University of Debrecen Faculty of Science Department of Theoretical Physics. Master s Thesis. Numerical Simulation of Earthquakes

Transcription

1 University of Debrecen Faculty of Science Department of Theoretical Physics Master s Thesis Numerical Simulation of Earthquakes Author: János Farkas physicist Supervisor: Ferenc Kun, PhD associate professor Debrecen 2008

2 Contents Introduction 1 Chapter 1 Earthquakes and SOC 3 11 The history of the research of earthquakes 3 12 Laws of earthquakes 7 13 Self-Organized Criticality 9 14 The Burridge Knopoff model of earthquakes 11 Chapter 2 Numerical simulation Differential equations Cellular automata The results of Kawamura et al 26 Chapter 3 The BlocK software package Theoretical preliminaries Technical background Simulational results 47 Summary 68 Acknowledgment 70 Appendix A Example of an earthquake 71 Appendix B History of versions 74 Appendix C Illustrations 75 Bibliography 79 Index 82 i

3 Introduction The research on catastrophic phenomena played an important role in the history of science It began with the observation and description of these phenomena and continued by developing basic models trying to explain their nature Today s research goes on by using the knowledge piled up in the different fields of physics The methods of statistical physics are used to provide a global description of catastrophic systems Chaos theory tells us that we cannot predict the long term behavior of a sufficiently complex phenomena Self-organized criticality (SOC) deals with the common statistical properties of the phenomena, like their scale-free nature SOC states that catastrophes are inherent part of critical systems Just as minor events, they arise from the dynamics of the system, and there is nothing special about them, except their impact on their environment In the present work we investigate the connection between earthquakes and SOC based on computer simulation of a widely used model of earthquakes, the Burridge Knopoff model The work can partially be treated as a report on the making and the early results of a simulational software package, called BlocK, which we developed to extend the future possibilities of earthquake and SOC research at the Department of Theoretical Physics of the University of Debrecen In Chapter 1 we give a brief historical background on the subject In Section 11 we show the important role of catastrophic phenomena in the past and present of mankind, and introduce the basic notions of the theory of plate tectonics Section 12 summarizes the most important observations on earthquakes In Section 13 we present the evolution of one of the most important recent idea in physics, self-organized criticality Section 14 describes the Burridge Knopoff (BK or spring-block) model of earthquakes, where the fields of earthquake research and self-organized criticality meet each other 1

4 Chapter 2 contains the background on the most important simulation methods used in earthquake modeling In Section 21 we describe the two numerical methods we used during the development of BlocK, the Euler method and the fourth-order Runge Kutta method Section 22 deals with cellular automata based simulations, which played an important role in the discovery of the SOC behavior of the BK model In Section 23 we summarize the recent work of Kawamura et al, which provided the main motivation of our work Chapter 3 presents the BlocK software package In Section 31 we review the theoretical preliminaries we need in the later sections Namely, we cover the problems of random number generation, the drawing of the initial state and the analytical solutions of the most simple settings of the model Section 32 is a detailed description of the parts of BlocK, the input it requires and the output it produces Section 33 presents the most important results of our model calculation Based on computer simulations, we recover the former results of the Kawamura group Novel results are obtained for the spatial correlation of consecutive earthquakes both by studying the distribution of interoccurence distance and the structure of the Mogi doughnut preceding major earthquakes 2

5 CHAPTER 1 Earthquakes and SOC 11 The history of the research of earthquakes Then the LORD rained down burning sulfur on Sodom and Gomorrah from the LORD out of the heavens Bible, Genesis 19:24 But afterwards there occurred violent earthquakes and floods; and in a single day and night of misfortune all your warlike men in a body sank into the earth, and the island of Atlantis in like manner disappeared in the depths of the sea Plato: Timaeus Earthquakes in history Natural catastrophes play an important role in the history of the Earth and mankind In the prehistoric ages, catastrophes caused mass extinctions, changed the climate and the whole geographical, geological and biological face of Earth In the era of modern man, they heavily affected the environment where our ancestors lived Besides of human interaction, they played a major role in the rise and fall of civilizations Earthquakes are part of these catastrophic events, and are connected with geophysical processes called the crust phenomena We can find reportings of earthquakes from as early as 1177 BC (China) The first known European earthquake reportings date back to 580 B C[34] Some earthquakes became part of human tradition Some believe, that the destruction of Sodom and Gomorrah in the Bible (see quotation) was the result of a powerful gas explosion following one or more earthquakes [41] Due to Plato, the island and the culture of Atlantis was eradicated due to a vast earthquake One possible explanation of the mystery of Atlantis is, that the thriving Minoan Civilization populating the island of Crete (Atlantis) and the islands of the Cyclades was brought to its knees by the hydromagmatic volcanic blast of the volcano on a nearby island (now Santorini), which caused several earthquakes and 3

6 11 HISTORY CHAPTER 1 EARTHQUAKES tsunamis [10] This explosion could have been similar to the well documented eruption of Krakatoa (Indonesia) in 1883 The nature of earthquakes remained a mystery until 1915, when Alfred Wegener published his idea of continental drifts in his book The Origin of Continents and Oceans [1] Though not all the statements of his idea was acceptable, continental drifts served as a solid base for the scientists of the 1950s and 1960s to create the theory of plate tectonics This theory covered many of the hardly explainible geological phenomena: orogenesis (mountain building), the shape of the continents, earthquakes, volcanism, rifting, the distribution of fossils on Earth, etc Modern explanation of earthquakes Plate tectonics offers a reasonable theoretical background for earthquakes According to the theory, the interior of the earth has both solid and liquid parts, and its material is being heated by the decay of radioactive elements (mostly by uranium, thorium and potassium) and by inner friction caused by movement due to tidal forces and convection The outmost part is the crust of the Earth, on which we live It is the upper part of the lithosphere If we go deeper into the Earth, we find the upper mantle, where materials exhibit a plastic behavior Due to the convection, the upper mantle (mostly its upper part, the asthenosphere) is in movement, breaking the lithosphere into parts we call plates A plate can be both a part of oceanic or continental crust The oceanic crust is usually thin (approximately 5 10 kms), and it carries the water of the oceans, while the continental crust is thicker (approximately kms), and it belongs to the land of the continents Since the lithosphere floats on the asthenosphere, the surface of the continents we see is only the tip of the iceberg The material of the plates are created at ridges or rifts, where the rising flow of magma (molten rock beneath the surface) reaches the surface The magma presses the two plates in the opposite direction, making ridges divergent plate boundaries If a ridge is created in continental crust, sooner or later water pushes in the pit of the boundary, and a new ocean borns (like the Red Sea, which is a future ocean) When the magma cools down, it becomes solid rock and forms the crust of the Earth The two plates diverge at ridges 4

