurrence of main shocks. With this relatively simple modification of a sandpile model for earthquakes, both the Gutenberg-Richter law for the distribution of earthquake magnitudes and the Omori law for the distribution of aftershock occurrence times are reproduced. Based on their results, Hainzl et al. (2000) conclude that a model of the Earth’s crust only involving elastic interactions in combination with postseismic transient creep self-organises into a statistically stationary state, as characterised by the Gutenberg-Richter and the Omori laws. The work of Hainzl et al. (2000) is indeed very interesting as it shows that a very simple model can reproduce the features of the Gutenberg-Richter and the Omori laws. Their results, however, should not be directly interpreted as evidence that the Earth’s crust is in a self-organised critical state, characterised by power laws. Firstly, the aftershock occurrence time distributions they have obtained show that the value of c must be non-zero as they only exhibit power law decay after a certain time from the main shock. Regardless of how one considers the Omori law, a power law which is limited to a certain range violates the requirement of self-organised criticality for pure power laws. Se- condly, if we consider an aftershock sequence as one event, the statistics of aftershock sequence durations and total number of aftershocks should exhibit power law distributions in order for the behaviour to be interpreted in terms of self-organised criticality. Hainzl et al. (2000) do not present such statistics for their modelling results. Thirdly, and perhaps most crucially, the authors take a step away from the fundamental characteristic of SOC-models which is that interactions only affect the system at one time, i.e. at the time of stress redistribution. Introducing a time dependent stress transient from each earthquake implies a higher degree of predictability of the model as the behaviour of the stress transient is known. The fundamental aspect of models resulting in self-organised criticality is that the model statistics cannot be predicted by the interaction rules in the models. In a model which incorporates time decaying postseismic stress transients, it is not surprising that activity rates which decay with time as in the Omori law can be obtained. It can therefore be questioned if this is a result of a self-organised critical state or whether it simply reflects the nature of the stress transients. Regardless of the conclusion, the work of Hainzl et al. (2000) shows that a physical model for the aftershock process, in

42 their case postseismic transient creep, is useful in providing a possible explanation for what we actually observe in nature.

2.3.3 Self-organised criticality and aftershock interevent time distributions When studying the distribution of aftershock interevent times, i.e. the time interval between two successive events in an aftershock sequence, the connection with the actual occurrence times of the events is lost. Just as an earthquake is associated with a measure of its magnitude, it can also be associated with a measure of its interevent time, i.e. the time that has passed since the previous earthquake occurred. Both the earthquake magnitude and the interevent time share the property that there is no link to the actual occurrence time of the earthquake. A power law distribution of earthquake magnitudes therefore only says that the actual distribution is what has been observed during a time period T. Similarly, a power law distribution of aftershock interevent times only says that this distribution has been observed during a time period T. Since there is no link to the occurrence time of events, only observing the interevent time distribution cannot be used to infer whether a short interevent time is associated with an early or late part of an aftershock sequence. From a view point of self-organised criticality, this implies that a power law interevent time distribution can be interpreted as an indication of a state where interevent times of all time scales can be initiated at any time. We must note, however, that long interevent times are correlated with late times as the rate of aftershocks decay with time. Aftershock interevent times of all time scales cannot, therefore, be initiated at any time in sequences of aftershocks. In section 2.2, we carefully studied the characteristics of interevent time distributions associated with Poisson processes as well as aftershock sequences obeying the Omori law. The interevent time probability density function for a Poisson process is given by μ · exp(−μΔτ), where μ is the constant underlying rate of events. For Omori law aftershock sequences, the associated interevent time probability density function is asymptotically constant in the range [0, (c)p/K], it decays asymptotically as a power law with exponent 2 − 1/p p in the range [c /K, ΔτLL=1] and it falls off for interevent times larger than ΔτLL=1. The value of ΔτLL=1 is primarily controlled by the length of the period of observation, T.Ifc = 0, the power law behaviour starts at Δτ = 0, otherwise at (c)p/K. Bak et al. (2002) concluded that a characteristic kink at large interevent times in area-aggregated interevent time distributions from California separates correlated and uncorrelated earthquakes, belonging to the same aftershock sequence. In my thesis, Paper II is a comment on the work of Bak et al. (2002), arguing that the characteristic kink does not represent a change in correlation between individual earthquakes. Instead, we argue that the characteristic kink represents the effect of ﬁnite aftershock time series. This ﬁniteness

43 is either due to a finite period of observation or to swamping of the time series caused by aftershock sequences of later main shocks. Our argument was later opposed by Corral and Christensen (2006), criticising that we did not consider area-aggregated interevent time distributions. Paper III consists of our reply to the comment of Corral and Christensen (2006). There, we show that aggregating Omori law interevent time distributions yields the shape of the interevent time curves presented by Bak et al. (2002). This implies that our interpretation of the kink is correct; it is only the effect of a finite period of observation of individual aftershock sequences. This interpretation has the important implication that the time interval between two subsequent earthquakes cannot be used to define them as either correlated or uncorrelated, i.e. belonging to the same aftershock sequence or not. For the full discussion of these issues, we refer to the papers themselves or their summaries in chapter 7. In the interevent time distributions presented by Bak et al. (2002), their figure 3 shows a characteristic kink also at short interevent times, separating an initially constant part from the power law decay regime. We have shown that the initially constant part reflects a non-zero value of c in the Omori law. The implication of this is that the interevent time distributions observed by Bak et al. (2002) do exhibit a characteristic scale at short interevent times which is not related to the finiteness of the system. Bak et al. (2002) argue, however, that this feature only reflects the inability of resolving individual earthquakes with interevent times shorter than 40 seconds. In Paper IV of this thesis, we conclude that an incomplete detection of aftershocks in Icelandic aftershock sequences does not fully explain the observed deviation from the power law decay at short interevent times. If a non-zero value of c does indeed reflect true properties of the aftershock process, the requirements of self-organised criticality for scale invariance and the lack of a characteristic scale are violated. If this is the case, it implies that interevent time distributions of aftershock sequences cannot be used to support the hypothesis of a self-organised critical state of the Earth’s crust. Pure power law interevent time distributions have also been used to argue against self-organised criticality as the underlying mechanism. Boffetta et al. (1999) argue that successive events in randomly driven sandpile systems are expected to be uncorrelated and to show Poissonian rather than power law interevent time statistics in the self-organised critical state. The reason for this is that the probability of triggering an event, due to the random loading at each time step, approaches a constant value as the steady state associated with self-organised criticality is obtained. Successive events will then be uncorrelated as the probability of triggering ceases to change with time. Boffetta et al. (1999) thus argue that the correlation between events expressed by power law interevent time distributions cannot be associated with a self-organised critical state as the underlying mechanism. In a nonconservative spring block model of earthquakes, however, Christensen and Olami (1992) find power law interevent time distributions only for events larger than a certain size. This

44 implies that self-organised criticality may still be the underlying mechanism as undetected events, due to the detection threshold of the measuring system, may account for the observed power law behaviour. Sánchez et al. (2002) shows that both random and correlated driving result in power law distributions of event durations that are indistinguishable from each other. The randomly driven system, however, results in a Poissonian interevent time distribution while the correlated drive results in a power law distribution. Sánchez et al. (2002) conclude that even if the measuring system is capable of detecting events of all sizes, the lack of a Poissonian interevent time distribution may simply imply a correlated driving of the system. The work of Sánchez et al. (2002) implies that two interpretations of power law interevent time distributions are equally possible: 1) Random, uncorrelated driving yields power law behaviour due to the incomplete detection of events, 2) A correlated driving of the system yields a power law distribution for interevent times by providing a correlation between events that the self-organised critical state itself cannot. It should be noted that it is not possible to distinguish between these two interpretations based on interevent time distributions only. Sánchez et al. (2002) thus conclude that interevent time distributions, regardless of their shape, cannot be used as indicators for self-organised criticality as the mechanism behind the distributions cannot be deduced. If we consider the ﬁrst interpretation of the results of Sánchez et al. (2002) to be correct, the observed interevent time distribution would then be a result of self-organised criticality whose Poissonian characteristics have been con- cealed through the incomplete detection of events. If we consider the second interpretation to be correct, however, tuning of the system by providing a correlated drive would then be necessary in order to yield power law interevent time statistics. The important implication of this is that the system, by itself, cannot organise into a state where power law interevent time distributions are obtained. In this case, the correlated drive represents a physical process that can be modelled and investigated in order to yield an understanding of what may govern the behaviour of aftershocks. As discussed earlier, it appears that a non-zero value of the Omori law parameter c is not the result of an incomplete detection of events. From this perspective, it also seems extremely unlikely that a power law interevent time distribution should be the result of incomplete detection. It is a very well established empirical observation that the rate of aftershocks is not a constant (such as in a Poissonian process) but does decay with time according to Omori’s law. Observing non-Poissonian interevent time statistics in aftershock sequences is therefore not surprising. If such an observation, however, were the result of incomplete detection of a Poisson process, this implies that there would be an abrupt and lasting increase in the regular and steady rate of earthquakes whenever a mainshock occurs. The new rate would then be at least as high as predicted by λ(0) (the Omori law rate immediately after the occurrence of the main shock) for the supposedly incompletely detected after-

45 shock sequence. If the seismological network is capable of detecting events with a rate λ(0), this rate is also what should be observed in the postseismic period rather than a rate that decays with time. The nature of the corresponding interevent time distribution which is observed should then also be Poissonian rather than a power law. Considering both the distribution of aftershock occurrence times and interevent times, it is thus clear that the observed behaviour cannot be caused by an incomplete detection of events. This implies that physical processes initiated by the main shock is the only viable explanation for the observed behaviour of aftershocks.

2.3.4 Summarising comment In this section, we have extensively discussed the hypothesis that the Earth’s crust is in a self-organised critical state, reﬂected by power law distributions of energy released in earthquakes (the Gutenberg-Richter law), aftershock occurrence and interevent times (the Omori law) and earthquake locations. The main argument supporting this hypothesis is the observation of a power law distribution of earthquake sizes (i.e. the Gutenberg-Richter law) in simpliﬁed models of loading and rupturing of the Earth’s crust. The essential feature of these models is that power law statistics of event sizes (based on ∼thousands of events) can neither be predicted from the loading nor the rules for stress redistribution. According to the theory of self-organised criticality, the crust has, by itself, organised into a critical state where long-range interaction between elements is possible during an individual event. Such a long range interaction implies a lack of characteristic scale and can thus only be expressed by power law distributions of event sizes. It is important to note that an event in these models represents a sequence of failed elements or fault patches. Within each sequence, the time to failure of the successive neighbouring elements is constant, governed by the interelement distance (constant) and the rupture velocity (assumed to be constant). The power law behaviour expressed by fractal distributions of earthquake locations and the Omori law for aftershocks is also considered to support the hypothesis of a self-organised critical state of the Earth’s crust. This has, however, not been modelled and discussed using self-organising criticality models to the same extent as for the Gutenberg-Richter law. Our discussion in this section has focused on the behaviour of aftershocks and the Omori law. We show that the observation of a power law decay in the rate of aftershocks shall not, and cannot, be used as an argument supporting the hypothesis of self- organised criticality. There are several arguments of different complexity for why this is the case. We will summarise these below: 1. An aftershock sequence of individual earthquakes can be considered as an analogue to the sequence of individual fault patches that rupture during an event in self-organised criticality models. Both types of sequences can thus be considered to represent one event in the system although the timing

46 between the individual aftershocks or fault patch failures is different. The size of an aftershock sequence would then be the number of aftershocks. From the viewpoint of self-organised criticality, this implies that hundreds to thousands of aftershock sequences are required in order to infer whether the total number of aftershocks is distributed according to a power law. From this perspective, an individual aftershock sequence provides no information on whether or not the crust is in a state of self-organised criticality. 2. There is a possibility that a non-zero value of c in the Omori’s law reﬂects a physical process rather than an incomplete detection of events. A non- zero value of c in Omori’s law implies a characteristic scale in both the distribution of aftershock occurrence times as well as interevent times. As a self-organised critical state is characterised by a lack of characteristic scales the Omori law does not support self-organised criticality if c is truly non-zero. 3. From the viewpoint of self-organised criticality, a power law aftershock interevent time distribution implies that interevent times of all time scales can be initiated at any time during an aftershock sequence. As the rate of aftershocks decays with time, long interevent times can not be correlated with early occurrence times. Therefore, interevent times of all time scales cannot be initiated at any time. 4. Random, uncorrelated, driving in a self-organised critical system where events are incompletely detected yields power law interevent time distributions. If all events were detected, the corresponding interevent time distribution would then be Poissonian. A power law interevent time distribution can also be obtained if a correlated driving of the system provides the correlation between events which the self-organised critical state itself cannot. Thus, based on observations of aftershock interevent time distributions in the form of power laws alone, it cannot be concluded whether the hypothesis of self-organised criticality can be discarded or shall be retained. 5. Omori’s law implies a power law decay in both the distribution of aftershock occurrence times and the distribution of interevent times. Taken together, this behaviour of aftershocks cannot be the result of incomplete detection in an underlying Poisson process resulting from a self-organised critical state of the Earth’s crust. 6. If we assume that a self-organised critical state of the Earth’s crust is characterised by a Poissonian earthquake process, argument 5 is very important. It tells us that the behaviour of aftershocks cannot be the result of such a state as this is inconsistent with a power law distribution of both occurrence and interevent times. Considering argument 4, it is thus possible to conclude that power law aftershock interevent time (and occurrence time) distributions must be the result of a correlated driving of the system and not an incomplete detection of events. Providing a correlated driving implies a tuning of the system in order to yield the required behaviour. Such tuning,

47 however, violates the basic principle of self-organised criticality, i.e. that no tuning is required to reach the self-organised critical state.

In the beginning of aftershock sequences, there is a possibility that the roughly constant rate, reﬂected by a non-zero value of c, may be due to an incomplete detection of events. Regardless of whether this is the case or not, it is however clear that the entire behaviour described by the Omori law for aftershocks is not the result of incomplete detection. Based on our discussion, we conclude that physical processes initiated by the main shock provide the correlated driving necessary to yield the Omori law, with or without a non- zero value of c. This conclusion provides a motivation for attempts to model the physical processes responsible for aftershocks in order to understand what controls their occurrence. Although we cannot discard the hypothesis of self- organised criticality, it is not useful as a modelling tool because the coupling between the resulting behaviour and the parameters of the models cannot be deduced. If we now consider argument 1 and model thousands of aftershock sequences based on a physical process, we may very well obtain a power law distribution for the total number of aftershocks in each sequence. We could then disregard all the details of the model and say that the result is an evidence of self-organised criticality. Such an interpretation, however, would only be of academic interest. The importance of the modelling does not lie in producing power law statistics but to understand the physical processes that control the occurrence of earthquakes in aftershock sequences. We conclude that a detailed understanding of the statistical distributions studied is necessary in order to make correct interpretations of the underlying physics and to model and interpret the physical process.

48 3. Statistical methods

In this chapter I discuss statistical methods which I apply to the study of aftershock sequences in my thesis. I will address simulation of aftershock temporal distributions, estimation of Omori law parameters for aftershock sequences and comparison of two distributions of the same property.

3.1 Simulation of aftershock sequences obeying the Omori law In this thesis I present an algorithm (called the stretching algorithm) for random simulation of aftershock occurrence times in aftershock sequences obeying the Omori law. The algorithm can be used in order to produce a large number of random realisations of aftershock occurrence times for an aftershock sequence specified by the Omori law parameters K, c and p. In Monte Carlo applications, for example, this is useful as we can estimate statistical properties from the obtained aftershock sequence distribution. In Paper V of my thesis, I use the Monte Carlo approach with the stretching algorithm in order to estimate confidence intervals for the number of aftershocks in certain time intervals of aftershock sequences observed within the South Iceland Seismic Zone (SISZ). In section 6.7 of the thesis, I also apply the Monte Carlo technique with the stretching algorithm in order to address the issue of data completeness at short times after main shocks in the SISZ. An algorithm called the thinning algorithm, introduced by Lewis and Shedler (1979), has been applied to the simulation of aftershock sequences obeying the Omori law (Ogata, 1999). The thinning algorithm may, however, be quite time-consuming as it needs to reject events from a uniformly distributed sequence of events with a rate corresponding to the largest rate of the aftershock sequence one wishes to simulate. The thinning algorithm must therefore simulate a large number of events that will never be used, and this may take considerable amounts of time. Given the Omori law parameters K, c, p and the time period T of a spe- cific aftershock sequence, the total number of aftershocks in the sequence can be calculated (equation 2.5 or 2.6). This represents the statistical expectation value of the number of aftershocks in the sequence. With the thinning algorithm the number of aftershocks produced is not predictable, but will vary around the expectation value given by equation 2.5 or 2.6. The stretching al-

49 gorithm, however, has the ability to produce sequences where the total number of aftershocks can be chosen to either agree exactly with or to vary around the expectation value. The idea behind the stretching algorithm in generating aftershock sequences obeying the Omori law is based on the behaviour of aftershock sequences where p = 1 and c = 0. In such aftershock sequences the relationship between the logarithm of aftershock occurrence times and their ordinal numbers in the sequence is linear. For p = 1 and/or c = 0 this relationship is nonlinear. We illustrate this in the semi-logarithmic plot in figure 3.1, where occurrence time is plotted versus aftershock number for four aftershock sequences with fixed K and T, but with varying values of c and p. The figure shows both theoretical curves and numerically simulated occurrence times. The stretching algorithm first generates a sequence of uniformly distributed numbers and then stretches the time axis in such a manner that the rate in the transformed series is consistent with the desired Omori law aftershock sequence. For p = 1 and c = 0, this is a simple transformation. For other values of p and c the operation is slightly more complex, but still straightforward.

5 10 C=0, p=1 C=200, p=1 C=0, p=1.5 C=0, p=0.5 4 10 Simulated

3 10

2 10 Occurrence time [s]

1 10

0 10 0 2000 4000 6000 8000 10000 12000 14000 Aftershock number Figure 3.1: Occurrence time versus aftershock number for Omori law aftershock sequences with a ﬁxed value of K (2000) and observation period T (10 000 s) but different values of c and p. Solid, dashed, dash-dotted and dotted lines: Theoretical curves. Crosses: Selected aftershock occurrence times versus their ordinal number from simulations using the stretching algorithm.

50 The stretching algorithm The stretching algorithm is also described in appendix A of Paper I in the thesis. Note though that there are typographical errors in equation A1 and A5 in the description provided in Paper I. Here I provide the correct equations (equations 3.1 and 3.5). The stretching algorithm can be used to simulate: 1. The expected number of aftershocks given the Omori law parameters K, c and p. 2. A fixed number of aftershocks by specifying N and two of the three Omori law parameters. The third parameter will be calculated and used in the algorithm. 3. Aftershock sequences specified by K, c and p with a random fluctuation of the number of aftershocks around the expectation value.