7 11 HISTORY CHAPTER 1 EARTHQUAKES by approximately the same speed in both directions, making an elder ridge the middle of the newborn ocean (that s why we call them mid-oceanic ridges ) When two plates are being pressed against each other, the thinner plate begins to go under the thicker one We call the area of this process a convergent plate boundary, or subduction zone At this boundary, a deep trench is created, while the thinner plate creases the thicker one, creating new mountains as folds in the crust As the subducting plate melts, its magma can find its way to the surface in the form of lava at volcanoes Figure 1 illustrates the plate boundaries Figure 1 Plate boundaries and phenomena associated with plate tectonics (Wikipedia) It follows that extreme mechanical stress is stored in the deformed plates at convergent plate boundaries Most of the earthquakes originate from the release of this stress, while others are caused by violent vulcanic eruptions If an earthquake occurs at one part of the world, mechanic waves are triggered due to the sudden movement of the crust These waves travel both inside and on the surface of the Earth, which enables us to detect them all around the world 5

8 11 HISTORY CHAPTER 1 EARTHQUAKES Appendix C provides additional information on earthquake occurence: the topography of the Earth with the most important fault zones are illustrated by Fig38, while Fig39 presents a map of the tectonic plates Measuring earthquakes The science of measuring the characteristic quantities of earthquakes dates back to 1880, when John Milne invented the first modern seismometer, the horizontal pendulum seismograph Today s seismometers use electronics to measure the displacement, velocity and acceleration of a mass placed in the instrument These data are recorded on a computer, and can be processed as digital time series Seismologists use different scales when they speak about the size of an earthquake The Mercalli intensity scale (1902) is based on the interviews of the witnesses and the damage caused by the earthquake [35] A more scientific and widely used scale is the Richter scale (1934), in which the magnitude µ of the earthquake is calculated as (1) µ = log 10 A + d, where A is the amplitude of the largest measured wave on a given seismograph, and d is a distance factor, both depending on the instrument we use [2] We can perceive earthquakes in a magnitude range of 25 9 on the Richter scale, where 9 is the magnitude of the largest earthquake observed (the Great Chilean Earthquake, 1960) A more recent scale is the seismic moment scale, based on the seismic moment of earthquakes The seismic moment is related to the amount of energy radiated by the given earthquake The Gutenberg Richter scale gives a relation between the energy E and the magnitude µ of the earthquake (2) µ = 2 3 log 10 E 787 Our magnitude scale will be similar to this (see [14] or section 14) If one observes the seismic waves by multiple seismometers placed on different places of the world, one can estimate the epicentre of the earthquake, and from the attributes of the waves, one can draw a map of the interior of the Earth These global projects are led by organizations like ISC (International Seismological Center) in the U K or IRIS (Incorporated Research Institutions for Seismology) in the US 6

9 12 EARTHQUAKE LAWS CHAPTER 1 EARTHQUAKES The science of earthquakes is an extended field, which is connected to physics, geo- and astrophysics, geology, vulcanology, seismology, paleoseismology and mineralogy, structural engineering and many other areas, showing the complexity and importance of the subject 12 Laws of earthquakes The crust of the Earth and earthquake faults are complex systems Their complexity mainly arises from their nonhomogenity and the nonlinear dynamics they obey The precise mapping of the materials in the Earth s crust is a very hard task, which can only be done with huge errors The investigation of the dynamics is also far from trivial, since it is very hard to reconstruct the extreme environment (temperature, density, stress, etc) of the rocks in an experiment At the same time it is very surprising that by investigating the statistical behavior of many earthquakes, one can find simple laws describing the phenomena quite precisely This was first recognized in 1956 by two seismologists, Beno Gutenberg and Charles Francis Richter They examined the frequency-magnitude statistics of earthquakes, and erected the Gutenberg Richter law It says that the relation between the frequency of occurence of earthquakes and their magnitude follows a power-law, resulting in a line when plotted in a log-log scale (3) log 10 Ṅ = bm + log 10 ȧ, where Ṅ is the number of earthquakes per unit time with a magnitude greater than m in the given area, and ȧ and b are constants [5] The value of b depends on the region, but it generally falls in the range [17], while ȧ characterizes the regional level of seismicity [5] As a consequence of the GR law, if we plot the R(µ)dµ rate of events in the magnitude range [µ, µ + dµ] against the magnitude, a similar power law is obtained In Fig2 an example of the Gutenberg Richter law is presented In 1894 Omori published an empirical law describing the temporal decay of aftershock rates We refer to this law as Omori s law Today a modified form of the law is used, 7

10 12 EARTHQUAKE LAWS CHAPTER 1 EARTHQUAKES due to Utsu [5, 8] (4) R(t) 1 (t 0 + t) ρ, where ρ 1 It can be seen that the aftershock rates decay as a power law (see Fig2) Figure 2 (left) Worldwide number of earthquakes per year, N, with magnitudes greater than m [5] The curve follows the Gutenberg Richter law (right) The temporal decay of aftershocks for the fault Omori originally investigated The results of Omori s measurements are included in the data (Fig from Prof Kawamura, Osaka University, Japan) We will concentrate on the laws above, but other proofs of the scale-free feature of earthquakes exist We only mention here, that one can explain the GR law by assuming that each fault has a power-law distribution of earthquakes, or by assuming that each fault has a caracteristic earthquake, but the faults follow fractal distribution Observations favor the latter case [5] It had also been observed, that the epicenters of earthquakes follow a fractal distribution This law is known as the Kagan law Figure 3 illustrates the spatial distribution of aftershocks and the Kagan law Most of the laws of earthquakes are of statistical nature regarding many events Since these laws allow statistical fluctuations, we cannot use them to predict earthquakes Geoscientists are in a continuous search for laws and methods that can help in the development of earthquake prediction These laws have to embrace general properties of 8

11 13 SOC CHAPTER 1 EARTHQUAKES Figure 3 (left) The spatial distribution of aftershocks of the Joshua Tree earthquake (1992, magnitude 61) [18, 5] The area represents a km volume in 160 days (right) Box-counting method applied to the Joshua Tree earthquake (JTS) and the Big Bear earthquake (BBS) [18, 5] The straight line belongs to the fractal correlation with D = 2 The curves follow the Kagan law earthquakes, which are not easy to perceive One of the candidates that can be used in prediction is the Mogi doughnut phenomena, raised by Kiyoo Mogi in 1969 He pointed out, that earthquake activity is suppressed at the point of a future large event, while there is increased activity in the doughnut-shaped vicinity of that site [11] 13 Self-Organized Criticality How can the dynamics of all the elements of a system as complicated as the crust of the Earth, conspire, as if by magic, to produce a law with such extreme simplicity? Per Bak [3] The theory of Self-Organized Criticality (SOC) is one of the most successful and promising theories of the late 20th century, that can explain many experimental phenomena from a very colorful variety of disciplines It can be compared to chaos theory, in a sense that the research of both started with experimenting with simple systems (like pendulums) showing intricate behavior Strange enough, SOC turned out to be a theory of complex systems having simple statistical properties SOC had been called to life by a handful of scientists, namely Bak, Tang and Wiesenfeld at Brookhaven National Laboratory around 1986 They intended to solve the problem 9

12 13 SOC CHAPTER 1 EARTHQUAKES of 1/f noise, a mystery emerging in many areas of physics and everyday life, from mass extinctions to stock prices [3] The statistics of these systems can be described with power laws, and their geometry often shows fractal properties The first model investigated by the group was a lattice of coupled pendulums Later, they developed the sandpile metaphor to describe their model more intuitively It turned out that the model consisting of several elements with the same simple, deterministic, local laws of dynamics, shows complex behavior, similar to the one of the critical state of phase transitions, with the features mentioned above In the case of the sandpile, there was no need to carefully adjust the parameters of the model to reach the critical state, but after a while, the system drived itself into this state, and remained there without external tuning That is why we call these systems Self-Organized Critical systems SOC systems are robust If we change the environment of the system on the fly, the system reaches its critical state after an intermediate period of transition SOC dynamics are often called sandpile dynamics, because of the first model showing SOC behavior One of the most important property of SOC systems are the power-law distribution of the characteristic quantities of events For example, if we plot the number of events against their magnitude (whatever it means in the given system), we see a straight line on a log-log scale The critical exponent the slope of the line depends on the specific system we examine Power laws indicate a scale-free behavior, which means that the system has no typical size [3] It implies that there is no need to examine such systems on different scales, because the driving mechanism behind the dynamics of the system remains the same Catastrophes are inherent parts of the system The value of the exponent is one of the most mysterious things in SOC phenomena systems from seemingly very different disciplines can have exactly the same critical exponent Up to now, there is no way to tell in advance whether a system shows SOC behavior We have some key properties though, which all the observed SOC systems share SOC systems tend to be complex, nonlinear systems having many degrees of freedom We call a system with a large variability complex [3] As the simulations showed, contingency is not relevant Both stochastic and deterministic systems can show SOC behavior [3] 10

13 14 THE BK MODEL CHAPTER 1 EARTHQUAKES 14 The Burridge Knopoff model of earthquakes The model The Burridge Knopoff (BK) model is a stick-slip model made by R Burridge and L Knopoff in 1967 [9] It is a relatively simple deterministic model, inspired by real earthquake faults It consists of blocks with the same size and mass, connected to each other and a plate (called the driving plate ) by springs The driving plate moves at a constant, very low speed, pulling the blocks via the springs connecting the blocks with the driving plate The blocks lie on another plate (the static plate ), which is fixed, and on which they can slide if the resultant force acting on them enables to do so If a block is stationary, it is sticked to the static plate The block can be pushed and pulled by the springs that connect them to the driving plate and to the neighboring blocks If the block moves, however, the static friction force that made it stick changes to a kinetic friction force Figure 4 illustrates the one-dimensional BK-model The model can be conceived as a cross section of the crust of the Earth where tectonic movements occur The two-dimensional model (where each block has 4 neighbors) represents the contact surface of the tectonic plates Figure 4 The one-dimensional Burridge Knopoff model The arrow indicates the motion of the driving plate, while the static plate is fixed 11

14 14 THE BK MODEL CHAPTER 1 EARTHQUAKES If we want to acquire the equation of motion of a block, we have to give the resultant force acting on it Let us use the following notations: X i : displacement of the block from the initial position, m: mass of a block, k c : elastic constant of the springs which belong to the neighboring blocks, k p : elastic constant of the spring which connects the block to the driving plate, t : time, v : speed of the driving plate, Φ: friction force In the model, the blocks are restricted only to move to the right (positive direction along the x-axis) This makes us possible to solve the equations more simply, since a negative resultant force can not only mean that the block is being deccelerated, but if the block is stationary it means, that the forces currently acting on it are unable to move it (so there is no need to test this case in a computer program) Since the velocity of a block can only be positive, the sign of the friction force is always negative The resultant force is the sum of the forces that act on the block For the i-th block one can write (see Fig5) (5) F i = F c + F p Φ, where F i is the resultant force acting on the i-th block, F c is the force that arises from the springs which connect the block to the neighboring blocks, F p is the force exerted by the spring which connects the block to the driving plate, and Φ is the friction force acting on the given block [16] According to Newton s Second Law of Motion, and with the formula of the classical elastic force, the equation of motion of block i can be written as F i = mẍi, F c = F i 1 + F i+1 = k c (X i 1 X i ) + k c (X i+1 X i ) = k c (X i+1 2X i + X i 1 ), F p = k p (v t X i ) 12