Having speciﬁed K, c and p, Tstart and T, where Tstart is the start time of the simulation (very close to zero) and T is the desired total length of the time series, the total number of aftershocks can be determined by integrating equation 2.4 between Tstart and T. Given the Omori law parameters the total number of aftershocks will be, for p = 1:

N = K · [ln(c + T) − ln(c + Tstart )] (3.1) and for p = 1: = − · 1 − 1 N K p−1 p−1 (3.2) (p − 1) · (c + T) (p − 1) · (c + Tstart ) where N is rounded off to the nearest lower integer. In the case of using ﬁxed N and two of the three parameters K, c and p, equation 3.1 and 3.2 can be rewritten to calculate the third parameter, which will then be used in the algorithm. Rewriting the above equations (3.1 and 3.2) the occurrence time for each of the aftershocks in the sequence can be determined. The occurrence time of aftershock number n (1, 2, ..., n, ..., N) will be, for p = 1: n t = exp + ln(c + T ) − c (3.3) n K start and for p = 1: ⎛ ⎞ 1 p−1 ⎝ 1 ⎠ tn = − c (3.4) ( − ) · − n + 1 p 1 p−1 K (p−1)·(c+Tstart )

Equations 3.3 and 3.4, as well as ﬁgure 3.1, show that the ordinal number n of an aftershock can be used to determine its occurrence time in Omori law aftershock sequences with any combination of the Omori law parameters K, c

51 and p. We use this, together with the linear behaviour for p = 1 and c = 0, to simulate any Omori law aftershock sequence through the following steps: 1. Given K, c and p, Tstart and T, calculate the number of aftershocks N in the sequence with equation 3.1 or equation 3.2, depending on the value of p. With ﬁxed N, calculate the missing Omori law parameter. 2. Use equation 3.3, with c = 0, p = 1 and the given value of K, to calculate the required time TR to simulate N aftershocks if the value of c is zero and the value of p is 1. 3. Now generate d · N uniformly distributed random numbers in the interval [ ( ), ( )+ · ( ( ) − ( ))] log10 Tstart log10 Tstart d log10 TR log10 Tstart , where d is an integer ≥ 1, and store them in a vector x.Ifd = 1, exactly N aftershocks will be produced by the algorithm (i.e. the expectation value for the sequence). Each random number corresponds to an aftershock. > ( ) 4. If d 1, discard all random numbers in the vector x larger than log10 TR . Due to the randomness in the generation of x, the number of aftershocks that is produced will then vary around the number N. 5. Determine the occurrence time for each (remaining) aftershock in the vector x by stretching the time between each event in this way: x tocc = 10 (3.5)

6. Now, if the specified values of c and p in step 1 actually are 0 and 1, respectively, the simulation is finished and tocc contains the simulated occurrence times of the aftershocks in the sequence. 7. However, if p = 1 and/or c = 0, we have to calculate the aftershock numbers corresponding to the occurrence times in the vector tocc.Astocc was simulated with c = 0 and p = 1, the corresponding aftershock numbers nocc are calculated using equation 3.1 with c = 0, the specified value for K and inserting tocc instead of T. Because of the randomness in the simulation of tocc, the calculated aftershock numbers nocc will have a random variation around the expectation values 1, 2, ...., n, ..., N. 8. As the final step, we utilise the fact that an aftershock number can be used to calculate the occurrence time of the corresponding aftershock for any Omori law aftershock sequence. We simply use K, c and p for the aftershock sequence we wish to simulate and insert nocc instead of n in equation 3.3 or equation 3.4, depending on the value of p. This gives us a set of simulated occurrence times of the aftershocks in the specified Omori law sequence.

52 3.2 Maximum likelihood estimation of Omori law parameters In a homogenous Poisson process, characterised by a constant rate λ, the probability of an event occurring in a time interval [t, t + Δt] is time-independent and given by: P(An event in [t, t + Δt]) = λ · Δt + o(Δt) (3.6) where o(Δt) is a function that goes to zero faster than Δt goes to zero. If Δt equals 1/λ an event will take place at some time between t and t + Δt as the corresponding probability is 1. For a nonstationary Poisson process, with a time dependent rate λ(t), the probability of an event in the interval [t, t + Δt] is given by: t+Δt P(An event in [t, t + Δt]) = λ(t) · dt + o(Δt) (3.7) t

The duration of the time interval having 100 % probability of an event is now dependent on the elapsed time t. With λ(t) given by the Omori law (equation 2.4), this duration is, for p = 1: 1 Δt = = exp + ln(c +t) − c −t (3.8) P 1 K and for p = 1: ⎛ ⎞ 1 p−1 ⎝ 1 ⎠ ΔtP=1 = −t − c (3.9) ( − ) · − 1 + 1 p 1 K (p−1)·(c+t)p−1

Now, we consider an inﬁnitesimally short interval [t, t +Δt] where Δt → 0 and approximate the probability of an event in this interval by:

P(An event in [t, t + Δt→0]) = λ(t) · Δt→0 + o(Δt) (3.10)

Given an observed aftershock sequence, with Omori law parameters K, c and p, and with occurrence times t1, t2,...,tN, we can use equation 3.10 to estimate the probability of an event in a small time interval centered at the occurrence time of each event in the sequence. Given the occurrence times and the Omori law parameters, we can also use equation 2.41 to determine the probabilities of no events occurring in between the aftershocks in the sequence. Having these probabilities, the likelihood that the observed aftershock occurrence times can be described by the Omori law with parameters K, c and

53 p is given by: N N−1 − ti+1 λ( )· t t dt L(t1, t2, ..., tN|K, c, p)=∏λ(ti) · Δti · ∏ e i (3.11) i=1 i=1 where the ﬁrst factor is the product of the probabilities of events in small intervals centered at each occurrence time, the second factor is the product of the probabilities of zero events in between the events of the sequence and λ(t) is the Omori law (equation 2.4). The multiplication of these probabilities with each other is justiﬁed by the basic assumption of a Poisson process, i.e. that events which occur in disjoint intervals are independent (Ogata, 1983). Taking the natural logarithm of equation 3.11 and simplifying yields: N N tN lnL(t1, t2, ..., tN|K, c, p)=∑ ln(λ(ti)) + ∑ ln(Δti) − λ(t) · dt (3.12) i=1 i=1 t1

We can maximise equation 3.12 in order to determine the Omori law parameters K, c and p which give the largest likelihood that the aftershock sequence can be described by the Omori law. Note, however, that it is only the ﬁrst and the third sums in equation 3.12 which contain K, c and p. The second sum depends only on the chosen interval sizes Δti. This term is therefore a constant which approaches −∞ if each Δti → 0. Being a constant, this sum can be excluded and the function to be maximised is: N t ∗ N lnL (t1, t2, ..., tN|K, c, p)=∑ ln(λ(ti)) − λ(t) · dt (3.13) i=1 t1

I have written a computer program to maximise equation 3.13 utilising a grid search over K and c in combination with conjugate gradients. The grid search is necessary as the function lnL∗ is nonlinear and several local maxima may exist. Having found the maximum likelihood estimates of K, c and p, the program also determines the standard deviation of the marginal error of each parameter as described in Ogata (1983). Figure 3.2 demonstrates the result of applying the program to two aftershock sequences where the underlying Omori law parameters are the same (K=100, c = 1000 and p = 1.1). The ﬁrst sequence represents the theoretically expected aftershock sequence with these parameters, i.e. where the aftershock occurrence times are calculated from the aftershock ordinal numbers 1, 2, .... ,N. The second sequence is a random realisation obtained with the stretching algorithm described in section 3.1. Both sequences cover a time period of 107 seconds after the main shock (∼ 3.8 months). In ﬁgure 3.2 a), the crosses show the rate of aftershocks (obtained using a histogram with logarithmically increasing bin sizes) for the theoretically expected aftershock sequence. As can be expected, the obtained rate matches the theoretically predicted rate given by the Omori law (solid line) very well. The

54 a) Theoretically expected sequence b) Random realisation 0 0 10 10

−2 −2 10 10

−4 −4 10 10

Histogram Histogram

Aftershock rate −6 Aftershock rate −6 10 λ (t) 10 λ (t) th th

0 2 4 6 8 0 2 4 6 8 10 10 10 10 10 10 10 10 10 10 Occurrence time [s] Occurrence time [s] c) (λthe−λ )/λ d) (λrnd−λ )/λ est th th est th th 20 20

10 10

0 0

−10 −10 Rel. difference [%] Rel. difference [%] −20 −20 0 2 4 6 8 0 2 4 6 8 10 10 10 10 10 10 10 10 10 10 Occurrence time [s] Occurrence time [s] Figure 3.2: Demonstration of the maximum likelihood approach in fitting the Omori law to an aftershock sequence specified by K=100, c = 1000 and p = 1.1. a) Crosses: Aftershock rate in the theoretically expected sequence (obtained through a histogram of the theoretically expected occurrence times). Solid line: Theoretically predicted rate given by the Omori law with K=100, c = 1000 and p = 1.1. b) Crosses: Af- tershock rate in a random realisation of the specified sequence (obtained through a histogram of the simulated occurrence times). Solid line: Theoretically predicted rate given by the Omori law with K=100, c = 1000 and p = 1.1. c) Relative difference in rate between the Omori law with estimated parameter values for the theoretically λ the expected occurrence times ( est ) and the Omori law with the given values K=100, c = 1000 and p = 1.1(λth). d) Relative difference in rate between the Omori law with λ rnd estimated parameter values for the occurrence times of the random realisation ( est ) and the Omori law with the given values K=100, c = 1000 and p = 1.1(λth). deviation that can be observed in the second bin is related to the discreti- sation in the histogram. When applying the program to the theoretically expected aftershock sequence, the Omori law parameters are recovered well, est = . ± . est = . ± . est = . ± . with Kth 93 8 35 3, cth 932 7 305 2 and pth 1 09 0 03. Figure 3.2 c) shows that the relative difference in the rate of aftershocks given by the Omori law with the estimated and the original parameters is less than ∼ 5%. In figure 3.2 b) the crosses represent the rate of aftershocks (obtained through a histogram with logarithmically increasing bin sizes) for the random realisation of the aftershock sequence while the solid line represents the theoretically predicted rate given by the Omori law. When compared with figure 3.2 a), the crosses are, due to the randomness, more scattered around the theoretical curve. The estimated Omori law parameters for this random realisation of the aftershock sequence are K = 84.8±32.4, c = 856.6±282.4 and p = 1.08 ± 0.03. Although these estimates are further away from the original Omori law parameters than the estimates for the theoretically expected sequence, the underlying values of K, c and p are still within the

55 uncertainty range. In figure 3.2 d) we can see that the relative difference in rate given by the Omori law with the estimated and the original parameters can now be as large as ∼ 12 %. We consider that this larger difference reflects the variability introduced by the randomness rather than difficulties in recovering the original Omori law parameters. For a real aftershock sequence, the underlying Omori law parameters are unknown and we cannot make comparisons as in figure 3.2 c) and d). More- over, the Omori law is only an empirical approximation for the behaviour of aftershock sequences. The capability to recover the intervals containing K, c and p in our example, however, indicates that the written program allows a reliable estimation of Omori law parameters that provide a good approximation to observed aftershock sequences.

3.3 Comparison of two datasets In my thesis I utilise the Kolmogorov-Smirnov test (K-S test) in order to investigate whether two different datasets can be considered as drawn from the same or from different underlying distributions. In my discussion on data completeness in section 6.7, I use the K-S test for comparing aftershock occurrence and interevent time distributions. Based on Press et al. (1992), I will here brieﬂy describe the K-S test. The K-S test is applicable to distributions of a single variable, for example aftershock occurrence times. The test is based on measuring the maximum absolute value of the difference between the two cumulative distributions de- ﬁned by the datasets that are to be compared. The K-S test statistic is given by: Dobs = max |SN (x) − SN (x)| (3.14) −∞

( ) ( ) where SN1 x and SN2 x are the cumulative distributions for the two datasets and N1 and N2 are the number of events in each dataset. In the K-S test, the null hypothesis is that the two different datasets are drawn from the same distribution. In such a case, the distribution of the test statistic D can be calculated approximately, giving the statistical signiﬁcance of any non-zero value of Dobs. The probability that the two datasets are drawn from the same distribution is then given by: √ √ P(Dobs < D)=QKS Ne + 0.12 + 0.11/ Ne · Dobs (3.15) where Ne is the effective number of data points given by:

N1 · N2 Ne = (3.16) N1 + N2

56 and QKS is a monotonic function deﬁned by the sum: ∞ j−1 −2 j2λ 2 QKS(λ)=2 · ∑(−1) · e (3.17) j=1

Taking the limiting values:

QKS(0)=1 QKS(∞)=0 (3.18)

If the probability P(Dobs < D) given by equation 3.15 is high, it implies a high likelihood of the same underlying distribution as D then can be larger than Dobs without implying two different distributions. If the probability P(Dobs < D) is small, however, this implies a high likelihood of the underlying distributions being different as D then is seldom larger than Dobs. Specifying a signiﬁcance level α, the null hypothesis can be rejected if P(Dobs < D) < α and the underlying distributions of the compared datasets can be considered different.

4. The physics of aftershocks

In this chapter, I will provide a brief overview of some of the physical mechanisms and processes that have been proposed to explain the occurrence of aftershocks and the decay in aftershock rate according to the Omori law. The mechanisms that I will discuss are rate and state friction, subcritical crack growth and transient deformation processes, such as viscoelastic relaxation, afterslip and poroelastic rebound due to postseismic pore pressure diffusion.

4.1 Physical models for the occurrence of aftershocks 4.1.1 Rate and state friction In seismology, constitutive laws for the frictional dependence on rate and state of earthquake faults have been widely applied to studying the mechanics of earthquakes and faulting. Reviews of this extensive subject can be found in Marone (1998) and Kanamori and Brodsky (2004). One formulation of a rate and state dependent friction law is (e.g. Dieterich, 1994): ˙ μ = μ0 + A · lnδ + B · lnθ (4.1) where μ is the coefﬁcient of friction of a fault surface; μ0, A and B are constants; δ˙ is the fault slip rate (i.e. the time derivative of the slip δ) and θ is a variable that describes the state of the fault surfaces in contact. Dieterich (1994) suggests a differential equation for the state variable θ which takes into account the history of fault sliding (through δ) and normal stress loading (through σ) of the fault: dθ 1 θ α · θ = − dδ − dσ (4.2) dt δ˙ Dc B · σ

In equation 4.2, Dc is a characteristic displacement and α is a parameter governing the dependence of the state variable θ on the normal stress σ. Dieterich (1994) applies rate and state dependent friction to the occurrence of aftershocks by relating steps in shear and normal stresses on faults to the failure times of a fault population. The fault population is characterised by a constant seismicity rate if the rate of tectonic loading is constant. However, if there is a sudden stress change, this will, through the evolution of slip rate (δ˙) and state (θ), modify the time to failure in the fault population. A key

59 characteristic of rate and state friction is that the earthquakes which occur are ones that would have occurred sooner or later; they are just advanced in time as a result of the stress change. How much they are advanced depends on the slip and loading history of the faults. In a rate and state friction model, static stress changes associated with a main shock result in a stepwise increase in the sliding speed and a shortening of the time to failure of pre-existing faults in the region surrounding the main shock. Dieterich (1994) shows, under the assumption of a constant normal stress, that rate and state friction implies a formulation for the rate of earthquakes following a sudden increase in shear stress that is in agreement with Omori’s law for the rate of aftershocks. Rate and state dependent frictional properties of faults can thus be considered as a possible explanation for the Omori law decay in the rate of aftershocks. Dieterich (1994) shows that rate and state friction also predicts an initially constant aftershock rate before the rate begins to decay as a power law with time. A fault population governed by rate and state friction is thus a physical model that predicts a non-zero value of c in the Omori law. Under the assumption of a stressing rate which is constant before and after the main shock, the parameter c is related to the magnitude of the static stress change in such a way that larger stress changes cause smaller values of c (Dieterich, 1994). This implies that the seismicity rate following the stress change approaches the power law decay faster for a large stress change than for a small. An explanation for this is that more earthquakes come close to failure faster for large stress changes, so that the seismicity rate must start to decay earlier than if the stress change had been small. It appears that rate and state friction predicts an inverse relationship between the Omori law parameter c and the magnitude of the main shock, with a larger main shock resulting in a smaller value of c for the resulting aftershock sequence. In Paper V I discuss this result in relation to a postseismic pore pressure diffusion model for aftershocks.

4.1.2 Subcritical crack growth Earthquakes are considered as frictional slip on faults or as dynamically propagating cracks in fundamental earthquake models (Scholz, 2002). Subcritical crack growth (also referred to as stress corrosion) has been suggested as the physical mechanism behind the Omori law for aftershocks (e.g. Yamashita and Knopoff, 1987; Shaw, 1993; Kanamori and Brodsky, 2004). Subcritical crack growth implies that cracks in a brittle material may grow spontaneously under certain conditions (such as high temperature or the existence of ﬂuids), even if the stress loading the crack is below the critical level. The growth of the cracks is then caused by a weakening of crack tips due to chemical corrosion, eventually resulting in catastrophic failures when cracks have reached a critical state (Kanamori and Brodsky, 2004). Considering the effect of a sudden static stress increase on a system of cracks, loaded at a constant rate and governed by stress corrosion, the stress

60 change has been shown to accelerate the crack growth, shorten the time to failure and to result in a formulation for subsequent seismicity rate in agreement with the Omori law (Gomberg, 2001; Kanamori and Brodsky, 2004). A subcritical crack growth model is thus similar to a rate and state dependent friction model in that a stepwise increase in stress yields a stepwise increase in sliding speed, a stepwise decrease in the time to failure and an Omori law rate of seismicity following the stress change. Subcritical crack growth also yields a nonzero value of c in the Omori law, implying an initially constant rate before power law decay takes over (Kanamori and Brodsky, 2004). Fol- lowing the derivation and the assumptions of Kanamori and Brodsky (2004), it can be shown that also a subcritical crack growth model results in an inverse dependence between c and the main shock magnitude.

4.1.3 Transient deformation processes Transient deformation of the Earth’s surface is commonly observed in the postseismic period of large earthquakes. The time scale of these deformation transients can vary widely, ranging from only a couple of months up to several years. The postseismic deformation associated with the June 2000 earthquakes in Iceland is an example where transients of different time scales have been observed. Following the June 2000 earthquakes, InSAR-data revealed a deformation transient lasting about two months (Jónsson et al., 2003). GPS- data obtained yearly between 2000 and 2004, however, have also revealed a slower deformation transient with a characteristic time scale of one year (Ár- nadóttir et al., 2005). This deformation transient is also captured by satellite radar images acquired between August 2000 and September 2005 (Jónsson, 2008). Based on a time-series analysis of satellite radar interferograms, Jóns- son (2008) infer a characteristic time scale of 1.4 to 2.7 years for the slow deformation transient following the June 2000 earthquakes in Iceland. Other examples of deformation transients lasting several years include postseismic deformation associated with e.g. the 1992 M(w)7. 3 Landers earthquake in California (Fialko, 2004) and the 2003, M(w)7. 2 Altai earthquake near the Russia-China-Mongolia border (Barbot et al., 2008). The most common suggestions for the mechanisms responsible for transient deformation in the postseismic period include viscoelastic relaxation of the lower crust and upper mantle (e.g. Deng et al., 1999; Freed and Lin, 2001), transient creep or afterslip (e.g. Benioff, 1951; Hearn et al., 2002; Perfettini and Avouac, 2004, 2007) and poroelastic rebound (e.g. Jónsson et al., 2003; Fialko, 2004). It appears, however, that a unique mechanism explaining postseismic deformation transients is difﬁcult to identify. Jónsson et al. (2003) point out that only poroelastic rebound, and neither afterslip nor viscoelastic relaxation, can explain the short time scale deformation transient associated with the June 2000 earthquakes in Iceland. Regarding the slower deformation transient, analysis of GPS-data is not able to distinguish between after-

61 slip and viscoelastic relaxation as the most plausible mechanism (Árnadóttir et al., 2005). Of these two mechanisms, Jónsson (2008) finds that the deformation associated with afterslip is incompatible with the postseismic deformation measured by satellite radar interferometry. Based on his analysis, Jónsson (2008) suggests that the slow deformation transient following the June 2000 earthquakes in Iceland was most likely driven by viscoelastic relaxation. Following the 2003 Altai earthquake, Barbot et al. (2008) find that most of the postseismic deformation can be explained by seismic moment release in aftershocks and that neither poroelastic rebound nor viscoelastic relaxation can explain the three-year long deformation transient. Deng et al. (1999), on the other hand, suggest that the postseismic deformation of the 1994 Northridge earthquake in California is mainly driven by viscous flow in the lower crust. The complexity of postseismic deformation transients is illustrated by the 1992 Landers earthquake in California. Perfettini and Avouac (2007) conclude that frictional afterslip down dip of the main shock fault plane well fits the observed postseismic deformation. Fialko (2004), however, finds that a combination of poroelastic relaxation and localised shear deformation on and below the Landers rupture explains most of the deformation. That several processes may be involved is supported by Freed et al. (2006), who conclude that no single mechanism can explain the postseismic deformation associated with the 2002 M(w) 7.9 Denali earthquake in Alaska. The processes of viscoelastic relaxation, poroelastic rebound and transient creep or afterslip have all been related to the occurrence of aftershocks. Deng et al. (1999) use a finite element model with an elastic upper crust, a weaker viscoelastic lower crust and a stronger upper mantle to model co- and postseismic deformation and stress change associated with the 1994 Northridge thrust faulting earthquake in California. Based on a good correlation between aftershock hypocenters and stresses transferred from the viscoelastic lower crust, Deng et al. (1999) conclude that viscoelastic relaxation could have a first-order influence on triggering of Northridge aftershocks. As the tail of the aftershock distribution in time matches the shear stress loading associated with viscoelastic relaxation in their model, it is argued that the long-term decay time of the Northridge aftershocks is controlled by the viscosity of the lower crust. Deng et al. (1999) point out, however, that the decay of aftershocks in the first several weeks appears too fast to be solely explained by viscoelastic relaxation of the lower crust and suggest that frictional weakening of the fault area, strength changes in fault zone materials or pore pressure changes may be the cause of the early aftershocks. Perfettini and Avouac (2004) investigate the role of coseismic stress changes on slip of a brittle creeping fault zone at midcrustal depth, characterised by a rate-strengthening rheology. Assuming that seismicity rate is proportional to the sliding velocity of the creeping fault zone, their model predicts a decay in the rate of aftershocks in agreement with the

62 Omori law. Perfettini and Avouac (2004) demonstrate that their model results in an inverse relationship between the Omori law parameter c and the magnitude of the main shock. This implies that the duration of the regime with an initially constant rate becomes smaller the larger the stress change. Applying the model to the 1999 Mw 7.6 Chi-Chi earthquake in Taiwan, Perfettini and Avouac (2004) find that the same deep afterslip model can explain the temporal evolution of both aftershocks and postseismic surface deformation. Extending the analysis to include postseismic stress changes, Perfettini and Avouac (2007) conclude that the spatial and temporal distribution of aftershocks of the 1992 Landers earthquake is consistent with the hypothesis that their occurrence is a result of frictional afterslip reloading the seismogenic zone. The shear strength of faults is significantly reduced by the existence of fluids within the pores of the Earth’s crust as the pressure of such fluids coun- teracts compressive normal stresses contributing to the shear strength. Evi- dence for this is supported by the observation of pore pressure induced seismicity in relation to, for example, reservoir loading (e.g. Beck, 1976; Bell and Nur, 1978; Talwani and Acree, 1985; Roeloffs, 1988; Pandee and Chadha, 2003), groundwater recharge (e.g. Saar and Manga, 2003), fluid injection (e.g. Zoback and Harjes, 1997; Shapiro et al., 2003), fluid extraction (e.g. Segall, 1985, 1989) and the occurrence of large earthquakes (e.g. Nur and Booker, 1972; Li et al., 1987; Bosl and Nur, 2002; Shapiro et al., 2003; Gavrilenko, 2005). Nur and Booker (1972) propose that the occurrence of aftershocks is a result of a pore pressure diffusion process initiated by main shock induced pore pressure perturbations. By postulating that the rate of aftershocks is proportional to an integral of the time derivative of pore pressure over the volume where pore pressure is increasing, they infer that the rate of aftershocks should decay with time as a power law with an exponent equal to 0.5. Nur and Booker (1972) show that this prediction holds initially for the aftershock sequence associated with the 1966 Parkfield-Cholame earthquake. At later times, however, the rate of aftershocks gradually shifts to a 1/t decay. Nur and Booker (1972) suggest that this may be caused by an increasing inability with time for the diffusion process to yield pore pressure increases able to overcome the failure threshold. The formulation for pore pressure induced aftershock rates of Nur and Booker (1972) does not predict an initially constant rate of aftershocks, as measured by a non-zero value of c in Omori’s law. Their model, however, is based on a very simple representation of an earthquake fault as an edge dislocation in an otherwise homogeneous and infinite elastic space. In practice, the distribution of induced pore pressures in a more realistic earthquake model depends in a complex way on the coseismic slip distribution of the main shock. For a realistic earthquake model it is thus not possible, analytically, to formulate an expression for the rate of aftershocks based on the postulation of Nur and Booker (1972). Bosl and Nur (2002), however, model pore fluid

63 diffusion following the Landers 1992 earthquake and evaluate the aftershock rate as described by Nur and Booker (1972) numerically. Their results show, not only that the predicted aftershock rate agrees with the decay in the rate of Landers aftershocks, but also that a pore pressure diffusion model can predict a non-zero value of c in Omori’s law.

4.2 Summarising comment I have discussed several physical processes and mechanisms that can explain the characteristic behaviour of aftershock sequences described by the Omori law. Of these, viscoelastic relaxation appears to influence the aftershock process only at long times after the main shock (Deng et al., 1999). The mechanisms of rate and state friction (Dieterich, 1994), subcritical crack growth (Kanamori and Brodsky, 2004), afterslip (Perfettini and Avouac, 2004) and main shock initiated pore pressure diffusion (Bosl and Nur, 2002) can all explain initially constant aftershock rates followed by power law decay with time. It is interesting to note that rate and state friction, subcritical crack growth and afterslip all predict an inverse relationship between the duration of the initially constant rate and the magnitude of the main shock initiating the aftershock process. This implies that it may be difficult to determine the relative influence of these different mechanisms on the aftershock process. In my thesis I have studied aftershock sequences in the south Iceland seismic zone, where poroelastic rebound associated with pore pressure diffusion is the only process that can explain the early postseismic deformation following the two June 2000 earthquakes in SISZ (Jónsson et al., 2003). Such a process is characterised by system-wide interaction, as the spatial variation in the coseismically induced pore pressure transients drives the diffusion process. This kind of interaction is neither included in the rate and state friction (Dieterich, 1994) nor the subcritical crack growth (Kanamori and Brodsky, 2004) models. In these models, the system consists of a population of faults which respond, individually, to a stress perturbation. As the faults do not interact, it is the rheological properties of the faults which governs how the rate of aftershocks decays following a main shock. In a pore pressure diffusion model, however, both the spatial interaction in the diffusion process and the rheological properties of faults may govern the decay in the rate of aftershocks. In Papers IV and V of my thesis, I investigate the implications of pore pressure diffusion for induced seismicity. Based on these results, I provide, in Paper V, an interpretation of characteristic features of SISZ aftershock sequences in terms of main shock initiated pore pressure diffusion processes. In the next chapter, I will discuss equations for pore pressure diffusion and poroelasticity that form the background of the work presented in Paper IV and Paper V.

64 5. Pore pressure diffusion

In this chapter, I will discuss the constitutive poroelastic equations that form the basis for the main shock initiated pore pressure diffusion processes modelled in my thesis. Based on these equations, I present calculations of induced stresses and pore pressures due to the coseismic crustal deformation associated with an earthquake. I will also present two diffusion equations, one considering the diffusion of pore pressures only and the other considering poroelastic adjustment of both stresses and pore pressures in the diffusion process. Following this, I will discuss how I evaluate seismicity induced by the main shock initiated diffusion processes.

5.1 Constitutive poroelastic equations A relationship between strain, stress, pore pressure and ﬂuid mass content per unit volume for an isotropic and ﬂuid saturated medium is given by Rice and Cleary (1976) through the coupled constitutive equations:

ν 3(νu − ν) 2Gεij = σij− σkkδij− Pδij (5.1) 1 + ν B(1 + ν)(1 + νu) 3ρ0(νu − ν) 3 m − m0 = σkk − P (5.2) 2GB(1 + ν)(1 + νu) B where the Einstein summation convention applies and εij are strains, σij stresses, P pore pressure, m − m0 change in fluid mass content per unit volume and ρ0 fluid density. The medium is described by four elastic constants, where G is the shear modulus, ν the drained Poisson’s ratio, νu the undrained Poisson’s ratio and B is Skempton’s coefficient. In my calculations I consider compressional stresses (volume decrease) as positive and dilatational stresses (volume increase) as negative. The sign convention of Rice and Cleary (1976), however, is the opposite, with compressional stresses negative and dilatational stresses positive. In equation 5.1, therefore, the pore pressure term is negative while it is positive in the equation of Rice and Cleary (1976).

65 Equation 5.1 can be rewritten in order to yield stresses rather than strains. From equation 5.1 we have:

1 + ν 9(νu − ν) σkk = · 2Gεkk + · P (5.3) 1 − 2ν B(1 − 2ν)(1 + νu) which inserted into equation 5.1 yields:

ν 3(νu − ν) σij = 2Gεij+ · 2Gεkkδij+ · Pδij (5.4) 1 − 2ν B(1 − 2ν)(1 + νu)

Given that the strains and pore pressures are known, the stresses in the medium can now be calculated using equation 5.4.

5.2 Earthquake induced stresses and pore pressures In order to calculate stresses and pore pressures induced by an earthquake, we assume that the associated elastic strain and stress changes throughout the medium take place so rapidly that there is no net ﬂuid ﬂow. The pore pressure induced by the earthquake can therefore be calculated by setting m − m0 = 0 in equation 5.2, yielding: B ΔP = · Δσ (5.5) 3 kk where ΔP and Δσkk indicate the coseismic change in pore pressure and mean stress, respectively. Inserting equation 5.5 into equation 5.1, the coseismic pore pressure change can be calculated from the coseismic change in volu- metric strain: B 1 + νu ΔP = · · 2GΔεkk (5.6) 3 1 − 2νu

By inserting the coseismic change in pore pressure into equation 5.4, the coseismic stress change is given in terms of the coseismic change in strain as: νu Δσij = 2GΔεij+ · 2GΔεkkδij (5.7) 1 − 2νu

Okada (1992) presents equations for calculating strains resulting from rectangular dislocations in an isotropic and homogeneous elastic half-space. In such a medium the constitutive equation relating stress and strain is:

σij = 2Gεij+ λεkkδij (5.8) where G is the shear modulus as before and λ is one of the Lamé parameters for the medium. The Lamé parameter λ is related to the shear modulus (G)

66 and Poisson’s ratio (ν) of the medium through the relationship: ν λ = · 2G (5.9) 1 − 2ν

By using the undrained Poisson’s ratio (νu) in equation 5.9, the response of a poroelastic medium to a coseismic change in strain (equation 5.7) can be written in the same form as for a purely elastic medium (equation 5.8). This implies that the equations of Okada (1992) can be used to calculate coseismic stress and pore pressure changes also for poroelastic media, provided that the undrained Poisson’s ratio is used to calculate the Lamé parameter λ.In my thesis, I use earthquake models consisting of several rectangular dislocations and calculate the total response by superposition of the stress and pore pressure changes given by the equations of Okada (1992).

5.3 Poroelastic diffusion equation In order to derive a diffusion equation for poroelastic adjustment of stresses and pore pressures in the postseismic period, we utilise Darcy’s law for fluid flow in the isotropic case, conservation of fluid mass, force balance, equilibrium conditions, and strain and stress compatibility equations. According to Darcy’s law, the fluid mass flow rate per unit area in the ith direction (qi)is related to the gradient of the pore pressure perturbation (ΔP) through: k ∂ΔP qi = −ρ0 · · (5.10) η ∂xi where k is the permeability of the medium and η is the dynamic viscosity of the fluid. The equation for the conservation of fluid mass is:

∂qi/∂xi + ∂m/∂t = 0 (5.11) where ∂m/∂t is given by the time derivative of the constitutive equation 5.2 as: ∂m 3ρ0(νu − ν) ∂ 3 = · Δσkk − ΔP (5.12) ∂t 2GB(1 + ν)(1 + νu) ∂t B

Applying equation 5.12 and Darcy’s law in the equation for conservation of ﬂuid mass, we get: (ν − ν) ∂ k 2 3 u 3 ∇ ΔP = 3 · · Δσkk − ΔP (5.13) η 2GB(1 + ν)(1 + νu) ∂t B

The above diffusion equation relates spatial derivatives of gradients in the pore pressure perturbation (ΔP) to temporal gradients of the term Δσkk − 3ΔP/B. Here we must note that Δσkk does not represent the coseismic stress perturba-

67 tion (equation 5.7) as the constitutive relationship between strains and stresses is described by equation 5.1 once the diffusion process has started. By utilising equation 5.1 to rewrite strain compatibility equations in terms of stresses, we can, using force balance and equilibrium conditions, derive equations describing the elements of the stress perturbation tensor involved in the diffusion process. The force balance equations (neglecting body forces) and equilibrium conditions that we use are:

∂Δσxx/∂x + ∂Δσxy/∂y + ∂Δσxz/∂z = 0 ∂Δσyx/∂x + Δ∂σyy/∂y + ∂Δσyz/∂z = 0 (5.14) ∂Δσzx/∂x + ∂Δσzy/∂y + ∂Δσzz/∂z = 0

Δσxy = Δσyx Δσxz = Δσzx (5.15) Δσyz = Δσzy

We then use interrelationships between elements of the strain tensor described by the following set of strain compatibility equations (see e.g. Wang, 2000; Neuzil, 2003): ∂ 2ε ∂ 2ε ∂ 2ε 2 xy = xx + yy ∂x∂y ∂y2 ∂x2 ∂ 2ε ∂ 2ε ∂ 2ε 2 yz = yy + zz ∂y∂z ∂z2 ∂y2 ∂ 2ε ∂ 2ε ∂ 2ε 2 xz = xx + zz ∂ ∂ ∂ 2 ∂ 2 x z z x ∂ 2ε ∂ −∂ε ∂ε ∂ε xx = yz + xz + xy (5.16) ∂y∂z ∂x ∂x ∂y ∂z ∂ 2ε ∂ ∂ε ∂ε ∂ε yy = yz − xz + xy ∂x∂z ∂y ∂x ∂y ∂z ∂ 2ε ∂ ∂ε ∂ε ∂ε zz = yz + xz − xy ∂x∂y ∂z ∂x ∂y ∂z

Using equation 5.1, the derivatives of the strain tensor components in the strain compatibility equations can be written in terms of derivatives of the stress tensor components. Inserting the corresponding derivatives into equation 5.16 and using the force balance equations (equation 5.14) and equilibrium conditions (equation 5.15), the following stress compatibility equations are eventually obtained:

68 ∂ 2Δσ 3(ν − ν) ∂ 2ΔP ∇2 ((1 + ν)Δσ − νΔσ )+ kk − u + ∇2ΔP = 0 xx kk ∂x2 B(1 + ν ) ∂x2 u ∂ 2Δσ 3(ν − ν) ∂ 2ΔP ∇2 ((1 + ν)Δσ − νΔσ )+ kk − u + ∇2ΔP = 0 yy kk ∂y2 B(1 + ν ) ∂y2 u ∂ 2Δσ (ν − ν) ∂ 2Δ ∇2 (( + ν)Δσ − νΔσ )+ kk − 3 u P + ∇2Δ = 1 zz kk 2 2 P 0 ∂z B(1 + νu) ∂z ∂ 2Δσ (ν − ν) ∂ 2Δ 2 1 kk 3 u P ∇ Δσxy + − = 0 1 + ν ∂x∂y B(1 + ν)(1 + νu) ∂x∂y ∂ 2Δσ (ν − ν) ∂ 2Δ 2 1 kk 3 u P ∇ Δσxz + − = 0 1 + ν ∂x∂z B(1 + ν)(1 + νu) ∂x∂z ∂ 2Δσ (ν − ν) ∂ 2Δ 2 1 kk 3 u P ∇ Δσyz + − = 0 (5.17) 1 + ν ∂y∂z B(1 + ν)(1 + νu) ∂y∂z

By summing the ﬁrst three of the stress compatibility equations a useful relationship between the perturbations in pore pressure ΔP and Δσkk is obtained (Rice and Cleary, 1976): (ν − ν) 2 6 u ∇ Δσkk − ΔP = 0 (5.18) B(1 − ν)(1 + νu)

Given that the pore pressure perturbation ΔP (equation 5.6) is known, equation 5.18 can be used to calculate the value of the perturbation in mean stress (Δσkk) when the stress perturbation tensor Δσij satisﬁes the stress compatibility equations above. Utilising equation 5.18, we can also rewrite the poroelastic diffusion equation (equation 5.13) as (Rice and Cleary, 1976): 3 ∂ 3 c ∇2 Δσ − ΔP = Δσ − ΔP (5.19) m kk B ∂t kk B where cm is the hydraulic diffusivity given by: 2 2 1 k 2G(1 − ν) B (1 + νu) (1 − 2ν) cm = · (5.20) 3 η (1 − 2ν) 9(νu − 1)(νu − ν)

Now, we have equations for all terms in the diffusion equation (equation 5.19) describing poroelastic adjustment of stresses and pore pressures in the postseismic period. To initiate the diffusion process, equations 5.6 and 5.18 are used to calculate the coseismic change in ΔP and Δσkk, respectively. The initial value of the diffusion term in equation 5.19 is then calculated by taking Δσ − 3 Δ Δ Δσ the difference kk B P. The initial values of P and kk can also be used in equation 5.17 in order to solve for the individual components Δσij of the stress perturbation.

69 5.4 Pore pressure diffusion equation A diffusion equation in terms of pore pressure perturbations only is common in hydrogeological applications (e.g. Roeloffs, 1996; Neuzil, 2003). Such an equation can be derived under the assumptions that strain is purely vertical and that the Earth’s surface is a free surface with no changes in the overburden pressure. Mathematically, these assumptions are:

Δσzz = 0 Δεxx = 0 (5.21) Δεyy = 0

Applying the conditions for purely vertical strain (Δεxx = Δεyy = 0) in the constitutive equation 5.1 yields:

ν 3(νu − ν) Δσxx = Δσyy = Δσkk + ΔP (5.22) 1 + ν B(1 + ν)(1 + νu)

With Δσxx and Δσyy given by equation 5.22 and Δσzz = 0, we get the following equation for Δσkk:

ν 6(νu − ν) Δσkk = Δσxx + Δσyy + Δσzz = 2 · · Δσkk + ΔP (5.23) 1 + ν B(1 + ν)(1 + νu)

From equation 5.23, we can see that Δσkk becomes:

6(νu − ν) Δσkk = ΔP (5.24) B(1 − ν)(1 + νu)

Comparing with equation 5.18, we can thus see that the stress compatibility equations (5.17) are satisfied under the assumption of a purely vertical strain and no changes in the overburden pressure. This implies that Δσkk given by equation 5.24 can be inserted into the diffusion equation 5.19, yielding: (ν − )( + ν) ∂ (ν − )( + ν) 2 3 u 1 1 3 u 1 1 cm∇ ΔP = ΔP ⇔ B(1 − ν)(1 + νu) ∂t B(1 − ν)(1 + νu) ∂ c ∇2 (ΔP)= (ΔP) (5.25) m ∂t By assuming an isodense and isoviscous fluid, no significant elevation changes over time and incompressible solid grains, equation 5.25 results in the following diffusion equation (Neuzil, 2003): ∂ D∇2 (ΔP)= (ΔP) (5.26) ∂t

70 where ΔP is the coseismically induced pore pressure transient (5.6). The hydraulic diffusivity D is given by: k D = (5.27) η 1+ν + Φ 1 3(1−ν)K Kf where k is the permeability, η dynamic viscosity of the ﬂuid, Φ the porosity, K drained bulk modulus of the porous medium and Kf the ﬂuid bulk modulus (Neuzil, 2003).