15 14 THE BK MODEL CHAPTER 1 EARTHQUAKES k p k c k c F c, i-1 F c, i+1 Fp Figure 5 Forces acting on a block For a detailed explanation see the text Substituting these equations into Eq(5), we obtain (6) mẍi = k p (v t X i ) + k c (X i+1 2X i + X i 1 ) Φ It can be seen from the above equation that the source of the excitation is the driving plate, which continuously makes energy flow into the system This energy does not remain in the system, but it is dissipated by the friction force, after some of the blocks slip and move along the x-axis We call the slip of the blocks an event An event occurs instantaneously, which means that the system time does not change during an event Because of this, no movement of the driving plate occurs during an event We describe the measuring of the attributes of events later Velocity dependent friction force In order to cast Eq(6) into a closed form, we need the functional form of the friction force between a block and the static plate As a simple approach, one can start with the classical Coulomb friction, which has the form F s = µ s F c, F k = µ k F c, where F s is the static friction, F k is the kinetic friction, µ s and µ k are the coefficients of the static and kinetic friction respectively, and F c denotes the contact force in the normal direction We assume that F c is constant in time, and it is the same for all the blocks 13

16 14 THE BK MODEL CHAPTER 1 EARTHQUAKES We can convert these two types of forces to one velocity dependent force µ s F c, for ẋ = 0, F f (ẋ) = µ k F c, for ẋ > 0 Recent measurements show that when the normal component of the contact force is large (as it is in the case of tectonic plates), the nature of the friction force depends on the relative velocity of the two contacting bodies More precisely, the force weakens as the velocity increases We use the friction force which had been introduced by Carlson, Langer, Shaw and Tang in [14], for ẋ < 0, Φ(ẋ) = 1, for ẋ = 0, 1 σ, for ẋ > 0 1+2αẋ/(1 σ) where σ and α are the two parameters of the force σ represents an instantaneous drop of the friction force at the onset of slip, and α represents the rate at which the friction force weakens with sliding velocity [23] It had been demonstrated recently, that with such a non-classical friction law, the BK model exhibits dynamical instability [23] Let us remark that one can improve the form of the velocity-weakening friction and use more sophisticated formulae, providing a more realistic description of the interaction of tectonic plates [32] The ẋ < 0 case in the form above expresses, that the block cannot move to the left Note that the force is normalized to Φ(ẋ = 0) We plotted the graph of the friction force in Fig6 Dimensionless equation of motion From the model one can derive N equations of motion, one equation for each block The equation of motion of the i-th block depends on the time t, the velocity of the block v, the mass m of a block, the position of the (i 1)-th, the i-th and the (i + 1)-th block, the elastic coefficients k p and k c, and the parameters σ and α of the friction force (7) mẍi = k p (v t X i ) + k c (X i+1 2X i + X i 1 ) Φ(Ẋi) (i = 1,,N) The value of the parameters m, k p, k c and v is constant Let us introduce some new notations for the characteristic quantities of the model: 14

17 14 THE BK MODEL CHAPTER 1 EARTHQUAKES σ α = 025 α = 100 α = 300 α = 1000 σ = 02 φ(ẋ) αẋ ẋ Figure 6 The velocity-weakening friction force, Φ(ẋ) σ determines the initial drop of the force when the block sets in motion, while α controls the rate of weakening with increasing velocity characteristic frequency, ω := k p m, which is the frequency of a block attached to a spring of the driving plate, characteristic distance, L := φ(0) k p, which is the maximum horizontal length of the spring of the driving plate before the block slips, if we neglect the elastic forces of the neighboring blocks acting on the block we examine, characteristic time, t 0 := T 2π = ω 1, where T is the period of a block attached to a spring of the driving plate, characteristic force, φ(0), which does not really matter in our case, since our friction force φ(ẋ) is normalized to φ(0), characteristic speed, v 0 = L t 0 = 2πL T = Lω, which is the ratio of the characteristic distance and the characteristic time, characteristic elastic coefficient, l 2 = kc k p 15

18 14 THE BK MODEL CHAPTER 1 EARTHQUAKES In order to make the equations dimensionless, the characteristic values have to be used as units of measurement [23] The dimensionless quantities can be introduced as follows block displacement: x i := X i L X i = Lx i = φ(0) k p x i, time: speed: force of friction: t = t t 0 = t ω t = t ω, v = v v 0 = v Lω v = Lωv, ϕ(ẋ i ) = φ(ẋi) φ(0) φ(ẋi) = φ(0)ϕ(ẋ i ) In our particular case, ϕ(ẋ i ) = φ(ẋ i ), but theoretically any form of the friction force can be used, so to keep our equations general, we make it dimensionless explicitly Substituting the new quantities into Eq(7), we get m φ(0) ẍ i = k p Lωv t k p ω k φ(0) φ(0) p x i + k c (x i+1 2x i + x i 1 ) ϕ(ẋ i )φ(0) k p k p This equation can be simplified using L = φ(0)/k p, and dividing the result by φ(0) This way we obtain m k p ẍ i = vt x i + k c k p (x i+1 2x i + x i 1 ) ϕ(ẋ i ) It can be seen from the left hand side that the coefficient of ẍ i is 1/ω 2, so we can write ẍ i = ω 2 [ vt x i + l 2 (x i+1 2x i + x i 1 ) ϕ(ẋ i ) ] Since the mass m of the blocks and its unit is arbitrary, we can set the value of m to be equal to k p This way the characteristic frequency ω = k p /m becomes ω = 1 Then the final form of the equation of motion of the i-th block follows as [23] (8) ẍ i = vt x i + l 2 (x i+1 2x i + x i 1 ) ϕ(ẋ i ) 16