5.5 Pore pressure diffusion modelling In my thesis, I model main shock initiated pore pressure diffusion processes in Paper IV and Paper V. In paper IV, I use equation 5.26 to model postseismic pore pressure diffusion initiated by the June 2000 earthquakes in Iceland. In Paper V, I investigate the dependency of pore pressure induced seismicity on the magnitude of the main shock initiating the diffusion process, again using equation 5.26. I also model the poroelastic diffusion process using equation 5.19 in order to investigate the effect of poroelastic adjustment of both stresses and pore pressures on seismicity induced by the diffusion process. In order to solve the poroelastic diffusion equation (5.19) and the pore pressure diffusion equation (5.26), I use a finite difference scheme. In both cases I use equations 5.6 and 5.7 to calculate coseismically induced pore pressure and stress perturbations in a three dimensional grid. The grid covers a volume surrounding the earthquake model used to calculate the coseismic strain perturbations. The calculations of the coseismically induced pore pressure perturbations provide the initial values when modelling the diffusion of pore pressures only, as described by equation 5.26. In this case the finite difference solution yields ΔP(x, y, z, t), i.e. the value of ΔP at each point in the grid for each time step t. When modelling poroelastic adjustment of stresses and pore pressures in the diffusion, I need to take additional steps in order to calculate initial values to use in the finite difference solution of the poroelastic diffusion equation (equation 5.19). First, I use equation 5.18 in order to calculate the mean stress perturbation Δσkk from the coseismic pore pressure perturbation ΔP.Given ΔP and Δσkk, I then calculate the term Δσkk −3ΔP/B at each point in the grid as initial values in the finite difference scheme. The finite difference solution (Δσ − Δ / )| now yields kk 3 P B x, y, z, t, and we can use equation 5.18 to calculate ΔP and Δσkk at each time step for each grid point. In order to fully evaluate the poroelastic adjustment of stresses in the diffusion process we then need, initially and at each time step, to solve the equation system 5.17 for each component of the stress perturbation tensor. This, however, requires several matrix inversions that, due to the size of the system

71 and the length of the modelled time period, is not feasible in terms of com- putation time. In order to investigate the effect of poroelastic adjustment of stresses, I therefore assume that relative proportions between elements of the stress perturbation tensor remain the same during the diffusion process. This implies that the stress perturbation tensor at time ti+1 is a scaled version of the tensor at time ti. The scaling factor at each time step n is calculated from Δσkk (tn)/Δσkk (tn−1). In order for the stress perturbation to satisfy force balance, equilibrium, and strain and stress compatibility equations, the coseismically induced stress perturbation is scaled with 2(νu − ν)/((1 − ν)(1 + νu)). For a more detailed description of the procedure I refer to Paper V.

5.6 Modelling of pore pressure induced seismicity To evaluate seismicity induced by the pore pressure diffusion process, I utilise the Mohr-Coulomb failure criterion (e.g. Scholz, 2002):

τ f = τ0 + μ · (σn − P) (5.28) where τ f is the shear stress required to overcome the shear strength of a fault, τ0 a cohesion term, μ the coefﬁcient of friction, σn the normal stress on the fault and P the pore pressure. Note that σn and P now refer to total stresses and pore pressures rather than a perturbation. In the modelling performed in Paper IV and Paper V, respectively, I assume that the following conditions apply prior to the main shock initiating the diffusion process: σ b 1. The background state of stress is strike-slip, with the principal stresses 1 σ b σ b σ b > σ b > σ b and 3 horizontal, 2 vertical and 1 2 3 . The direction of the max- σ b imum horizontal stress in the background state of stress ( 1 ) is assumed to be N30◦E. For the SISZ, this direction is consistent with the results of Lund and Slunga (1999) and Bergerat and Angelier (2000). 2. Optimally oriented faults (according to the Mohr-Coulomb failure criterion, equation 5.28) are on the point of failure in the background state of stress. 3. The regional pore pressure Pb is hydrostatic.

By adding Δσij and ΔP to the background state for stresses and pore pressures, respectively, total stresses and pore pressures are obtained. When modelling the diffusion of pore pressures only (equation 5.26), the total state of stress does not vary with time and is equal to the sum of the background stress state and the coseismically induced stress perturbation (equation 5.7). Mod- elling poroelastic adjustment of stresses in the diffusion process, however, implies a temporal variation also in the total state of stress. Given the total stress state, the critical pore pressure of faults that are optimally oriented for

72 failure is (according to the Mohr-Coulomb failure criterion, equation 5.28): 1 / / P = −σ (1 + μ2)1 2 − μ + σ (1 + μ2)1 2 + μ (5.29) crit 2μ 1 3 where μ is the coefﬁcient of friction and σ1 and σ3 are the largest and smallest principal stresses of the total stress, respectively. Note that Pcrit varies with time when modelling poroelastic adjustment of stresses in the diffusion process. Having calculated Pcrit, we then consider seismicity to be induced by the pore pressure diffusion process if ΔP + Pb becomes larger than Pcrit at a given time step. In that case, the occurrence time is found by linear interpolation of when, in the time interval since the previous time step, ΔP + Pb = Pcrit. I then assume that this point cannot be triggered again by the diffusion process. For speciﬁc details on how I evaluate the background state of stress, pore pressure and induced seismicity, I refer to Paper IV and Paper V of the thesis.

6. Tectonics of Iceland and the South Iceland Seismic Zone

My thesis focuses on the South Iceland Seismic Zone (SISZ), an on-land transform plate boundary segment of the Mid-Atlantic ridge. In this section I will describe the tectonic setting of Iceland and the SISZ, the Icelandic seismological network (SIL) and the two M6.5 earthquakes occurring within SISZ in June 2000. The aftershocks associated with these two earthquakes constitute a major part of the dataset that I study in my thesis. I will, therefore, also discuss the issue of completeness in the register of recorded aftershocks shortly after the occurrence main shocks within SISZ.

6.1 Tectonic setting of Iceland Iceland is situated on the Mid-Atlantic ridge, the divergent plate boundary that separates the Eurasian and North American lithospheric plates north of the Azores triple junction (38◦N) (figure 6.1). North of the major Charlie-Gibbs fracture zone at 52◦N, the spreading has been offset to the west and occurs at right angles to the plate boundary. The ridge trends N-S, has a well developed rift valley and a high rate of seismicity, typical for the Mid-Atlantic ridge. The characteristics of the ridge and the seafloor spreading, however, change systematically as Iceland is approached. The trend of the ridge changes to N35◦E, oblique spreading appears to take place north of 56◦N, the seismicity rate becomes lower and the central rift valley gives way to a central horst north of 58.5◦N (Einarsson, 1986). Thus, although the half spreading rate is small (1 cm/year), the Mid-Atlantic ridge just south of Iceland (the Reykjanes ridge) has some characteristics associated with fast spreading ridges. North of Iceland, the plate boundary follows the Kolbeinsey ridge, having a relatively high elevation, low seismicity and an asymmetric position with respect to the distance to adjacent continents (Einarsson, 1986). Further north, after being offset to the east by the major Jan Mayen fracture zone, the ridge continues in the Arctic, first northeastwards and later on northwards. Iceland is believed to owe its existence above sea level, with the unique situation of spreading taking place on land, to a plume of hot, partly molten material in the mantle beneath the island. Based on investigations of the seismic velocity structure in the mantle beneath Iceland, the plume center has been inferred to be located below the western part of Vatnajökull and the mid-

75 -50˚ -40˚ -30˚ -20˚ -10˚ 0˚ 10˚ 20˚ 75˚km 75˚ 0 500 JMFZ 70˚MAR 70˚ KOL 65˚ICELAND 65˚ RR

60˚ 60˚ NA 55˚CGFZ 55˚ 50˚ 50˚ 45˚EUR 45˚ MAR 40˚AZO 40˚ 35˚ 35˚ -50˚ -40˚ -30˚ -20˚ -10˚ 0˚ 10˚ 20˚ Figure 6.1: Tectonics in the vicinity of Iceland. NA: The North American plate, EUR: The Eurasian plate, MAR: The Mid-Atlantic ridge, AZO: The Azores triple junction, CGFZ: The Charlie-Gibbs fracture zone, RR: The Reykjanes ridge, KOL: The Kol- beinsey Ridge and JMFZ: The Jan Mayen fracture zone. Plate boundaries are collected from Cofﬁn et al. (1998) and Jóhannesson and Sæmundsson (1998).

Icelandic highlands (e.g. Wolfe et al., 1997; Darbyshire et al., 1998; Allen et al., 2002) (ﬁgure 6.2). This hypothesis is consistent with the region being the most volcanically active in Iceland during at least the last 800 years (Larsen et al., 1998) and the fast ridge spreading characteristics of the Mid- Atlantic ridge close to Iceland (Einarsson, 1986). Figure 6.2 illustrates the location of the presumed plume center in relation to the plate boundary zones that accommodate the onland spreading of the Mid-Atlantic ridge in Iceland. We can see that the inferred center of the plume is intersected by the Eastern and Northern Volcanic Zones. Beginning in the south of the island, however, the present day spreading initially follows the Reykjanes Peninsula. The spreading then becomes partitioned between two parallel and roughly northeast trending rift zones, the Western and East- ern Volcanic Zones (WVZ and EVZ). GPS measurements in south Iceland (Sigmundsson et al., 1995; La Femina et al., 2005) suggest that most of the spreading takes place in the EVZ and that spreading is dying out in the WVZ. Transform motion between the two rift zones is accommodated by the east- west trending South Iceland Seismic Zone (SISZ), which is described in more detail below. In the north of Iceland spreading is conﬁned to the Northern

76 Volcanic Zone (NVZ), which extends out to the north coast. Off the coast, the broad Tjörnes Fracture Zone (TFZ) offsets the spreading to the west in order to transfer motion to the ocean ﬂoor continuation of the Mid-Atlantic ridge north of Iceland. Figure 6.2 shows that the average geological spreading rate across Iceland, predicted by the NUVEL-1A global plate velocity model (DeMets et al., 1994), is 19 mm/yr in the direction N75◦W. In Iceland, the part of the present day plate boundary formed by the EVZ and the NVZ is offset to the east relative to the offshore, ocean bottom, continuation of the Mid-Atlantic ridge just south and north of the island. Two previously active rift zones have been found in western Iceland, the youngest on the Snæfellsnes Peninsula and the oldest in the Westfjords (Hardarsson et al., 1997) (ﬁgure 6.2). This implies that the plate boundary, with time, has migrated westwards relative to the plume center. It appears, however, that discrete eastward ridge jumps occurs when the plate boundary has migrated some critical distance, maintaining spreading over the Icelandic mantle plume (Sæmundsson, 1979). Such a process can explain the present-day low rate of spreading in the WVZ, the easterly offset of the plate boundary and the formation of the SISZ as spreading in the EVZ propagates southwards from the mantle plume (Einarsson and Eiriksson, 1982).

Figure 6.2: Geodynamics of Iceland. RP: Reykjanes Peninsula, H: Hengill triple junction, SISZ: South Iceland Seismic Zone, WVZ, EVZ and NVZ: Western, Eastern and Northern VolcanicZones, TFZ: Tjörnes Fracture Zone, WF: West Fjords, SP: Snæfell- snes Peninsula, V: Vatnajökull. Original ﬁgure supplied by Páll Einarsson, University of Iceland.

77 6.2 The South Iceland Seismic Zone 6.2.1 Mechanics of faulting The SISZ is a left-lateral transform plate boundary between the North Ameri- can and the Eurasian lithospheric plates, stretching 70-80 km eastwards from the Hengill triple junction to the EVZ. The relative motion between the plates is, however, not accommmodated by a major E-W oriented strike-slip fault. In- stead, the SISZ consists of a belt of 10-20 km long N-S trending faults (figure 6.3) where recent M>6 earthquakes (Stefánsson et al., 2000) and the surface style of faulting (Einarsson and Eiriksson, 1982) indicate right-lateral strike- slip motion during earthquake ruptures. Different explanations have been suggested for this way of accommodating the transform motion along the SISZ. Einarsson et al. (1981) and Einarsson and Eiriksson (1982) hypothesised that this behaviour may reflect that the SISZ is relatively young and that a major through-going E-W oriented fault has not been able to form, perhaps due to a southward propagation of the SISZ in response to the southward propagation of spreading in the EVZ. Another hypothesis is the bookshelf tectonic model (e.g. Cowan et al., 1986; Tapponnier et al., 1990; Phipps Morgan and Kleinrock, 1991; Reiser Wetzel et al., 1993), suggesting that transform motion between overlapping ridge segments is accommodated by strike-slip faulting on an array of faults that are oriented at high angles relative to the shearing transform motion and rotation of the blocks in between the faults. In the SISZ, the left-lateral transform motion can thus be accommodated by N-S right-lateral strike-slip faults with coun- terclockwise rotation of the blocks in between the faults (Einarsson, 1991). Boundary element modelling by Hackman et al. (1990) shows that the east- west left-lateral transform motion in south Iceland can be accommodated by right-lateral motion in an array of N-S trending faults if the fault spacing is less than 5 km and their lengths are of the order of 20-25 km. Stefánsson and Halldórsson (1988) propose a dual mechanism model where the N-S trending faults are formed in an interplay between the general E-W tensional stress associated with the spreading of the Mid-Atlantic ridge and horizontal compressional stresses at depth caused by outflow of the mantle plume. In this model, release of compressional stresses at depth by E-W oriented transform slip may, in the shallow parts of the crust, cause E-W tensile stresses (due to spreading) to dominate over compressional stresses (due to plume outflow). This enables N-S oriented vertical dikes to intrude at depths of 5-10 km from a thin layer of low electrical resistivity with a 10-20 % melt fraction (Stefánsson and Halldórsson, 1988). Shear stresses exerted from below are then taken up by the low rigidity dikes formed by these intrusions, resulting in earthquakes on N-S vertical faults when these stresses are released. Crustal deformation studies in the SISZ (Sigmundsson et al., 1995) suggest that about 85 %±15 % of the present-day relative motion between the Eurasian and North American lithospheric plates is accommodated by the

78 SISZ, and that left-lateral shear strain is accumulating across the zone. This interpretation is supported by an analysis of GPS-data between 1992 and 2004, suggesting that the observed N-S faulting in the SISZ is driven by E-W left- lateral shear below 15-20 km depth (Árnadóttir et al., 2006). To accommodate left-lateral shear strain with an array of N-S striking faults spaced 1-5 km apart requires an average slip rate on each fault of 0.5-5 mm/yr, implying that the present-day SISZ cannot be more than a few tens of thousand years old if bookshelf faulting is the mechanism (Sigmundsson et al., 1995). If the SISZ only takes up part of the motion, it implies that the remaining part, up to 15 %± 15 %, may be taken up by the WVZ. This indicates that the formation of the fault pattern in the SISZ may be more complicated than if only the EVZ is active. Gudmundsson and Brynjólfsson (1993) speculate that the fault pattern and faulting behaviour within the SISZ is the response to a stress ﬁeld associated with both the WVZ and the EVZ being active. Based on analysis of GPS-data between 1994 and 2003, La Femina et al. (2005) conclude that an along-strike, from southwest to northeast, decrease in spreading rate in the WVZ and increase in the EVZ is consistent with spreading being accommodated by the EVZ and the WVZ being deactivated. This supports the hypothesis of Einarsson and Eiriksson (1982) that the SISZ is formed as spreading in the EVZ propagates southwards from the mantle plume.

6.2.2 Historical seismicity Since the settlement of Iceland in the ninth century, several of the largest earthquakes in the country have taken place within the SISZ. Reports of earthquakes before the year 1700 are, however, regarded as incomplete (Einarsson et al., 1981). Typically, large earthquakes in the SISZ occur individually or in sequences, generally progressing from east to west and lasting from a few days to a few years. Figure 6.3 shows approximate locations and years for some of the known earthquakes/earthquake sequences in the SISZ. Sequences of very destructive earthquakes took place in 1784 and 1896, with the largest areas of destruction in the east, indicating a decrease in magnitude with the westward propagation of the sequence (Einarsson et al., 1981). The magnitude of the largest earthquake in the 1896 sequence has been estimated to 7.1 (Stefánsson and Halldórsson, 1988). The largest earthquake within the SISZ during the 20th century was a magnitude 7 earthquake in 1912, near the farm Selsund in the easternmost part of the zone. This earthquake was the ﬁrst in the SISZ to have an instrumentally determined magnitude (Karnik, 1969). In may 1987, a magnitude 5.8 earthquake at the Vatnafjöll volcano within the EVZ showed right-lateral strike-slip on a north-south striking fault, indicating an extension of the SISZ into the EVZ (Bjarnason and Einarsson, 1991; Agustsson et al., 1999). In the western- most part of the SISZ, high seismic activity took place at the Hengill triple junction between 1994 and 1998. It has been suggested that an increase in

79 pressure in the roots of central volcanoes in the area and the corresponding crustal deformation triggered the seismic activity (Sigmundsson et al., 1997; Feigl et al., 2000). In June 2000, two M6.5 earthquakes took place in the SISZ (see ﬁgure 6.3). These earthquakes are discussed further in section 6.4 as the associated aftershocks form a part of the dataset utilised in my thesis. In the SISZ, large earthquakes or sequences of large earthquakes have taken place with a 45 to 112 year interval (Einarsson et al., 1981). The two June 2000 earthquakes match this pattern as they took place almost 90 years after the 1912 earthquake in the SISZ. On May 29, 2008, a large earthquake that was felt over most of southwest Iceland occurred in the western part of the SISZ. The earthquake was caused by the rupture of two N-S oriented vertical strike-slip faults just outside the town of Hveragerdi, with a combined magnitude of M6.3 (Vogfjord et al., 2008). The occurrence of this earthquake, west of the June 2000 earthquakes, agrees with the pattern of a westward migration in sequences of large earthquakes in the SISZ. As this earthquake took place during the ﬁnal stage of my thesis work, I have not made any attempt to investigate the associated aftershock sequence. I believe, however, that such an analysis will yield additional insight into the earthquake process in Iceland.

-21.4˚ -21.2˚ -21˚ -20.8˚ -20.6˚ -20.4˚ -20.2˚ -20˚ -19.8˚ 64.2˚ 64.2˚ J21 J17

1896

64˚HVG 1784 64˚ 1784 1630 1896 1734 1896 SEL 1896 1912

HLA km 0 5 10 63.8˚ 63.8˚ -21.4˚ -21.2˚ -21˚ -20.8˚ -20.6˚ -20.4˚ -20.2˚ -20˚ -19.8˚ Figure 6.3: Active faults in the SISZ and historical seismicity. Black numbers: Years for known large earthquakes/earthquake sequences at their approximate locations (Einarsson et al., 1981). Thin black lines: Mapped faults in SISZ (Jóhannesson and Sæmundsson, 1998). Thick black lines: Faults of the June 17 and June 21, 2000, earthquakes. J17 and J21: Harvard CMT focal mechanisms of the June 17 (J17) and the June 21 (J21), 2000, earthquakes. HVG, SEL and HLA: The towns Hveragerdi, Selfoss and Hella. The inset shows Iceland and the location of the SISZ.

80 6.3 The SIL network The SIL network in Iceland was initiated in 1988 as a joint effort of the Nordic countries towards earthquake prediction research in the SISZ through the cre- ation of a high sensitivity seismic network (Stefánsson et al., 1993). The phi- losophy behind the design of the SIL network is to enable accurate locations, dynamic source parameter determinations and fault plane solutions for very small earthquakes in order to monitor the physical processes leading up to earthquakes. In particular, microearthquake information can provide a detailed view of the concentration and redistribution of stresses within the crust. This is highly valuable for attempts at earthquake prediction and understanding the earthquake process.

-23˚ -22.5˚ -22˚ -21.5˚ -21˚ -20.5˚ -20˚

J21 J17 64.2˚ 64.2˚

64˚ 64˚

63.8˚ 63.8˚ km 0 10 20 63.6˚ 63.6˚ -23˚ -22.5˚ -22˚ -21.5˚ -21˚ -20.5˚ -20˚ Figure 6.4: The SIL network (grey triangles) and seismicity (black dots) in the SISZ and on the Reykjanes Peninsula between 2000-06-17 and 2006-02-13. The focal mechanisms of the June 2000 earthquakes are shown (J17 and J21). Stations in the SIL network experiencing problems following the June 2000 earthquakes are circled (Kristin Vogfjord, Icelandic Meteorological Ofﬁce, personal communication). The inset shows SIL station locations for the whole of Iceland (white triangles).

The SIL network has collected data since the beginning of 1990, and fully automatic routine analysis has been in operation since July 1991 (Stefánsson et al., 1993). By the end of the year 2000, the SIL network had grown from the initial 8 stations in the SISZ to 42 stations, now covering also the Reykjanes peninsula, the Eastern and Northern Volcanic Zones and the Tjörnes Fracture Zone plus a few stations in the highlands (Jakobsdottir et al., 2002) (see ﬁgure 6.4). In the SISZ, the SIL network records earthquakes with a Mw ∼ 0 magnitude of completeness (Wyss and Stefánsson, 2006) and automatically determines the location, magnitude and focal mechanism for each recorded event (Ste- fánsson et al., 1993; Bödvarsson et al., 1999). Based on synthetic tests with the original conﬁguration of the network with 8 stations, Rögnvaldsson and

81 Slunga (1993) conclude that the focal mechanism inversion routine gives fault ◦ plane solutions for local events down to ML=0.5 correct to within ±15 in strike, dip and rake. Of great value for studies of aftershock processes, the automated analysis of the SIL network has shown that the system is capable of recording and detecting more than 1200 events occurring in a single day (Bödvarsson et al., 1999). By the end of 2007, the network had located and analysed more than a quarter of a million earthquakes in Iceland.

6.4 The June 2000 earthquakes in the SISZ Of the June 2000 earthquakes in the SISZ, the first struck in the afternoon the 17th of June, the Icelandic National Day. The epicentre of the earthquake was approximately 10-15 km north of the town Hella, near the farm Skamm- beinsstadir. Three and a half days later, about one hour after midnight on the 21st of June, the second earthquake struck about 15 km west of the first, with an epicentre located south of the mountain Hestfjall and about 10 km east of the town Selfoss. Due to good weather and the fact that many people were outside to celebrate the National Day when the first earthquake took place no one was seriously injured. However, several houses were severely damaged, especially in the town Hella, located south of the active fault in the first earthquake (Stefánsson et al., 2000). The second of the two earthquakes caused less damage due to the larger distance to more densely populated areas. The June 17th earthquake took place in an area where it was expected that the next large earthquake within the SISZ could occur. No short term precursors were recognized prior to the earthquake. A short term warning was, however, issued to the authorities 26 hours before the June 21 earthquake. The warning stated the likely location and that preparations should be made for an impending earthquake within a short period of time (Stefánsson et al., 2000). Figure 6.3 shows the main orientation of the fault traces and focal mechanisms for the two June 2000 earthquakes, referred to as the J17 and J21 events, respectively. Both focal mechanisms and aftershock spatial distribution (figures 6.4 and 6.6; Hjaltadóttir and Vogfjord (2005)) indicate N-S oriented strike-slip faulting on vertical faults, consistent with the typical faulting behaviour in the SISZ. Locally, however, the style of faulting is more complex, and mapping of the surface fractures of the two faults has revealed conjugate faulting taking place in both earthquakes (Bergerat and Angelier, 2003; Clifton and Einarsson, 2005). In the southernmost rupture zone of the J21 earthquake, left-lateral strike-slip faulting has been mapped on a 2.5 km long segment oriented N77◦E as well as right-lateral strike-slip faulting on a 500 m long segment oriented N30◦E (Clifton and Einarsson, 2005). Based on a joint inversion of InSAR and GPS-data measuring the coseismic deformation, Pedersen et al. (2003) have derived distributed slip models for each of the June 2000 earthquakes, assuming that rupture is confined to a

82 single plane (figure 6.5). Although these models are a significant simplication of the complex faulting behaviour, I use them in Paper IV to model the stress changes induced by the June 2000 earthquakes. Following the June 2000 earthquakes, a coseismic rise in water level was observed in geothermal wells in the compressional quadrants of the two faults and a drop in geothermal wells in the dilatational quadrants. In the postseismic period, the opposite pattern was observed with increasing water levels in the dilatational quadrants and decreasing levels in the compressional quadrants (Björnsson et al., 2001). Modelling of InSAR data has shown that the postseismic crustal deformation during the first two months after the June 2000 earthquakes could be explained by poroelastic rebound rather than by viscoelastic relaxation or afterslip (Jónsson et al., 2003). I investigate the implications of a pore pressure diffusion process for aftershocks within SISZ in two of the papers in this thesis (Paper IV and Paper V).

a) J17, M = 6.4 b) J21, M = 6.43 w w 0 250 0 250 200 −5 −5 200 150 150 100 −10 −10 100 Strike−slip [cm] Strike−slip [cm] Along dip 15 km 50 Along dip 15 km 50

−15 −15 −10 0 10 −10 0 10 Along strike 21 km Along strike 21 km Figure 6.5: Slip models of a) the June 17 and b) the June 21, 2000 earthquakes in Iceland (Pedersen et al., 2003). The largest slip is in the order of 2.5 m in the June 17 model and 2.9 m in the June 21 model. The models correspond to moment magnitudes Mw = 6.40 (June 17) and Mw = 6.43 (June 21), respectively.

6.5 Studied aftershock sequences In my thesis, I study aftershock sequences associated with main shocks occurring within the SISZ. Figure 6.6 shows earthquakes located in the hypocentral region of the two M6.5 June 2000 earthquakes. The main map shows seismicity from the occurrence of the June 17 (J17) main shock until 2004-02-29, plotted together with the focal mechanisms of the J17 and June 21 (J21) main shocks. The two interior boxes within the main map are used in Paper V to de- ﬁne the June 2000 aftershocks as aftershocks of the J17 and J21 main shocks, respectively. Slightly different divisions are used in Paper I and Paper IV but are not shown here for clarity. The external box deﬁnes the limit of a region in which the lower right inset shows seismicity occurring from 1992-01-01 until

83 just before the J17, 2000, main shock. The lower right inset also shows the location and focal mechanism of a M4.5 earthquake of September 27, 1999 (S27), whose aftershock sequence is also studied in Paper V. The S27 main shock is located about 5 km west of the J21 main shock and the S27 aftershocks are all located within the J21 box.

-21˚ -20.5˚ -20˚ km 0 10 20 64.2˚J21 J17 64.2˚

64˚ 64˚

S27

63.8˚ 63.8˚

-21˚ -20.5˚ -20˚ Figure 6.6: Spatial distribution of seismicity in the hypocentral region of the two M6.5 earthquakes in the south Iceland seismic zone on June 17 and June 21, 2000. Main map: Seismicity taking place from immediately after the occurrence of the June 17 (J17) main shock until 2004-02-29, plotted together with the focal mechanisms of the J17 and June 21 (J21) main shocks. The two interior boxes are used in Paper V to define the June 2000 aftershocks as J17 and J21 aftershocks, respectively. Lower right inset: Seismicity within the region defined by the largest box in the main plot, occurring from 1992-01-01 until just before the J17, 2000, main shock. In the lower right inset the focal mechanism of a M4.5 earthquake of September 27, 1999, is shown. The associated aftershock sequence is studied in Paper V. Upper right inset: Iceland and the location of the region shown in this figure.

Figure 6.7 a) shows the cumulative number of earthquakes versus time (since January 1, 1992) within the J17 and J21 boxes shown in the main plot of figure 6.6. These plots clearly indicate the S27, J17 and J21 main shocks through the rapid increase in the number of earthquakes immediately following their respective occurrence. In figure 6.7 b), c) and d), I show histograms of the number of earthquakes versus time for the first day of the S27 aftershock sequence (b)) and, following the J17 main shock, for the first 10 and 14 days in the J17 (c)) and J21 (d)) boxes, respectively. Figure 6.7 b) - d) also

84 show the number of earthquakes predicted by the Omori law, using the maximum likelihood estimates of the Omori law parameters for each aftershock sequence. The estimates of the Omori law parameter c for the S27, J17 and J21 aftershock sequences are all non-zero, implying that the rate of aftershocks is roughly constant initially.

4 x 10 a) J17+J21 boxes c) J21 box, S27 aftershocks 2 80 J17 box J21 box 1.5 60

1 40

S27 0.5 20

J17, J21 No. of earthquakes

Cum. no. of earthquakes 0 0 0 1000 2000 3000 4000 0 0.5 1 Days after 1992−01−01 Days after M4.5 S27 earthquake c) J17 box d) J21 box 150 300 J21 J21 100 200 J17

50 100 No. of earthquakes No. of earthquakes 0 0 0 1 2 3 4 5 6 7 8 9 10 −4 −2 0 2 4 6 8 10 Days after M6.5 J17 earthquake Days after M6.5 J21 earthquake Figure 6.7: a) Cumulative number of earthquakes versus time within the J17 (solid curve) and J21 (dashed curve) boxes, respectively (figure 6.6). The arrows indicate occurrence times of the S27 as well as the J17 and J21 main shocks. b) Number of earthquakes versus time during the first day of the S27 aftershock sequence. The curve represents Omori’s law with parameters K = 265.7, c = 5295.8 and p = 1.04. c) and d) Number of earthquakes versus time within the J17 and J21 boxes, during the first 10 and 14 days after the J17 main shock, respectively. The occurrence times of the J17 and J21 main shocks are indicated by the vertical lines. The curves represent the Omori law, with parameters K = 116.7, c = 104219.0 and p = 0.84 for the J17 aftershock sequence, and K = 104.8, c = 49306.0 and p = 0.82 for the J21 aftershock sequence.

6.6 Data completeness in studied aftershock sequences In studies of aftershock sequences, it is important to assess whether the roughly constant rate in the beginning of aftershock sequences only represents an incomplete detection of earthquakes. Although I address this issue in Paper IV, I will here provide an additional discussion on the topic. In my thesis, the important issue is whether incomplete detection of aftershocks is entirely responsible for the non-zero value of the Omori law parameter c in

85 the studied aftershock sequences. In Paper IV, I find that this is not the case, and that the initially constant rate of aftershocks in SISZ reflects a physical process. If the true value of c is indeed zero, we can use Omori’s law to estimate the number of aftershocks that have been missed during the time period with an initially constant rate of aftershocks. Given the maximum likelihood estimates of the Omori law parameters (see figure 6.7), we can use equation 2.6 in order to estimate the number of missed events. Between the occurrence time of the first aftershock and the estimated value of c for each sequence, the number of undetected aftershocks would then be 3790 in the M4.5 S27 aftershock sequence, 3804 in the M6.5 J17 aftershock sequence and 3295 in the M6.5 J21 aftershock sequence. In figure 6.7 c), we can see that the number of aftershocks within the J17 box is very low during the first 10 hours following the J21 main shock. In the study of Daniel et al. (2008), they consider that this is caused by the J21 main shock disguising aftershocks taking place within the J17 fault zone region. Shortly after a large main shock, intense activity is likely to cause some aftershocks to be undetected by the seismic network. We know, however, that the Gutenberg-Richter law is applicable to many earthquake datasets. The number of large earthquakes having the potential for swamping smaller events is thus quite small, and the number of earthquakes increases rapidly with decreasing magnitude. As small earthquakes exhibit high frequency wave trains that are of short duration (e.g. Brune, 1970; Boatwright, 1980; Lay and Wallace, 1995), we thus consider that individual earthquakes should become identi- fiable and separable within a relatively short time following a main shock. By analysing seismograms following the Mw6.6 2004 mid-Niigata Prefecture earthquake in Japan, Enescu et al. (2007) find that a small, but non-zero, value of c is a reliable result. In a study of four aftershock sequences in Japan, Nanjo et al. (2007) infer that a non-zero value of c cannot always be the result of incomplete detection at short times following a main shock. Following the June 2000 main shocks, some of the stations in the SIL network experienced problems with the recording of data (Kristin Vogfjord, Ice- landic Meteorological Office (IMO), personal communication). In figure 6.4 we can see that these stations (marked with circles) are located on the Reyk- janes Peninsula, quite far from the J17 and J21 epicentres. We may thus consider it rather unlikely that these stations played a significant role in causing incomplete detection of aftershocks. The analysis of the data recorded by the SIL network following the June 2000 earthquakes is still not completely finished, but has been progressing over the years (Kristin Vogfjord, IMO, personal communication). The dataset used in my thesis was extracted from the IMO database in February 2004. Extracting the same spatial and temporal window again in December 2008 reveals 413 and 257 additional earthquakes within the J17 and J21 boxes, respectively. These numbers are rather low in relation to the number of aftershocks estimated to be missing if the true value

86 of c is zero. We thus consider it unlikely that a completion of the data analysis following the June 2000 main shocks will result in additional aftershocks that can account for those that are still “missing”. If c is equal to zero, the estimated number of missing aftershocks is of the same order for the S27 and the J17 aftershock sequences. The magnitudes of the S27 (M4.5) and J17 (M6.5) main shocks, however, are significantly different. The M4.5 S27 main shock is, due to its significantly smaller magnitude, not expected to cause problems for the SIL network to detect aftershocks to the same degree as the larger M6.5 J17 main shock. That c is non-zero also for the S27 aftershock sequence thus supports the hypothesis of Paper IV that an initially constant rate of aftershocks in SISZ reflects a physical process. So far, my discussion on data completeness has been mainly qualitative. In the next section, however, I will describe an attempt to assess whether systematic detection problems may result in a non-zero value of c in the studied aftershock sequences within SISZ.

6.7 Assessment of systematic detection problems If the true value of c is zero for an aftershock sequence, we may suspect that systematic detection problems of the seismological network is the cause of a non-zero value of c. In order to assess this possibility for the studied SISZ aftershock sequences, I use different ways of systematically removing events from simulated, “complete” (c=0), aftershock sequences. The “complete” aftershock sequences are simulated using the stretching algorithm (section 3.1) with c = 0 and the estimated values of K and p for the S27, J17 and J21 aftershocks, respectively. The difference in the number of aftershocks within the first c seconds of the simulated and the observed sequences then define the number of events to be removed in each case. In this analysis, I use increasingly complex ways to remove, at random, events from simulated aftershock sequences. The different alternatives include random removal of events during the first c seconds, within differently chosen time intervals from the main shock (always < c) and with interevent times below differently chosen interevent time limits. In these alternatives, the removal is systematic as the prescribed rules define the events where random removal shall take place. An end member in the range of increasingly complex removal alternatives is to use the recorded data as a filter to define the events to be removed. This is done by dividing the first c seconds of the S27, J17 and J21 aftershock sequences into equally sized bins and counting the number of aftershocks in each bin. These numbers determine the number of events that are to be kept from the corresponding bins of the simulated sequence. The events to be kept in each bin are then selected at random. With this alternative, the removal of events is no longer systematic as the data exhibit non-systematic

87 deviations from the Omori law in the beginning of respective aftershock sequence (see figure 6.7 b), c) and d)). After the removal of events, I utilise the Kolmogorov-Smirnov test (section 3.3) in order to assess the following two null hypotheses: A0: During the first c seconds, the resulting distribution of simulated occurrence times after removal cannot be distinguished from the distribution of observed occurrence times. B0: The interevent time distribution associated with the first c seconds of the simulated sequence after removal cannot be distinguished from the corresponding distribution of interevent times for the observed aftershock sequence. If neither the null hypothesis A0 nor the null hypothesis B0 can be rejected, we can neither rule out the possibility of incomplete detection nor a physical process as responsible for a non-zero value of c. However, if we can reject either, or both, of the two null hypotheses, we can conclude that the initial behaviour of aftershock sequences cannot represent incomplete detection in the manner given by the removal alternative. For each removal alternative, I perform 1000 simulations of “complete” aftershock sequences and determine the number of times when neither A0 nor B0 can be rejected at 5 % significance level. For the S27, J17 and J21 aftershock sequences, this analysis shows that either one or both of A0 and B0 can be rejected for a large proportion of the simulations when using the systematic removal alternatives. We can thus conclude that the systematic removal of events reflected by these alternatives are unlikely to result in aftershock sequences matching the observed data. When using the observed aftershock sequences as a filter to remove events, however, the null hypotheses A0 and B0 can never be rejected in the simulations. We may interpret this in terms of a non-linear and unsystematic potential for the seismological network to detect aftershocks at short times after a main shock. We may also, however, make the interpretation that systematic detection problems are an unlikely cause of the initial behaviour in the S27, J17 and J21 aftershock sequences. In this case, we can interpret the roughly constant rate in the beginning of the studied aftershock sequences as a reflection of a physical process. Such an interpretation is the topic of Papers IV and V in my thesis.