19 14 THE BK MODEL CHAPTER 1 EARTHQUAKES Time evolution of the model The BK model describes a deterministic nonlinear dissipative system with stick-slip dynamics This model was the first example of Self-Organized Criticality in a deterministic dynamical system [13] The BK model is deterministic, because it contains only quenched disorder The only reason why we cannot solve the model analytically are the nonlinearity of the friction force, and the large number of degrees of freedom due to the relatively large number of blocks The time evolution of the model starts from a random stable initial state Then, through the continuous displacement of the driving plate, energy begins to flow into the system, increasing the stress stored in it When the resultant force acting on a block becomes so big that the force of static friction cannot withstand, the block slips Note that this can only occur when the resultant force is directed to the right, since slipping to the left is prohibited by the friction force When a block moves, the forces acting on its neighboring blocks change, so that the slipping of a block can trigger the slipping of its neighbors In most of the cases, slipping is only a local phenomenon, and it only affects a few blocks While a block is on the move, the force of kinetic friction acts on it Kinetic friction is a dissipative force, acting always against the movement of a block, so when it is active, energy flows out of the system This makes it possible to locally release the stress in the system Excitation of the system is a global phenomenon As the driving plate moves slowly, a large amount of energy is being pumped into the system, but since excitation is distributed uniformly among the blocks, for a long period of time it has no spectacular effects This way the system of the blocks and springs accumulates energy very slowly On the other hand, the only means of relaxation is the dissipation of energy through the force of kinetic friction during the slipping of a block This is a local phenomenon The slipping and moving of the blocks is called an event, which happen suddenly, in zero system time Events go off in such a short time, that during an event, the driving plate is practically stopped That s why we call the moving of the driving plate quasi-static Even if a small amount of energy is dissipated, it is released in a small area of the system, which can make the density of the released energy quite large Since the stress stored in the system is relatively large, and the magnitude of dissipation is usually small, the energy begins 17

20 14 THE BK MODEL CHAPTER 1 EARTHQUAKES to pile up in the system When it reaches a critical amount, large events can occur, in which many of the blocks are involved The random initial state of the system is stable, which means that no local stress reaches the threshold where a block slips Since this state is randomly generated, it is very likely that the stress stored in the system is much lower than the threshold stress it can stand As a consequence, in the beginning of the process, large events rarely occur Later on, the stress of the system fluctuates around a characteristic value as energy flows through the system In this state events are more frequent, and sometimes even large events can arise, the size of which is only limited by the system size This state of the system is called a critical state The model we discuss here drives itself into this state inevitably, and as it reaches this state, it does not leave it any more Because of this feature, the model shows an example of Self-Organized Criticality (SOC) When the system reaches the state of SOC, its behavior becomes statistically more regular, so that we can examine it The mean elongation of the springs and the energy stored in the system versus system time can be seen in Figs7 and ǫ(t) Plate springs Block springs t Figure 7 Mean elongation of the springs It can be seen that the plate springs are much more elongated than the springs between the blocks The parameters were set as N = 100, l = 3, α = 1, σ =

21 14 THE BK MODEL CHAPTER 1 EARTHQUAKES E(t) Plate springs Block springs t Figure 8 Energy stored in the system The plate springs store most of the potential energy, since they are more elongated The beginning of the curve has a quadratic shape due to the quadratic nature of the elastic potential Parameter values: N = 100, l = 3, α = 1, σ = 001 It can be seen that, although the model is fairly simple, it exhibits complex time evolution, which cannot be predicted from the equation of motion of a single block (which plays a very important role in the model, since the model is deterministic) Measuring the events To compare the characteristics of the model to real earthquakes, we have to introduce concepts that can be paralleled to the concepts used in geology and seismology The terminology we use here is the same as used by Kawamura et al [23, 32] The moment of an event is defined as the total displacement of the blocks occured in an event, M = i u i, where u i is the total displacement of the i-th block during the event The magnitude of an event is the 10 base logarithm of the moment, µ = log 10 M [14] The total length on which the two tectonic plates contact can be expressed as the total number of blocks N We regard the critical block of an event as the epicenter We measure the distance of the epicentres of two events as the difference of the serial numbers of their critical blocks, ranging from 0 to N 1 The area where the earthquake takes place is represented by the number of blocks involved in the event The real life 19

22 14 THE BK MODEL CHAPTER 1 EARTHQUAKES time is represented as the system time in the model The recurrence time of two given events is the system time elapsed between them By measuring these values, we can investigate by computer simulation if the model shows the well known characteristics of real earthquake faults (see section 33) 20

23 CHAPTER 2 Numerical simulation 21 Differential equations Finite difference methods The time evolution of the BK model can be determined from the equations of motion of the blocks (Eq(8)) As it has been mentioned in the previous chapter, this differential equation cannot be solved analytically In order to examine the behavior of the model system, the equations should be solved numerically by computer simulation Our main problem in dynamical simulation is to solve first order differential equations given in the following form (9) dx dt = f(x, t) If we need to solve a second order differential equation like Newton s equation of motion, we will have to write it in a form of two first order differential equations, usually in the form of (10) dv dt = a (x(t), v(t), t), dx dt = v(t) Equation (8) is in fact Newton s equation of motion, applied to the i-th block of the model The common approach of the finite difference methods in solving the problem is to discretize the time axis t into distinct intervals with length t Then we have to determine the values of v n+1 and x n+1 at the time t n+1 = t n + t Our aim is to obtain the solution of this discretized problem that best approximates the continuous solution [4] Of course, this cannot be done by decreasing t under any limit due to data representation problems In the rest of this subsection we discuss the concepts needed to decide which finite difference method to use Then we will take a closer look on the two methods we used 21