88 7. Summary of papers

7.1 Paper I Modelling fundamental waiting time distributions for earthquake sequences Overview The temporal behaviour of earthquake sequences can be examined by investigating the distribution of occurrence times for earthquakes from a defined geographical area. An alternative approach is to investigate the distribution of waiting times, i.e. the time interval between successive events in the sequence. The term waiting time can also be referred to as e.g. interevent time. Empirical waiting time distributions for earthquake sequences have been used to look for universal distribution parameters in different earthquake datasets (Corral, 2003, 2004a,b; Davidsen and Goltz, 2004) and as a basis for proposing a “unified scaling law”, relating earthquake magnitude, the size of the area investigated, waiting time, the b-value in the Gutenberg-Richter law and the epicentral fractal dimension (Bak et al., 2002; Christensen et al., 2002). It has been claimed that a characteristic falloff from power law behaviour at large waiting times is related to a change from correlated aftershocks of the same sequence to uncorrelated events (Bak et al., 2002). This feature is considered to reflect important physics of the system (Bak et al., 2002). We examine waiting time distributions for two simple but important models of temporal earthquake distributions: The homogeneous Poisson distribution for main shock occurrence and a distribution of occurrence times in an aftershock sequence obeying the Omori law. The Poissonian and Omori law waiting time distributions are analysed theoretically and numerically (utilising random simulations of the corresponding earthquake sequences), and compared with earthquake sequences observed in Iceland. For an observed sequence of events, an empirical waiting time distribution can be obtained by counting the number of waiting times in histograms with bins increasing logarithmically in size. The shape of the underlying probability density function is then given by normalising the number of waiting times in each bin with the corresponding bin size. The probability density function of the Poissonian waiting time distribution is: −μΔ f (Δt)=μ · e t (7.1)

89 a) Fig. 3 a), Paper I b) Fig. 3 e), Paper I c) Fig. 3 f), Paper I 4 5 5 10 10 10

3 10 64 yr 105

2 0 0 10 10 10 104 Count 16 yr

1 Scaled pdf 3 10 10

0 −5 −5 10 10 10 0 2 4 6 0 2 4 6 0 2 4 6 10 10 10 10 10 10 10 10 10 10 10 10

d) Fig. 4 a), Paper I e) Fig. 4 e), Paper I f) Fig. 4 f), Paper I 5 5 10 10

4 10

105 2 10 64 yr 0 0 10 10 Count 0 10 Scaled pdf 104 103 4 yr

−2 −5 −5 10 10 10 0 2 4 6 8 0 2 4 6 8 0 2 4 6 8 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 Waiting time (s) Waiting time (s) Waiting time (s) Figure 7.1: Summary of figure 3 and 4 in Paper I. Crosses: Waiting time distributions and probability density functions for numerical realisations of Poisson and Omori law sequences. Solid lines: Theoretical curves for the corresponding distribution and probability density functions. The plots are regenerated and may therefore (due to randomness in the numerical realisations) exhibit slight differences compared to the plots in Paper I. a) Figure 3 a) in Paper I: Waiting time histogram for two Poisson sequences with rate μ=625 events/year observed for 16 and 64 years, respectively. b) Figure 3 e) in Paper I: Probability density functions for Poisson sequences with 100 000, 10 000 and 1000 events over 16 years, respectively. c) Figure 3 f) in Paper I: Aggregated Poisson waiting time distribution obtained by aggregation of data from figure b). d) Figure 4 a) in Paper I: Waiting time histogram for two Omori sequences (K = 90, C = 0 and p = 1) observed for 4 and 64 years, respectively. e) Figure 4 e) in Paper I: Probability density functions for three Omori sequences with 100 000 (p = 1, K = 5000, C = 0), 10 000 (p = 1, K = 500, C = 0) and 1000 (p = 1, K = 50, C = 0) events over 16 years. f) Figure 4 f) in Paper I: aggregated Omori waiting time distribution obtained by aggregation of data from figure e).

where μ is the underlying rate of the sequence and Δt is the waiting time. Wait- ing time histograms with logarithmically increasing bin sizes for two Poisson sequences of different duration but the same underlying rate are shown in ﬁg- ure 7.1 a). The ﬁgure demonstrates that the location of the peak in the distributions, corresponding to a waiting time 1/μ, has no dependence on sequence duration. Figure 7.1 b) demonstrates the shape of probability density functions for Poissonian sequences of the same duration but with different underlying rates. The probability density functions are characterised by a constant regime that is followed by a rapid falloff. The point of separation between the two regimes appears at shorter and shorter waiting times the larger the underlying rate is. In the analysis of Bak et al. (2002), waiting time distributions from

90 different geographical areas are added together, or aggregated. Aggregation of Poissonian waiting time distributions with different underlying rates results in an initially constant part that is followed by an apparent power law decay (figure 7.1 c)). The point of separation between the two regimes is governed by the Poissonian sequence with largest rate. The Omori law describes the rate of decay of aftershocks as (Scholz, 2002): dn K = = λ(t) (7.2) dt (C +t)p where n is the number of events following the main shock, K, C and p are empirical constants, and t is the time since the main shock. In order to derive a theoretical equation for the Omori law waiting time distribution, we assume that the Omori law defines continuously changing rates of homogeneous Pois- son processes that are only valid momentarily, for infinitesimally short periods of time. The waiting time probability density function is then given by: T 2 1 K − K ·Δt (Δ )= · (C+t)p · f t 2p e dt (7.3) Ntot 0 (C +t) where Δt is waiting time, K, C and p are the Omori law parameters, T is the length of the time period of observation of the aftershock sequence and Ntot is the total number of waiting times. This equation is analysed further in order to investigate the characteristic features of the Omori law waiting time distribution. This analysis shows that the theoretical waiting time distribution for Omori law aftershock sequences is asymptotically constant in the range [0, (2C)p/K], decays asymptotically as a power law with exponent 2 − 1/p in the range [(2C)p/K, ((C + T)p)/K] and falls off rapidly for waiting times larger than ((C + T)p)/K. Figure 7.1 d) shows that the number of waiting times for Omori sequences (with identical parameters) is roughly constant at first (when C = 0) and then starts to fall off rapidly at a waiting time controlled by the time for which the sequence has been observed. Waiting time probability density functions for Omori law sequences (when C = 0) exhibit a power law decay followed by a rapid fall off, as shown in figure 7.1 e). The exponent of this power law is 2−1/p. This figure also demonstrates that the location of the fall off is not only controlled by the duration of the sequence, but also the Omori law parameters. When aggregating Omori law waiting time distributions with different parameters, figure 7.1 f) demonstrates that this results in a transition from the power law decay with exponent 2 − 1/p into an apparent power law decay with another exponent. Figure 7.1 d-f) also illustrate waiting time distributions for Omori law aftershock sequences that are numerically simulated with an algorithm that we introduce. We refer to this algorithm as the stretching algorithm as it is based on stretching the time axis in order to yield Omori law behaviour. In the pre- sentation of the algorithm in the appendix of Paper I, however, there are typo-

91 graphical errors in equations A1 and A5 that I wish to point out. In equation A1 the natural logarithm should be used, i.e.:

N = K · [ln(C + T) − ln(C + Tstart )] (7.4)

In equation A5, the stretching of the time between events should take place by raising 10 to the power of x (not by multiplying 10 with x), i.e. : x tocc = 10 (7.5)

We apply the characteristic features of Poissonian and Omori law waiting time distributions to the analysis of earthquake sequences recorded in Iceland (figure 7 in Paper I). Aftershock sequences following the June 2000 earthquakes in Iceland show clear correspondence with the theoretical waiting time distribution of a single sequence of events governed by the Omori law. In the Hengill volcanic region, the waiting time distribution agrees relatively well with the Omori law distribution at short and intermediate waiting times. The long waiting time tail, however, does not show a rapid falloff as for a single aftershock sequence. Instead, it appears to reflect waiting time distributions of several Omori aftershock sequences of different rates, that are largely separated in time and therefore become aggregated. A final data example from the Katla volcanic area shows more of a Poissonian character of the corresponding waiting time distribution.

Discussion and conclusions Understanding the properties of waiting time distributions for fundamental earthquake sequences is important for drawing conclusions regarding the physics of the underlying process. Earthquake time series can be complex and reﬂect different processes. However, as many large earthquake datasets are dominated by aftershock activity for considerable periods of time, they can be expected to be well modelled by a single Omori aftershock sequence and the corresponding waiting time distribution. Our analysis shows that the Omori law waiting time distribution is characterised by an initially constant part, a power law decay with exponent 2 − 1/p and a rapid falloff controlled by the Omori law parameters and the length of the time period for which the aftershock sequence has been observed. A rapid falloff in waiting time distributions for real earthquake sequences is therefore a natural consequence of time-limited Omori law earthquake sequences. The investigated earthquake sequences from Iceland illustrate that waiting time analysis can provide insight into earthquake processes without the neces- sity of identifying main shocks and aftershocks. Although the data is largely consistent with Omori law and Poissonian behaviour there are signiﬁcant deviations from these simple models. We consider that these deviations may be useful in elucidating the underlying physics of earthquake processes. Statisti- cal analysis of earthquake data, however, should be regarded as just one of the

92 components in the development of physical models of earthquake processes. We conclude that derived statistical models may ultimately become misleading without a suitable physical understanding.

7.2 Paper II Earthquakes descaled: on waiting time distributions and scaling laws Overview Large earthquake datasets, such as from California, have been used in order to study the distribution of waiting times between successive earthquakes (e.g. Bak et al., 2002; Christensen et al., 2002; Corral, 2003, 2004a). Based on waiting time distributions for an earthquake dataset from California, Bak et al. (2002) propose a unified scaling law relating the Omori law, the Gutenberg- Richter law (Gutenberg and Richter, 1944) and the fractal dimension for the α −bm D distribution of earthquake epicenters such that T Pm,L = f (T × 10 L ).In the proposed scaling law T is waiting time, Pm,L the waiting time distribution (a histogram with logarithmic bins, normalised with bin size), α the slope of the power law regime of the distribution, m magnitude threshold, L cell size, b the exponent in the Gutenberg Richter law and D the fractal dimension of the epicentral distribution. Applying this scaling law, the observed waiting time distributions collapse onto a single curve consisting of an approximately constant part and a rapidly decaying part, separated by a sharp kink. Bak et al. (2002) claim that the approximately constant part represents “correlated” aftershocks (i.e. belonging to the same aftershock sequence) while the decaying part represents “uncorrelated” earthquakes, and that the slope of the power law regime (α) corresponds to the exponent p in the Omori law. Large earthquakes can generate aftershock sequences containing many thousands of events which dominate the seismic activity in an area for long periods of time. During such periods the empirical Omori law for aftershocks (Utsu, 1961) can, alone, give a good description of the earthquake activity and, thus, the distribution of waiting times between successive events. We therefore utilise waiting time distributions of a single aftershock sequence obeying the Omori law in order to investigate the unified scaling law and the claims of Bak et al. (2002) regarding the physics of earthquake processes. The threshold magnitude rescaling (10−bm) of the waiting time axis is found to be a direct consequence of the Gutenberg-Richter law and does not provide new information on earthquake processes. Moreover, we show that α in the unified scaling law is, generally, not identical to the parameter p in the Omori law. We also discuss some problems with the spatial scale factor (LD) and suggest some caution in interpreting D in the unified scaling law as a proof of a spatial fractal dimension of the data. Next, we discuss the physical significance of the bend (or kink), interpreted by Bak et al. (2002) as marking a change in the statistics from “correlated” to

93 “uncorrelated” events. We demonstrate that the existence of a rapid decay or falloff is a fundamental characteristic of aftershock sequences that have been observed for limited periods of time. With the definition of successive events as correlated if they belong to the same aftershock sequence this implies that there is no change in correlation at the bend. As real earthquake datasets can be complex we investigate whether this conclusion is valid for more realistic datasets, consisting of several main shock - aftershock sequences and possibly some degree of background seismicity. Our analysis show that the bend and falloff observed for a single aftershock sequence is also present for realistic earthquake datasets. The deviation of these distributions from that of a single aftershock sequence, however, mainly appear to the left of the bend. More- over, with the same definition of correlated events as above, we find that a mixture of correlated and uncorrelated events contribute to the waiting time distribution over a wide range of waiting times, both to the left and to the right of the bend.

Discussion and conclusions In order to draw conclusions regarding the physics of earthquake processes from waiting time distributions it is important to understand the basic characteristics of typical distributions, such as those given by e.g. the Omori law. The Omori law for aftershocks deﬁnes a reference frame for many earthquake datasets, including the data from California studied by Bak et al. (2002). If empirical waiting time distributions deviate from this reference frame this may provide new information on the physics of earthquake processes. However, in order to extract this information, the observed waiting time distributions must be properly modelled. Our modelling of realistic earthquake datasets results in waiting time distributions that appear consistent with data presented by Bak et al. (2002). This suggests, as we might expect, that the waiting time distribution for California is dominated by different aftershock sequences rather than Poisson distributed seismicity with only a few aftershock sequences. Our modelling also show that there is no distinct waiting time between two successive events that can be used to deﬁne them as either correlated or uncorrelated. In order to gain a deeper understanding of earthquake processes using waiting time distributions, we conclude that we should not only look for generalised and scale-independent laws but also, using suitable modelling, study deviations from “self similarity” and power law behaviour.

94 7.3 Paper III Reply to Comment on Earthquakes Descaled: On waiting time distributions and scaling laws Overview In a comment to Lindman et al. (2005) (Paper II in my thesis) Corral and Christensen (2006) claim that the kink in observed waiting time distributions does correspond to a change in the correlation regime, and that our discussion of the uniﬁed scaling law is inadequate as it does not consider the aggregation procedure of Bak et al. (2002). In order to address these comments we consider aggregation of Omori law p (dn/dt = C1/(C2 +t) ) waiting time distributions where the values of C1 are distributed according to a power law as suggested by Corral and Christensen ( ) ∝ / 0.8 (2006) (P C1 1 C1 ). Here, C1 and C2 denote the Omori law parameters that are usually referred to as K and c, respectively. Our analysis illustrates that: 1) the position of the kink in aggregated Omori waiting time distributions is controlled by the position of the kink in the included distribution with the largest value of C1 and the period of time for which the corresponding aftershock sequence has been observed (Jónsdottir et al., 2006), 2) the kink does not represent a change in correlation as the aggregation procedure mixes waiting times both from before and after the kink in the individual distributions. Altogether, this implies that it is not possible to draw the general conclusion that the kink represents events being at the "onset of correlation with the mainshock", as claimed by Corral and Christensen (2006). This understanding is important as the uniﬁed scaling law proposed by Bak et al. (2002) focuses on the near power law regime at shorter waiting times and the kink where this regime ends.

Discussion and conclusions The effect of Poissonian distributed background seismicity on aftershock sequences can, for single as well as aggregated distributions, be noticed over a wide range of waiting times, both before and after the kink, and not only in the tail as Corral and Christensen (2006) appear to claim. Aggregated waiting time distributions does not change our conclusions in Lindman et al. (2005), namely that the kink does not represent a change in correlation regime, and that no distinct waiting time can be used to deﬁne two successive earthquakes as either correlated or uncorrelated. We conclude that earthquake waiting time distributions may indeed contain new and important information about earthquake processes. However, to access this potential information it is necessary to also model the data, rather than only seeking simplifying and potentially misleading generalisations such as “uniﬁed scaling laws”.

95 7.4 Paper IV Physics of the Omori law: Inferences from interevent time distributions and pore pressure diffusion modeling Overview In aftershock sequences it is commonly observed that an initially (roughly) constant rate of aftershocks is followed by a power law decay in the rate. This observation is the basis for the empirical Omori law, describing the decay in p the rate of aftershocks as dn/dt = C1/(C2 + t) , where n is the number of aftershocks, C1, C2 and p are empirical constants, and t is the time since the main shock (Utsu, 1961). Here, C1 and C2 denote the Omori law parameters that are usually referred to as K and c, respectively. The onset of the power law decay in the Omori law is controlled by the empirical constant C2, measuring the duration of the period with an initially constant rate. In studies of aftershock sequences it is a debated issue whether the initially constant rate reflected by the Omori law parameter C2 relates to the physics of aftershock generation at short times after a main shock or if it is related to incomplete detection of aftershocks. The aim of this paper is to propose a physical model that can explain a non-zero value of C2 (i.e. an initially constant rate) for aftershock sequences following two M6.5 earthquakes, taking place within the south Iceland seismic zone (SISZ) on June 17 and June 21, 2000. At very short times after a main shock some aftershocks are inevitably missed due to the high rate of activity and the swamping of small events by larger ones. It is therefore important to assess the extent of aftershocks missed by the recording seismic network. Utilising interevent time distributions for the aftershocks associated with the June 17 and the June 21 main shocks, respectively, aftershocks with interevent times less than 100 seconds appear to be incompletely recorded (figure 7.2 a) and b)), if the true value of C2 is indeed zero. The difference between the interevent time distributions with zero and non-zero C2 in figure 7.2 a) and b) tells us that the number of missed aftershocks would then be about 7000 and 7600, respectively. To assess these numbers, we consider the distribution of earthquake magnitudes according to the Gutenberg-Richter law (Gutenberg and Richter, 1944) for aftershocks with interevent times less than and larger than 100 seconds, respectively (figure 7.2 c) and d). The magnitude of completeness is about 0.7 for aftershocks with interevent times less than 100 seconds. Considering a zero magnitude of completeness for the seismic network in SISZ (Wyss and Stefánsson, 2006), an extrapolation from 0.7 to 0 of the frequency-magnitude distribution implies that roughly 800 and 1500 aftershocks (with interevent times less than 100 seconds) following the June 17 and the June 21 main shock, respectively, have been missed. The number of aftershocks inferred to be missing in the two aftershock sequences is thus much lower than predicted by a zero value of C2. Based on interevent time distributions and complete-

96 a) b)

6 All eqs 6 All eqs 10 C =0 10 C =0 2 2 C =334166 C =206314 2 2 4 4 10 10 Count Count 2 2 10 10

0 0 10 10 0 2 4 6 8 0 2 4 6 8 10 10 10 10 10 10 10 10 10 10 Interevent time [s] Interevent time [s] c) d)

6 Interevent time > 100 s 6 Interevent time > 100 s 10 Interevent time < 100 s 10 Interevent time < 100 s Extrapolation to m =0 Extrapolation to m =0 c c 4 4 10 10 N>=m N>=m 2 2 10 10

0 0 10 10 −2 0 2 4 6 −2 0 2 4 6 Magnitude Magnitude Figure 7.2: Figure 2 in Paper IV. a) and b) Interevent time distributions (squares) for the J17 and J21 aftershock sequences, respectively. The curves represent theoretical interevent time distributions for Omori law aftershock sequences that have been ﬁtted to the data. The parameters of the curves are a) C1 = 1628, p = 1, C2 = 0 (solid curve) and C2 = 334166 (dashed curve) respectively, and in b) C1 = 1853, p = 1, C2 = 0 (solid curve) and C2 = 206314 (dashed curve). c) and d) Frequency- magnitude distribution according to the Gutenberg-Richter law for June 17th (c) and June 21st (d) aftershocks having interevent times above (dashed curve) and below (solid curve) 100 seconds, respectively. The dotted curve is an extrapolation (according to the Gutenberg-Richter law) of each frequency-magnitude distribution down to magnitude zero, marked by a vertical line.

ness arguments, we thus infer that a non-zero value of C2 cannot be explained solely by an incomplete detection of events but must also reflect a physical process in operation shortly after the main shock. Following the June 2000 earthquakes in Iceland, water level changes in geothermal wells (Björnsson et al., 2001) and postseismic poroelastic rebound observed by InSAR (Jónsson et al., 2003) demonstrated that diffusion of pore pressure transients played a significant role in the postseismic crustal deformation. We know that the coseismic deformation associated with earthquakes modifies the stress and strain fields in their vicinity, leading to temporary changes in pore pressure in the surrounding crust. In areas of coseismic di- latation, pore pressure will decrease while an increase will take place in areas of coseismic compression. The induced pore pressure transients will then diffuse through the crust and equilibrium will eventually be reached. During the diffusion process it can be expected that aftershocks will occur in areas of increasing pore pressure.