24 21 DIFFERENTIAL EQUATIONS CHAPTER 2 NUMERICAL SIMULATION to solve the equations of the BK model The source of the following definitions and the descriptions of the methods is the work of Gould and Tobochnik [4] We call a roundoff error the error in the computation caused by number representation With the number of mathematical operations, the roundoff error may increase greatly Truncation error is the error associated with the choice of algorithm This type of error mainly comes from the discretization of a continuous problem Even if we would use our algorithm on an idealized computer, that can store numbers with infinite precision, this error would occur Theoretically, truncation error decreases with decreasing time steps, but in this case the roundoff error and the computation time increases The local error is the error we have in one step of the computation Its sources are the roundoff and truncation errors The global error is the sum of the local errors in all the steps of the computation A given method is of n-th order, if the global error of the method is proportional to ( t) n If we denote the total time by T and the time step by t ( t T), then the number of steps is T t 1 t This means, that if the local error is of order ( t) n, then the global error is of order ( t) n 1 t = ( t)n 1 A method is unstable if it increases its error greatly after a number of steps Unstable methods can yield accurate solutions for short times, but diverge from the true solution for longer times Methods that are not unstable are stable We used only self-starting methods This means, that we needed only the velocity and the position of the blocks at the start of the time interval of the simulation We denoted the initial time by t 0, and the initial position and velocity of the i-th block by x i,0 := x i (t 0 ) and v i,0 := v i (t 0 ), respectively We start from a stable state, where no slipping occurs Thus i {1,,N} : v i,0 = 0, where N is the number of the blocks The Euler method This was the first finite difference method in history The method s iteration equations are the difference equations belonging to the problem s 22

25 21 DIFFERENTIAL EQUATIONS CHAPTER 2 NUMERICAL SIMULATION differential equations v n+1 := v n + a n t, x n+1 := x n + v n t Since the local error is of order ( t) 2, the global error is of order t, which implies that this is a first-order method both in velocity and in position In spite of this method is unstable, it can still be used to acquire first-approach solutions We used this method in the first version of our simulation program Because it is very easy to implement, it was useful in testing all the other parts of the program When we knew all the other parts were functioning well, we replaced the function of the Euler method with the one of the Runge Kutta method The fourth-order Runge Kutta method This method consists of the following steps: (1) We estimate the slope of the solution at the beginning of the time interval (k 1 ) (2) We give two estimations of the slope at the middle of the interval (k 2 and k 3 ) (3) We estimate the slope at the end of the time interval (k 4 ) (4) We gain the new position using a slope constructed from the weighted mean of the estimated slopes, giving twice the weight to the center slopes This can be formulated as k 1 := f(x n, t n ) t, k 2 := f(x n + k 1 2, t n + t 2 ) t, k 3 := f(x n + k 2 2, t n + t 2 ) t, k 4 := f(x n + k 3, t n + t) t, x n+1 := x n (k 1 + 2k 2 + 2k 3 + k 4 ) The solution of Newton s equations of motion is the following k 1v := a(x n, v n, t n ) t, k 1x := v n t, k 2v := a(x n + k 1x 2, v n + k 1v 2, t n + t 2 ) t, 23

26 22 CELLULAR AUTOMATA CHAPTER 2 NUMERICAL SIMULATION k 2x := (v n + k 1v 2 ) t, k 3v := a(x n + k 2x 2, v n + k 2v 2, t n + t 2 ) t, k 3x := (v n + k 2v 2 ) t, k 4v := a(x n + k 3x, v n + k 3v, t + t) t, k 4x := (v n + k 3v ) t, v n+1 := v n (k 1v + 2k 2v + 2k 3v + k 4v ), x n+1 := x n (k 1x + 2k 2x + 2k 3x + k 4x ) Although other higher order methods would have also been suitable to solve the equations of the BK model, we have chosen to implement this algorithm not only because it is a quite precise fourth-order method, but also because Mori and Kawamura used this method in their paper [23], on which our work is mainly based 22 Cellular automata The behavior of the sandpile model had an important effect on the idea of SOC The original sandpile model of Bak, Tang and Wiesenfeld was a cellular automaton [3] Cellular automata were invented by Stephen Wolfram as discrete deterministic systems with a complex time evolution The elements of a cellular automaton are arranged on a discrete grid, where each element stores some local information The cellular automata is defined if the data stored by the elements, and the rules of the evolution are given These rules show when and how an element s data shall be updated in each step of the discrete time The rules are usually local, based on the values of the given element, and some other elements in its neighborhood 24

27 22 CELLULAR AUTOMATA CHAPTER 2 NUMERICAL SIMULATION The sandpile model The first sandpile simulations ran on a square grid In the basic model, the element with the coordinates (x, y) stores a non-negative integer value z(x, y), which denotes the height of the sand pile at the given point, in units of sand grains The initial state is drawn randomly, assigning values from the set {0, 1, 2, 3} to all the elements, using uniform distribution The rule to be applied is the following: if z(x, y) > 3, then z(x, y) := z(x, y) 4 and z(x ± 1, y ± 1) := z(x ± 1, y ± 1) + 1 At each time step, this rule is applied to every element of the lattice simultaneously This rule is able to introduce avalanches to the system, because if z(x, y) exceeds 3, and we distribute the top 4 sand grains to the four neighbors, they can become unstable, triggering the rule again The resolving of the events is quasi-static, which means the system time does not elapse when the state contains at least one unstable element If there is no position where we need to redistribute the grains, the time is incremented by one In each time step, we drop a grain of sand in a random, say (i, j) position, increasing the height of the pile by one at that position This can, of course, trigger an event The continuous dropping of grains is the source of the excitation of the model, while dissipation occurs if one or more grains fall off the edge of the table, thus if a grain is placed outside of the lattice [3] Similar models exist for forest fires, evolution, interaction of agents in social networks, etc The Olami Feder Christensen model Olami, Feder and Christensen published their idea of cellular automaton based model of earthquakes in 1992 [16] This model had been derived from the two-dimensional continuous Burridge Knopoff model The elements are arranged in an L L square grid, where each element represents a block Every block is connected to its four-neighbors via elastic springs We store the displacement of a block from its relaxed position and the resultant force acting on it The movement of the driving plate increases the strain stored in the springs The difference of the OFC and BK models arises from the handling of slipping Slipping is discrete in the OFC model, which means that the blocks subject to slipping first jump to their relaxed position (where the resultant force acting on them becomes 0) After this, the strain on 25