97 We model the diffusion process following the June 2000 earthquakes through a ﬁnite difference solution of the diffusion equation 2 D∇ Pind = ∂Pind/∂t, where D is the hydraulic diffusivity and Pind is the coseismically induced pore pressure transient. We use the equations of Okada (1992) with the fault slip models of Pedersen et al. (2003) to calculate stresses and pore pressures induced by the two June 2000 earthquakes in two three-dimensional grids, each covering a volume surrounding the June 17 and June 21 fault, respectively. In order to evaluate where and when earthquakes in the two grids will be triggered by the diffusing pore pressure transients we utilise the Mohr-Coulomb failure criterion and an assumption of a critically stressed crust prior to the two June 2000 earthquakes.

a) b) −1 −1 10 10 Triggered points Triggered points −2 Theoretical curve −2 Theoretical curve 10 10

−3 −3 10 10

−4 −4 10 10 Histogram count Histogram count

4 5 6 7 4 5 6 7 10 10 10 10 10 10 10 10 Occurence time [s] Occurence time [s] c) d) 4 4 10 10 Triggered points Triggered points 3 Theoretical curve 3 Theoretical curve 10 10

2 2 10 10 Count Count

1 1 10 10

0 0 10 10 0 2 4 6 0 2 4 6 10 10 10 10 10 10 10 10 Interevent time [s] Interevent time [s] Figure 7.3: Figure 5 in Paper IV. a) and b) Rate of occurrence of triggered points (circles) above 12 km depth when modelling the diffusion process for a) the June 17th earthquake and b) the June 21st earthquake, respectively. c) and d) Interevent time distribution (crosses) for triggered points in a) and b). Theoretical curves for occurrence rate (a and b) and interevent time distribution (c and d) are plotted for the Omori law sequence that best ﬁts the data. For the June 17th sequence the parameters are C1 = 410576, p = 1.4, C2 = 304289, and for the June 21st sequence C1 = 330750, p = 1.4 and C2 = 176354.

Figure 7.3 a) and b) show that the resulting rate of triggered grid points exhibits temporal behaviour consistent with the Omori law, with a roughly constant rate initially followed by a power law decay. The subsequent transition into an apparent exponential decay after about 106.5 seconds (35 days) tells us that the timescale of pore pressure diffusion processes can be relatively short. This, however, will depend on the permeability, which in our case is chosen to make a reasonable match between modelled pore pressure and observed water level changes in geothermal wells. In ﬁgure 7.3 c) and d) it can be seen that

98 theoretical Omori law interevent time distributions match well the interevent time distributions for the triggered sequences at short interevent times. At long interevent times, however, there is some deviation between theoretical and observed interevent time distributions. For both distributions in ﬁgure 7.3 c) and d) we can see that the behaviour for interevent times less than about 100 seconds is consistent with a non-zero value of C2 in the real data presented in ﬁgure 7.2 a) and b).

Discussion and conclusions We infer that an initially constant rate of aftershocks in aftershock sequences is not only a result of missed events at short times after a main shock but also representative of a physical process. Based on modelling of induced seismicity following the two M6.5 June 2000 earthquakes in Iceland, we suggest that this process is the diffusion of pore pressure transients induced by the main shocks. We also suggest that the initially constant rate of aftershocks, governed by C2 in the Omori law, is a characteristic signature of postseismic pore pressure diffusion. Physically, we consider that the initially constant rate of aftershocks may be related to the reduction of high pore pressure gradients across a fault zone at short times after a main shock. We note, ﬁnally, that our relatively simple pore pressure diffusion model cannot alone explain the large degree of aftershock activity in the fault zones of the June 2000 earthquakes. Based on our analysis, however, we believe that diffusion of pore pressure does play a role in the process of aftershock generation.

7.5 Paper V Spatiotemporal characteristics of aftershock sequences in the south Iceland seismic zone: interpretation in terms of pore pressure diffusion and poroelasticity Overview Following two M6.5 earthquakes in the south Iceland seismic zone (SISZ) in June 2000, signiﬁcant water level changes took place in geothermal wells (Björnsson et al., 2001) and crustal deformation measured by InSAR indicated poroelastic rebound (Jónsson et al., 2003). These observations demonstrate that diffusion of main shock induced pore pressure transients played a signiﬁ- cant role in geophysical and hydrological processes in the postseismic period of these two earthquakes. The SISZ thus forms a natural laboratory for investigations of the relationship between main shock initiated diffusion processes and the occurrence of aftershocks. We utilise aftershock sequences recorded in the SISZ and pore pressure diffusion modelling of induced seismicity in order to investigate and interpret the physical origin behind characteristic features in the temporal distribution of aftershocks.

99 −1 −1 10 a) M4.5 S27 10 b) M6.5 J17

−3 −3 10 10

−5 Interval 1 −5 Interval 1 10 Interval 2 10 Interval 2

Aftershock rate [1/s] Interval 3 Aftershock rate [1/s] Interval 3

2 4 6 8 2 4 6 8 10 10 10 10 10 10 10 10 Occurrence time [s] Occurrence time [s] d) All sequences −1 c) M6.5 J21 0 10 10

−3 10 −5 10

−5 Interval 1 M4.5 S27 10 Interval 2 M6.5 J17

Aftershock rate [1/s] Interval 3 Aftershock rate [1/s] M6.5 J21 −10 10 2 4 6 8 2 4 6 8 10 10 10 10 10 10 10 10 Occurrence time [s] Occurrence time [s] Figure 7.4: Figure 2 in Paper V. Rate of aftershocks versus time (black dots) for a) the September 27, 1999 M4.5 aftershock sequence (S27), b) the June 17, 2000 M6.5 aftershock sequence (J17) and c) the June 21, 2000 M6.5 aftershock sequence (J21). d) The S27, J17 and J21 sequences plotted together, but shifted vertically for easier comparison. The solid black curves in a) - d) represent the Omori law for each sequence. In a), b) and c) the circles, squares and diamonds indicate bins in the temporal distribution spanning or containing a time interval during which spatial snapshots of the aftershocks are shown in figure 3, 5 and 7 in Paper V, respectively. The circles span a C2 seconds long interval and the squares and diamonds contain one-day intervals for the S27 sequence and five-day intervals for the J17 and J21 sequences, respectively. Upward-pointing arrows: Bins exhibiting significant drops in aftershock rate. Downward-pointing arrows: Bins exhibiting an increase in aftershock rate.

Figure 7.4 shows the temporal variation in the rate of aftershocks for 3 selected aftershock sequences recorded in the SISZ. They are taken from limited regions spanning fault zones around a M4.5 earthquake on September 27th, 1999 (a)) and the M6.5 earthquakes on June 17th (b)) and June 21st (c)), 2000, respectively. In the subsequent discussion we refer to these as the J17, J21 and S27 aftershock sequences, respectively. For comparison, figure 7.4 d) shows the aftershock rate of the three sequences in the same plot. From figure 7.4, and from our spatiotemporal analysis described in Paper V, we briefly summarise five characteristic features of these aftershock sequences: 1. The aftershock sequences obey, in general, the Omori law for aftershocks p (dn/dt = C1/(C2 + t) ) (Utsu, 1961). Here, C1 and C2 denote the Omori law parameters that are usually referred to as K and c, respectively. 2. The duration of the initial period with a roughly constant rate of aftershocks, measured by C2 in Omori’s law, is significantly shorter for a smaller main shock than for a larger.

100 3. Signiﬁcant deviations from the general power law decay in aftershock rate can be seen, with distinct and temporary drops and increases in aftershock rate (marked by upward and downward pointing arrows in ﬁgure 7.4 a) to c)). 4. In general, the aftershocks in the S27, J17 and J21 sequences tend to mi- grate towards the respective main shock fault zone with time. The aftershocks also appear to be concentrated in the dilatational quadrants of the main shock, where pore pressure is reduced coseismically and increases in the postseismic period. 5. Aftershocks responsible for distinct rate increases form distinct spatial clusters in areas where pore pressures are expected to rise in the postseismic period, enhancing the potential for aftershock triggering.

To investigate the potential of pore pressure diffusion processes in explaining characteristic features of the S27, J17 and J21 aftershock sequences, we have modelled induced seismicity due to pore pressure diffusion following main shocks with moment magnitudes Mw = 2.2 and Mw = 4.6, respectively. The pore pressure diffusion process is modelled with the diffusion equation 2 D∇ ΔPind = ∂ΔPind/∂t, where D is the hydraulic diffusivity and ΔPind is the coseismically induced pore pressure transient. The hydraulic diffusivity D is the same in both cases. Seismicity triggered by the diffusing pore pressure transients is evaluated utilising the Mohr-Coulomb failure criterion and an assumption of a critically stressed crust prior to the respective main shock. Figure 7.5 a) shows that the growth rate in the crustal volume of pore pressure induced seismicity following the two hypothetical main shocks agrees well with the Omori law, with an initially constant rate followed by a subsequent power law decay. The figure also shows that a larger main shock indeed results in a larger value of C2, i.e. a longer duration of the period with constant rate than for a smaller main shock. For the Mw = 2.2 case, we can also see that both the enabling of crustal volumes for triggering and the end of the power law decay regime take place earlier than for the Mw = 4.6 case. This indicates that less time is required for the diffusion process to trigger seismicity following a smaller earthquake and that the associated sequence is likely to be over faster if the influence of the main shock on the surrounding crust is smaller. Our pore pressure modelling demonstrates that the diffusion process associated with a larger main shock affects a larger crustal volume than the process associated with a smaller one. Induced seismicity also contracts towards the fault zones as the diffusion process progresses. This implies that triggering gradually migrates to volumes with larger coseismic influence from the main shock on stresses and pore pressures, where longer time will be needed for pore pressures to recover and trigger seismicity. The initially constant rate thus implies that the volume where pore pressure recovery is enough to overcome the failure threshold is initially roughly constant. As the diffusion process pro-

101 a) b) 0 0 10 10 T dev −2 −2 10 10 /dt /dt

trig −4 M =2.2 trig −4 10 w 10

dV M =4.6 dV w C =694 M =4.6, poroelastic −6 2 −6 w 10 C =88281 10 M =4.6, pore pressure 2 w

−2 0 2 4 6 8 0 2 4 6 8 10 10 10 10 10 10 10 10 10 10 10 Time after main shock [s] Time after main shock [s] Figure 7.5: Figure 10 a) and b) in Paper V. a) Rate of change in the volume of triggered points per time unit, pore pressure diffusion. Solid circles: Diffusion initiated by the Mw = 2.2 main shock. Open squares: Diffusion initiated by the Mw = 4.6 main shock. Solid curves: The Omori law with parameters C1 = 410.7, C2 = 694, p = 0.82 (Mw = 2.2) and C1 = 164.7, C2 = 88281, p = 0.82 (Mw = 4.6). Dashed and dash- dotted vertical lines: Locations for the respective values of C2. b) Rate of change in the volume of triggered points per time unit, Mw = 4.6 diffusion process. Open squares: Pore pressure diffusion. Open diamonds: Poroelastic diffusion. Solid curves: The Omori law. Dashed vertical line: Indicates the timing, Tdev, of a deviation from the Omori law in the poroelastic diffusion process.

gresses, however, there is a transition into a power law decay of the rate as this volume begins to decrease when approaching the main shock fault zone. We have also investigated the effect of allowing poroelastic adjustment of both stresses and pore pressures by modelling the diffusion process following 2 the Mw = 4.6 main shock with the diffusion equation cm∇ (Δσkk − ΔPind)= ∂ (Δσkk − ΔPind)/∂t. The resulting sequence of pore pressure induced seismicity (ﬁgure 7.5 b)) also shows Omori law behaviour, with the additional feature of a distinct and temporary rate increase in the tail of the sequence. In- cluding poroelastic adjustment of stresses relaxes high failure thresholds that pore pressure recovery only would be unable to overcome. Our analysis shows that the poroelastic adjustment allows the contraction of induced seismicity towards the fault zones to continue and that this seismicity is responsible for the distinct rate increase observed in ﬁgure 7.5 b).

Discussion and conclusions By modelling pore pressure diffusion and induced seismicity for different types of faulting and permeabilities, Gavrilenko (2005) finds that decreasing the permeability increases the duration of the period with an initially constant seismicity rate, as measured by the Omori law parameter C2. In the SISZ, however, we believe that it is not different permeabilities that are reflected in the significantly different values of C2. Moreover, different permeabilities do not provide an understanding of the physical origin behind the roughly constant rate. Instead, we consider that C2 is not only related to the permeability

102 of the medium but also to the magnitude and fault slip distribution of the main shock initiating the diffusion process and the associated aftershock sequence. Main shocks of different magnitudes influence the surrounding crust to different degrees, both in terms of the spatial extent of the affected volume and in terms of the magnitude of the coseismic stress and pore pressure changes. We suggest that the initially constant rate in aftershock sequences governed by pore pressure diffusion reflects that the growth rate of the crustal volume where pore pressure recovery is enough to overcome the failure threshold is roughly constant initially. During the pore pressure diffusion process, pore pressure recovery takes place in the entire crustal volume affected by the main shock. Approaching the fault zone, however, the coseismic influence of the main shock on the failure threshold increases in a nonlinear way. Coupled with a larger coseismic influence on pore pressure, this implies that longer time is required for pore pressure recovery to exceed the failure threshold. Moreover, as coseismic pore pressure changes equilibrate in the diffusion process, the speed of pore pressure recovery will slow down as the diffusion process progresses. Altogether, this implies that a constant growth rate for the volume where pore pressure recovery overcomes the failure threshold can only be sustained for a limited period of time. A physical interpretation of the power law decay in the rate of aftershocks is thus a transition into a regime where increasingly longer time is necessary for pore pressure recovery to overcome the failure threshold. As a larger main shock affects a larger volume, with a larger magnitude of the coseismic stress and pore pressure changes, this transition will take a longer time. Piombo et al. (2005) evaluate the effect of poroelastic adjustment of stresses and pore pressures on the Coulomb failure function, measuring the change in failure potential. Their analysis shows that the failure potential can change from inhibitation to promotion of failure due to poroelastic adjustment during the diffusion process. In our modelling, such a phenomenon is represented by the rate increase in the tail of the Mw = 4.6 sequence when we take poroelastic adjustment into account. In the SISZ, we consider that the observed rate increases, following the distinct rate drops, may reflect the same physical process. As possible alternative physical mechanisms, we discuss secondary triggering of aftershocks as in the statistical ETAS model (Ogata, 1988), afterslip and viscoelastic relaxation of the lower crust and upper mantle. Of these models, we consider it unlikely that secondary triggering and afterslip can explain these features. We cannot, however, entirely rule out viscoelastic relaxation as the timing of the seismicity rate increases for the M6.5 J17 and J21 sequences yields viscosities in the range estimated for Iceland. If it is the same physical process, however, that is responsible for this behaviour in all of the studied aftershock sequences, then viscoelastic relaxation cannot explain the distinct rate increases in the M4.5 S27 sequence, the first taking place only ∼1.5 days following the main shock.

103 To summarise, we conclude that pore pressure diffusion and poroelastic stress relaxation provides a physical model that can explain characteristic features of aftershock sequences in Iceland, i.e. a magnitude dependent duration of the period with an initially constant rate of aftershocks and a general power law decay in aftershock rate that can be interrupted by distinct and temporary rate increases and decreases. Our conclusions, however, are based on aftershock sequences associated with two main shock magnitudes only (the M4.5 S27 and M6.5 J17, J21 sequences) and very simple modelling of pore pressure induced seismicity. Despite this, we consider the ability of pore pressure diffusion and poroelastic adjustment to capture the characteristic features of aftershock sequences in the SISZ as promising for our understanding of the physics of the aftershock process. We recommend further and more detailed studies of real aftershock sequences, coupled with more realistic modelling, in order to assess the role played by main shock initiated pore pressure diffusion processes in driving aftershock activity within the south Iceland seismic zone. The aftershock sequence associated with the M6.3 earthquake within SISZ on May 29, 2008 provides an opportunity to investigate our conclusions in this paper.

104 8. Discussion and conclusions

The work in my thesis is concerned with improving our physical understanding of the earthquake process. In my attempt to achieve this aim I have limited myself to studying processes driving the occurrence of aftershocks after large main shocks. I have approached the problem from both a statistical and a physical point of view through various kinds of modelling. With the results of the modelling at hand I have utilised aftershock sequences recorded within the south Iceland seismic zone as a basis for my interpretation of the aftershock process. The thesis contains an extensive discussion of statistical seismology with a particular focus on interevent time distributions of aftershock sequences obeying the empirically well established Omori law for aftershocks. In a sequence of earthquakes, we can consider earthquakes as correlated if they occur as a result of the same underlying process. This is the view taken by Bak et al. (2002), arguing that earthquake interevent times can be used to distinguish between correlated and uncorrelated events in earthquake sequences. Theo- retical equations for Omori law interevent time distributions are derived in Paper I of my thesis, with more details provided in section 2.2.2. One of the findings of this analysis is that the duration of the time interval between two successive earthquakes does not provide any information on whether or not they are a part of a sequence of earthquakes that are driven by the same underlying process. Earthquake interevent times can thus neither be used to define earthquakes as correlated nor uncorrelated in this sense. This understanding is important in our discussion in Paper II and III, where we address a unified scaling law for seismicity proposed by Bak et al. (2002). The proposed scaling law is based on empirically observed interevent time distributions, a combination of the Gutenberg-Richter law for the distribution of earthquake magnitudes, the Omori law for aftershocks and a fractal distribution of earthquake epicentres. In Papers II and III we conclude that interevent time distributions of typical earthquake sequences must be properly understood and modelled before conclusions regarding the physics of the earthquake process are drawn. The theory of self-organised criticality (SOC) as a model for earthquake occurrence implies that the Earth’s crust is in a critical state, characterised by power law distributions. In seismology, the Gutenberg-Richter law for the distribution of earthquake magnitudes is the primary argument supporting the SOC-hypothesis, but also the Omori law for aftershocks has been discussed in this context. In section 2.3 I discuss the SOC-hypothesis with a particular

105 focus on Omori law behaviour of aftershock sequences and the associated interevent time distributions. Based on a review of selected papers, I conclude that the power law behaviour in the Omori law shall not, and can not, be taken as evidence of self-organised criticality. The primary argument for this conclusion is that the Omori law behaviour can only be obtained by providing a correlated driving mechanism, representing an ongoing physical process. The basic idea behind SOC is therefore violated, i. e. that a state characterised by power law distributions has to be obtained through uncorrelated driving. If we are interested in understanding the physical process driving aftershocks, we must consider that the SOC-hypothesis does not provide any motivation for obtaining such an understanding. When the SOC-hypothesis is applied, obtaining power law distributions is commonly regarded as having achieved satisfactory results. In order to achieve a better understanding of the actual process, it is necessary that we, instead, focus our attention on how physical processes result in an occurrence of aftershocks in accordance with the Omori law. In Paper IV of the thesis, we infer that an initially constant rate in aftershock sequences following the two M6.5 June 2000 earthquakes in the south Iceland seismic zone is representative of a physical process. Based on modelling of pore pressure induced seismicity, we suggest that an initially constant aftershock rate is a characteristic signature of postseismic pore pressure diffusion initiated by a main shock. In Paper V we analyse, in detail, the spatial and temporal properties of aftershocks following the two M6.5 June 2000 earthquakes and a M4.5 earthquake in September, 1999. Using pore pressure diffusion modelling we show that main shock initiated diffusion processes can provide an explanation for characteristic features in the spatiotemporal distribution of aftershocks within the south Iceland seismic zone. These features include an initially constant aftershock rate, whose duration is larger following a larger main shock, and a subsequent power law decay that is interrupted by distinct and temporary deviations in terms of rate increases and decreases. In Paper V we interpret the physical origin of these features in terms of a nonlinear coseismic inﬂuence of the main shock on the surrounding crust and poroelastic adjustment of stresses and pore pressures during main shock initiated diffusion processes. In order to assess the role played by main shock initiated diffusion processes in driving the aftershock activity within the south Iceland seismic zone, we need to extend our analysis through more detailed modelling and investigations of more aftershock sequences. The M6.3 earthquake taking place on May 29, 2008 in the western part of the south Iceland seismic zone provides an opportunity for investigating the conclusions of this thesis. The modelling can also be extended by allowing more heterogenity in the background stress state, frictional properties etc. To include rate and state dependent friction in the failure criterion would be interesting in order to investigate its inﬂuence

106 on aftershocks induced by main shock initiated diffusion processes. This will be left for future work. The ability of main shock initiated pore pressure diffusion processes to capture characteristic features in the spatiotemporal distributions associated with aftershock sequences within the south Iceland seismic zone is promising for our understanding of the aftershock process. Starting out with a statistical approach to the problem, I move, as suggested in Paper I to III, into modelling with the aim of gaining a physical understanding of the statistical distributions. The modelling that I have performed is very simple and does not provide an explanation for all of the complexities in the real data. An important conclusion of this thesis, however, is that statistical analysis and physical modelling are complimentary. I argue that it is only through modelling of the physical process and a thorough statistical analysis of data that we will gain a better understanding of the processes governing the occurrence of earthquakes. Rather than being content with a general statistical model such as self-organised criticality, I recommend such modelling and analysis in order to gain as much insight as possible. The work presented in this thesis is, in this respect, only the beginning.