28 23 FORMER RESULTS CHAPTER 2 NUMERICAL SIMULATION their neighbors are recomputed by distributing the strain originally stored in the blocks slipped Because the slipping itself is discrete, no real movement occurs, which makes the use of the friction force needless The dissipation comes from a parameter which controls the portion of the strain to be redistributed The OFC model shows robust SOC behavior and can reproduce the Gutenberg Richter law for magnitude distribution [16] Due to Mori and Kawamura, there are some drawbacks of the cellular automaton models of earthquakes Altough they usually try to mimic the original BK model, the BK model is much more versatile and can be connected to experiments more easily, because it contains the velocity-weakening friction force, which can be measured to different types of materials at different environmental conditions [27] Besides, the statistical properties of the BK model and the cellular automata are not always identical, even if cellular automata can qualitatively reproduce the well known properties of real earthquake faults [27] Because of this, in this work we use the original BK model A detailed comparison of the BK and OFC models can be found in [22] 23 The results of Kawamura et al Our work is mainly based on the papers of Hikaru Kawamura and his collaborators, Takahiro Mori and Akio Ohmura [23, 27, 32] They performed numerical simulations of the one-dimensional BK model with dimensionless equations of motion detailed in section 13 They used the velocity-weakening friction force introduced by Carlson, Langer, Shaw and Tang in [14] They extensively examined the model s dependence on its parameters They found that the results depend most spectacularly on the parameter α, which controls the strength of velocity-weakening They also examined the effects of varying the parameters σ (drop of friction), l (which gives the kc k p elastic constant ratio) and N (the number of blocks) They implemented periodic boundary conditions and generated 10 7 events, leaving 10 4 events out of the statistical investigations to eliminate transient effects [23] They used the Runge Kutta method of the fourth order (see section 21) with tν = 10 6 discrete displacement The size of the system had been set in the range 26

29 23 FORMER RESULTS CHAPTER 2 NUMERICAL SIMULATION N {800,, 6400} These settings resulted in the most precise simulation of the BK model at that time, requiring remarkable computational power [27] Mori and Kawamura examined the magnitude distribution, the mean displacement of blocks, the local and global recurrence-time distribution, the magnitude correlations between successive events, the time and spatial correlations of events before and after the mainshock and the time-dependent magnitude distribution They also compared the time-predictable and size-predictable models [27] We used the results of Mori and Kawamura to compare with the results generated by the BlocK program We managed to reproduce all the numerical results important for the future work with BlocK Because of this, we will refer to the results of Kawamura et al at section 33, where we examine the test process on which BlocK went through 27

30 CHAPTER 3 The BlocK software package 31 Theoretical preliminaries Random numbers In order to draw the initial state, we need random numbers with a uniform distribution There are a large number of methods to generate random numbers on a computer These methods can be divided into two main types The methods of the first type try to acquire numbers from a real random physical process, like the decay of radioactive atoms On an ordinary desktop computer, similar methods probe the system clock, using the last few digits of the current system time to generate a random number Though the system clock is based on a precisely regular physical phenomenon, like piezoelectricity, the running of softwares on a sufficiently complex platform can be treated as a random process, since, for example time sharing operating systems cannot switch between processes exactly after the time slice of the current process had elapsed These methods usually work well for most of the applications (like computer games), but their behavior is quite unpredictible to use them in scientific applications, since nothing can guarantee that the numbers returned will follow a uniform distribution The other problem is, that these methods are too slow because of the continuous probing, and in scientific applications we usually generate zillions of random numbers The other type of random number generators return pseudo-random numbers The methods of this type are deterministic, and are based on mathematical algorithms This way, the numbers returned by them are far from being random, but they are still highly usable in scientific computation, because one can take a look at their attributes, which remain always the same A pseudo-random number generator must have the following properties [6]: 28

31 31 THEORETICAL PRELIMINARIES CHAPTER 3 BLOCK good statistics the relative frequency of the numbers in the generated sequence has to approximate the density function of the uniform distribution, large period the more random numbers the generator can acquire, the better (periodicity of the sequence cannot be avoided due to the limits of the precision of the number representation on a computer), fast the time taken by the random number generator has to be much shorter than the runtime of the program, reproducable the generator shall be able to give the same sequence of numbers from time to time, which is important for the testing of programs We implemented a linear congruent modulo pseudo-random number generator Its algorithm is simple and fast, and it has characteristics which makes it adequate for our purposes The iteration formula of the generator is based on the remainder of integer number division [6] x i = f(x i 1 ) mod m, where x i, x i 1 and m are integers This method needs the value of m and x 0 as parameters It generates numbers until it reaches x k = x 0, and from that it repeats the numbers in the sequence We have to give parameters to a given function f with which the length k of the period will be the longest Because of the presence of the modulo in the formula, it is guaranteed that k {1,,m} The most simple case is the f(x) = ax + c linear case x i = (ax i 1 + c) mod m It can be shown that we achieve the maximum period length, if m is a Mersenne prime m = 2 p 1, and a is chosen to fulfill the condition a m mod m = 1 [6] Using 32-bit signed integers, the maximum period length is k = , when m = and a = or a = To get numbers in the interval [0, 1], we have to use the ratio u i = x i /m The implementation of the algorithm above is the following Note, that random t is the type of the random numbers, which is always a 32-bit signed integer, both on a 32- and on a 64-bit computer random t seed = ; 29

32 31 THEORETICAL PRELIMINARIES CHAPTER 3 BLOCK double uniform ( void ) { seed = 16807; } return 05 + ( double ) seed 05 / RANDOMMAX; We perform three simple tests to make sure of the quality of the sequence In the first test, we ascertain the distribution of the sequence is close to the uniform distribution In Fig1 the histogram of 10 6 random numbers in the interval (0, 1) is presented The bin size is 0025 The density function belonging to a continuous uniform distribution are shown in grey P(ξ) ξ Figure 1 Distribution of the generated pseudo-random sequence After this, we examine the correlation of the successive numbers in the sequence This can be done by plotting consecutive points with (x i, x i+1 ) coordinates on the plane The distribution of the points on the [0, 1] 2 rectangle is uniform, so we can conclude that no significant correlations are present In Fig2 the result of this test is shown for 10 4 numbers Finally, we examine the mean of the sequence The mean approximates 05 when the number of rolls increase, so the generated sequence behaves the same way as if it were a 30