107

9. Summary in Swedish

Efterskalvsfysik i södra Islands seismiska zon: Insikter om jordbävningsprocessen genom statistik och numerisk modellering av efterskalvssekvenser Förekomsten av kraftiga jordbävningar såsom magnitud 7.9 jordbävningen i Sichuan i Kina i maj 2008 är en påminnelse om behovet av att förbättra vår förståelse av processen bakom jordbävningar. Jordbävningsprocessen kan beskrivas som en cykel bestående av olika faser, vars tidsförlopp är väsentligt olika mellan de olika faserna. De olika fasernas tidsskalor sträcker sig från sekunder till minuter under den seismiska fasen (själva brottprocessen i en jordbävning), månader till år i de pre- och postseismiska faserna samt ﬂera decennier i den interseismiska fasen (uppbyggnad av spänningar i jordskorpan mellan jordbävningar). De pre- och postseismiska faserna är förknippade med processer som på grund av närheten i tid till en jordbävning (ett huvudskalv) skiljer sig åt från de interseismiska processerna. Min avhandling behandlar en del av detta pussel då jag försöker få en bättre förståelse av den postseismiska fasen och processer som pågår efter att stora jordbävningar eller huvudskalv ägt rum. Genom att förbättra förståelsen av efterskalvsprocessen kan vi erhålla viktiga insikter om jordbävningsfysik i allmänhet. Att förstå fysiken bakom efterskalvsprocesser har också en stor praktisk betydelse eftersom kraftiga efterskalv kan orsaka ytterligare skador och dödsfall genom att infrastruktur som försvagats på grund av huvudskalvet kollapsar. Jag har studerat efterskalvssekvenser i södra Islands seismiska zon (SISZ) vilken utgör en del av plattgränsen mellan de nordamerikanska och eurasiska litosfäriska plattorna. SISZ är en transform zon i öst-västlig riktning där platt- rörelserna, generellt sett, tas upp genom kraftiga jordbävningar som äger rum på vertikala förkastningar i nord-sydlig riktning. År 2000, den 17:e respektive 21:a juni, ägde två sådana jordbävningar med magnitud 6.5 rum inom SISZ. En del av det data som jag har studerat utgörs av efterskalvssekvenserna as- socierade med dessa två skalv. I min forskning har jag använt mig av både av ett statistiskt och ett fysikaliskt angreppssätt för att bättre försöka förstå efterskalvsprocessen. Statistik har den fördelen att stort antal olika fall eller stora mängder data kan studeras på ett överskådligt sätt. Modellering av fysikaliska processer, å andra sidan, har fördelen att det möjliggör en förståelse av vad som kan eller inte kan förklara statistiska egenskaper i det data som man studerar.

109 Statistiska studier och modellering kompletterar därför varann och är båda viktiga vid studier av jordbävningsprocesser. Den första delen av min avhandling behandlar statistisk seismologi. Där diskuterar jag statistiska fördelningar inom seismologin: Gutenberg-Richters lag för fördelningen av jordbävningars magnitud, Omoris lag för hur efterskalvsfrekvensen avtar med tid efter ett huvudskalv samt fraktala fördelningar av förkastningar och jordbävningars epicentrum eller hypocentrum (jordbävningars startpunkt). Utifrån Omoris lag härleds i artikel I, samt i avsnitt 2.2.2, ekvationer för fördelningen av intereventtider mellan två på varandra följande efterskalv i efterskalvssekvenser styrda av Omoris lag. En förståelse av egenskaperna för sådana intereventtidsfördelningar är viktig för att korrekt tolka slutsatser dragna ifrån empiriska interevent- tidsfördelningar bestämda från jordskalvsdata. Ett av resultaten av denna analys är att ingen distinkt intereventtid kan användas för deﬁniera jordbävningar som korrelerade eller okorrelerade. Detta innebär att längden av tidsintervallet mellan på varandra efterföljande jordbävningar inte innehåller någon information om huruvida de är eller inte är en del av en jordbävningssekvens driven av samma underliggande process. Denna förståelse är viktig i vår diskussion i artikel II och III i min avhandling. Dessa två artiklar behandlar en statistisk skaleringslag för seismicitet föreslagen av Bak et al. (2002). Den föreslagna skaleringslagen är baserad på intereventtidsfördelningar och kombinerar Gutenberg-Richters lag för fördelningen av jordbävningars magnitud med Omoris lag för efterskalv samt en rumslig fraktal fördelning av jordbävningars epicentrum. I artikel II och III är en slutsats att intereventtidsfördelningar för typiska jordbävningssekvenser måste förstås och modelleras på ett riktigt sätt innan man drar slutsatser om fysiken bakom jordbävningsprocessen. Utifrån Gutenberg-Richters lag går det att härleda ett uttryck i form av en potenslag för den statistiska fördelningen av den energi som löses ut i jord- bävningar. En potensfördelning av en kvantitet relaterat till storleken av en händelse är ett karaktärsdrag för självorganiserade kritiska system. Att till- ståndet kallas självorganiserat innebär att sådana potensfördelningar måste er- hållas utan att systemet ställs in eller justeras för att åstadkomma önskat resultat. Ett självorganiserat kritiskt system måste därför kännetecknas av regel- bunden drivning och enkla villkor för samverkan mellan element i systemet. Detta innebär att det utifrån drivning och villkor för samverkan inte går att förutse att storleken för händelser i ett självorganiserat kritiskt system kommer att följa en potensfördelning. Denna potensfördelning kan bara fastställas genom observation. En sådan potensfördelning innebär att händelser av alla storlekar kan förekomma och att det inte går att förutsäga när en händelse av en viss storlek kommer att äga rum. Dessa egenskaper hos potensfördelningen är anledningen till att systemet anses vara i ett kritiskt tillstånd.

110 Att Gutenberg-Richters lag funnits gälla i vitt skilda tektoniska miljöer anses av vissa forskare vara ett uttryck för att jordskorpan beﬁnner sig i ett självorganiserat kritiskt tillstånd. Ackumulation av spänning som kan lösas ut i jordbävningar sker på ett regelbundet sätt genom de litosfäriska plattor- nas rörelse relativt varandra. När en jordbävning startar sker en omfördel- ningar av spänningar och spänningstillståndet i omgivande skorpa avgör om jordbävningen fortsätter eller dör ut. Ytterst detaljerad information om jordskorpans spänningstillstånd är därmed nödvändig för att kunna förutsäga jord- bävningars storlek, tidpunkt och läge. Hypotesen om självorganiserad kritikalitet är därför omdebatterad inom seismologin eftersom den innebär att det inte är möjligt att förutsäga jordbävningar. Det är helt enkelt inte praktiskt eller ekonomiskt möjligt att mäta jordskorpans spänningstillstånd i så detaljerad grad som krävs. Förkastningssystem samt jordbävningars epi- och hypocentrum har, i alla fall inom vissa intervall, kunnat beskrivas med potenslagar i form av rumsliga fraktala fördelningar. Omoris lag för efterskalv har dessutom, efter den inledande perioden med konstant efterskalvsfrekvens, också formen av en potenslag. Dessa observationer utgör, enligt vissa forskare, ytterligare ett stöd för teorin om självorganiserad kritikalitet som modell för förekomsten av jordbävningar. I avsnitt 2.3.2 och 2.3.3 diskuterar jag teorin om självorgan- iserad kritikalitet i förhållande till Omoris lag samt intereventtidsfördelningar. Utifrån en studie av valda artiklar av andra författare drar jag slutsatsen att Omoris lag inte kan användas som ett stöd för teorin om självorganiserad kritikalitet. Den främsta anledningen till det är att Omoris lag bara kan erhål- las genom en korrelerad drivning av systemet, det vill säga genom att justera systemet för att erhålla önskat resultat. Detta strider alltså mot en av de grund- läggande principerna i teorin om självorganiserad kritikalitet. Genom sin natur erbjuder teorin om självorganiserad kritikalitet ingen motivation att studera vad som ligger bakom statistiska potenslagar eftersom dessa lagar “bara” uppstår. Att en korrelerad drivning av ett jordbävningssystem är nödvändig för att ge Omoris lag är därför en viktig insikt i detta samman- hang. Detta innebär att det är meningsfullt att försöka modellera den eller de processer som denna korrelerade drivning representerar. Genom sådan modellering kan vi förbättra vår förståelse av fysiken bakom efterskalvsprocesser och vad som ger upphov till de statistiska egenskaperna hos det data som vi har tillgång till. I min avhandling diskuterar jag i avsnitt 4 olika fysikaliska processer som kan ge upphov till Omoris lag för efterskalv. Mekanismer som jag tar upp innefattar hastighets- och tillståndsberoende friktion för förkastningar, subkritisk spricktillväxt samt transienta deformationsprocesser såsom viskoelastisk återhämtning, efterrörelser i förkastningszonen och poroelastisk återhämtning orsakad av portrycksdiffusion. Vattennivåförändringar i geotermiska borrhål samt InSAR-mätningar av jordskorpans deformation inom SISZ demonstrerade att poroelastiska processer pågick under de

111 närmsta månaderna efter de två M6.5 jordbävningarna i juni år 2000. I artikel IV i denna avhandling sluter vi oss till att en inledningsvis konstant frekvens av efterskalv i efterskalvssekvenser inom SISZ är ett uttryck för en fysikalisk process. Utifrån modellering av portrycksinducerad seismicitet föreslår vi att en inledningsvis konstant efterskalvsfrekvens är ett kännetecken för postseismisk portrycksdiffusion initierad av ett huvudskalv. Artikel V i avhandlingen innehåller en detaljerad analys av rumsliga och tidsmässiga egenskaper för efterskalvssekvenser efter juni 2000 jordbävningarna (M6.5) samt en M4.5 jordbävning i september 1999, vilken också ägde rum inom SISZ. En av dessa egenskaper utgörs av en inledningsvis konstant efterskalvsfrekvens vars varaktighet är längre efter ett större huvudskalv. En annan egenskap är ett efterföljande avtagande enligt en potenslag, där avtagandet dock avbryts av distinkta och tillfälliga avvikelser i form av ökningar och minskningar av efterskalvsfrekvensen. I artikel V visar vi att portrycksdiffusion initierade av huvudskalv kan ge en förklaring till de karakteristiska egenskaperna för efterskalvssekvenser inom SISZ. Vi tolkar att det fysikaliska ursprunget till dessa egenskaper utgörs av ickelinjär påverkan av huvudskalv på den omgivande jordskorpan samt poroelastisk justering av spänningar och portryck under diffusionsprocesser initierade av huvudskalv. För att utvärdera den roll som diffusionsprocesser initierade av huvudskalv spelar för efterskalvsaktiviteten inom SISZ är det nödvändigt att utvidga analysen till mer detaljerad modellering och undersökningar av ﬂer efterskalvssekvenser. En M6.3 jordbävning den 29:e maj 2008 utgör en möjlighet att undersöka slutsatserna i denna avhandling. Modelleringen kan också utvidgas genom att tillåta mer variation i bakgrundsspänningar, friktionsegenskaper m.m. Att inkludera hastighets- och tillståndsberoende friktion i det brottkriterium som jag använt vore intressant för att undersöka dess påverkan på efterskalv inducerade genom portrycksdiffusion. Detta har jag lämnat för framtida arbete. Det är lovande för vår förståelse av efterskalvsprocessen att diffusionsprocesser initierade av huvudskalv har en förmåga att förklara karakteristiska egenskaper i de rumsliga och tidsmässiga fördelningarna hos efterskalvssekvenser inom södra Islands seismiska zon. Från att ha börjat med ett statistiskt angreppssätt går jag, som föreslaget i artikel I till III, till modellering med målet att erhålla en fysikalisk förståelse av de statistiska fördelningarna. En viktig slutsats i denna avhandling är att statistisk analys och fysikalisk modellering kompletterar varandra. Det är bara genom modellering av fysikaliska processer som vi kan tolka företeelser i den observerade jordbävningsstatistiken. Hellre än att nöja sig med en generell statistisk modell såsom självorganiserad kritikalitet rekommenderar jag en noggrann dataanalys kopplat med fysikalisk modellering för att erhålla så stor insikt som möjligt. Det arbete som jag presenterar i denna avhandling är, i det avseendet, bara en början.

112 10. Acknowledgements

In hindsight, I could not have predicted what I would learn, both in terms of science and in terms of life, when I embarked on the journey as a Ph. D student. I highly appreciate my supervisors, Roland Roberts and Björn Lund, whose input has been crucial for finding a route leading to the writing of this thesis. I am grateful for Roland’s critical mind and his questioning of our initial work on interevent times. After recovering from the initial (main) shock, I highly appreciate the lesson learned: the importance of critical assessment and reflection. I am grateful for the close cooperation on interevent times that followed this event and for the rewarding and open discussions we have had throughout my thesis work. To Roland I would also like to say that thinking about your favourite (?) scientific principle, Occams Razor, was of great help at the end of my thesis, when I was assessing what I had actually achieved. Throughout the work with my thesis, I appreciate the time taken by Björn when I needed another point of view, or needed to discuss various practical and theoretical issues. The practical help of Björn has also been valuable, especially when my computer broke down at an inconvenient time. Björn’s critical review of my manuscripts has also been a great benefit. I highly appreciate his attention to details and ability to find the weak points in what I have written. Many times during my thesis work, I have been making bold claims without having enough on my feet. To Björn I would like to say that I am grateful for your stopping me from going down such roads, especially when my argumentation was not enough to bear the claims that I made. For my work in Paper IV of the thesis, I am grateful to Thóra Árnadóttir for providing me with slip models of the June 2000 earthquakes in the south Iceland seismic zone. I wish to thank all past and present students in the Geophysics group that I have had the privilege of following during part of your studies. “Takk” to Kristin for the cooperation when we struggled with the waiting (interevent) time distributions. Our scientifically extremely important mud cracking coffee cake experiment at the home of you and Pálmi was great fun. I also appreciate that I have been able to practise Icelandic with you and Pálmi. I owe many thanks to Tuna, my office mate and Blues Brother for the most part of my studies. Thank you for the memorable moments we have shared. Tuna, Nazli, Sawasdee and Lijam, you have a special place in my heart as you attended mine and San’s wedding and endured the cold! I hope that you will never forget your first skiing experience! Thanks to Thomas for talks during breaks

113 and thanks to Eva for your interest in my Christian faith and my personal testimony. I highly appreciate the unexpected gift from the people in the Geophysics group at my wedding. Thank you all! Among the Geophysics staff I wish to thank Reynir Bödvarsson and Laust Pedersen for your proposal of the initial study on interevent times, Dan Dyrelius for your advise on teaching, Christ- opher Juhlin for your courses on seismics and fractals, Pálmi Erlendsson for the practical help with all computer issues and the encouragement from your cheerful and happy mood, Conny Holmkvist for the fieldwork and the enjoy- able chats that I had with you, providing a good break in the academic work, Hasse Palm for teaching me about seismic network surveying and letting me use part of your office while writing my thesis, Lasse Dynesius for your hints on fixing my car and turning out to be my relative, Siv Pettersson for help- ing me with documents and papers, Hossein Shomali for listening to what I learned during my Bible school year and Arnaud Pharasyn for our talks and never giving up in trying to get me take a break. An important process has been going on in my life during my studies, leading to my coming to belief in God in the autumn of 2004 and my baptism in the name of Jesus in the spring of 2007. The Bible verse quoted on the dedication page of the thesis is a good characterisation of this (ongoing) process. I thank God for what I have learned about myself, but would also like to thank all the people that have been involved, mentioned and unmentioned. Many thanks to Anders Holmbom, for your first prayer for me in 2002, and for being the priest at mine and San’s wedding in 2005. I owe many thanks to the Chinese Bible study group in Uppsala and David and Maj-Britt Bingham for prayers and Bible teaching. I am very grateful for the practical help and friendship of Ren Ping, Yang Tao, XiongYu and SiuFun, LianZi and AGui, Wu Dan and Markus, and RuiYan & Kristin. I thank Anders, Helena, Sofia, Martin and Anna-Karin for the Alpha course in the Baptist Church in 2004, where I began to learn what Christianity was about. During a year at Teen Challenge Bible School I learned many things about myself and about God. For this, I am grateful to Martin Lindroos and Eva Lindberg, all guest teachers, the TC bible school class 2007/2008, and all the people me and San met during our stay in Göteborg. It is also nice to know a Christian in the same department; I appreciate the friendship with you and Yvonne a great deal. I would like to thank Staffan for your “Exciting!” comment in a tough situation, and thanks to God for having prepared a way out! Thanks to Jonas & Sandra and Katarina & Berry for your help. The writing of my thesis has been a struggle many times, but now it is finally done. Thank you, San, for your faith in God, your encouragement and for sharing your life with me, in good times, in bad times. For being able to finish my thesis in a balanced way, and for everything that has been achieved, I give my thanks to God.

114 A. Errata

In Paper I there are typographical errors in the caption of Figure 6 and two of the equations in Appendix A, equation A1 and A5. In the caption of ﬁgure 6, the coordinates of area b) and c) are incorrect. The correct caption for Figure 6 is:

Fig. 6. A map of the study area in southern Iceland. Triangles show station locations in the SIL network. a) Hengill volcanic area (64.0N-64.15N, 21.0W- 21.4W). b) Area around the June 21st (year 2000) main event (M6.5) (63.85N- 64.1N, 20.6W-20.9W). c) Area around the June 17th (year 2000) main event (M6.5) (63.85N-64.1N, 20.2W-20.6W). d) Katla volcanic area (Godabunga) (63.49N-63.8N, 19.25W-19.45W).

In equation A1, the natural logarithm should be used, i.e. :

N = K · [ln(C + T) − ln(C + Tstart )] (A.1)

In equation A5, the stretching of the time between events should take place by raising 10 to the power of x (not by multiplying 10 with x), i.e.: x tocc = 10 (A.2)

115

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