33 31 THEORETICAL PRELIMINARIES CHAPTER 3 BLOCK x i xi+1 Figure 2 Correlation map of the successive elements of the random sequence The completely diffuse pattern demonstrates the fine quality of the generator real random sequence with uniform distribution The value of the mean as a function of the number of rolls can be seen in Fig3 Since the function uniform() passed the tests we mentioned here, we found it suitable to generate the initial state of the system The setup of the initial state is described in the next subsection Setup of the initial state In order to get the initial state, we have to arrange the blocks on the fixed plate We assume, that the distribution of the blocks is uniform The only thing we have to take into account, that the initial state has to be static, which means that no block can move, and there can be no block which is accelerated to the right Let us consider the special state, when the first block is placed arbitrarily, and the other blocks are placed in a way that all the springs connecting the neighboring blocks are fully relaxed We will refer to this state as the state of tranquility The system 31

34 31 THEORETICAL PRELIMINARIES CHAPTER 3 BLOCK mean number of rolls Figure 3 The mean of the pseudo-random sequence converges to 05 with increasing number of function calls stores no energy in this configuration A special case of this state is the reference state, when the first block is placed in the origin of the coordinate system We denoted the dimensionless displacement of the i-th block with x i From now on, x i means the displacement of the i-th block from its position in the reference state Thus, the state of tranquility can be formalized as i {0,, N 1} : x i = c, where c is a constant In the case of the reference state, c = 0 If we use this notation, we do not need to know the relaxed length of the springs Its only drawback is that the state cannot be accurately illustrated by drawing the blocks When the system can be found in a state that is not tranquil, it is in a tense state Using these notations, we distinguish two other types of the states of the system The state is static, if i {0,,N 1} : ẋ i = 0 ẍ i 0, and it is non-static, if i : ẋ i > 0 ẍ i > 0 We call the process of connecting the blocks to the driving plate gripping During this process, we connect/reconnect the springs of the driving plate exactly above the blocks This way, a snapshot of the current state will be imprinted on the driving plate We can use three different methods to grip the blocks: 32

35 31 THEORETICAL PRELIMINARIES CHAPTER 3 BLOCK (1) grip the reference state: each of the springs are connected above the position where their block would be in the reference state, (2) grip the initial state: first we draw the initial state, then we grip the blocks, (3) re-grip the blocks after every event: similar to the previous one, but we vertically reconnect all the springs of the driving plate after every event, too We implemented the second method For this, we do not need to take the springs of the driving plate into account when we draw the initial state, and there is no need to re-grip the state after each event, which makes the program run faster We deploy the blocks from the left to the right The first block can be placed anywhere, so we put it to x 0 = 0 In order to make the initial state a static one, we have to assign a value to the displacement x i from a given interval For this, we have to give the boundaries of the interval of the possible values of x i The equation of motion of the i-th block is Eq(8) ẍ i = vt x i + l 2 (x i+1 2x i + x i 1 ) ϕ(ẋ i ) Immediately after the deployment of the i-th block, this equation can be simplified to the following equation ẍ i = l 2 (x i 1 x i ) 1, since the block is not connected to the driving plate yet (which implies, that vt x i = 0), the new block does not move (so ϕ(ẋ i = 0) = 1), and there is no block on the right of the block we examine The state will be static only if the new block is far enough from the previous one, so the new block is not pushed away by the spring of the previous one, ie ẍ i 0 From this, we get the lower boundary of the interval l 2 (x i 1 x i ) 1 0, x i 1 x i 1 l 2, (11) x i x i 1 1 l 2 The upper boundary of the interval is the highest displacement, with which the spring between the i-th and the (i 1)-th block does not pull away the previous block, ie 33

36 31 THEORETICAL PRELIMINARIES CHAPTER 3 BLOCK ẍ i 1 0 To get the value of the upper boundary, we have to write the equation of motion of the (i 1)-th block ẍ i 1 = l 2 (x i 2x i 1 + x i 2 ) 1 0, x i 2x i 1 + x i 2 1 l 2, (12) x i 2x i 1 x i l 2 We obtained that when we place the new block the following condition has to be satisfied x i [x i 1 1l 2, 2x i 1 x i 2 + 1l ] 2 We assume that the distribution of the displacement is uniform in this interval For the first and second block we apply x 0 = 0, x 1 [x 0 1l 2, 2x o + 1l ] = 2 [ 1 ] 1 l 2, l 2 According to the above arguments, we can now draw the initial state After the deployment of the blocks, we apply the gripping function In the implementation, the deployment of the blocks is done by the function initialize(), and the gripping function is called grip() Note that if l = 0 the program makes the reference state the initial one It is important for the testing of the code (see the next subsection) The program can write any static state into an output file by the function writestate() We can illustrate the position of the blocks by plotting their displacement as a function of their serial number For the initial state, this can also be used to test whether the drawing method was successful If it was, we have to see a graph similar to the one generated by a classical random walk problem with a uniform distribution We show the initial state of 6400 blocks in Fig4 Figure 5 shows the result of a binary random walk after 6400 steps We generated it by using the random function described above If it returned a number below 05, we decreased the position by one, otherwise we increased it In Fig6 the random walk generated by adding the numbers returned by the random function to the position is presented In this case, we scaled and shifted the number returned to the interval [ 1, 1] 34

37 31 THEORETICAL PRELIMINARIES CHAPTER 3 BLOCK xi i (block serial number) Figure 4 Displacement of 6400 blocks in the initial state x step Figure 5 1-dimensional binary random walk Analytical solutions In order to verify the program, we need to solve the most simple cases analytically We discuss two cases in detail 35

38 31 THEORETICAL PRELIMINARIES CHAPTER 3 BLOCK x step Figure 6 1-dimensional random walk The position of a random walker is statistically the same as the initial state of the model Harmonic oscillator We get the model of the harmonic oscillator, if we make the following assumptions: N = 1, there is only one block, ϕ(ẋ > 0) = 0, there is no kinetic friction To accomplish these terms, the program has to be run with the following parameters l = 0 k c = 0, there are no springs connecting the neighboring blocks, this way, the blocks do not feel the effect of each other, behaving as if they were independent, σ = 1, α = 0 ϕ(ẋ > 0) = 0 If l = 0, no random initial state can be drawn (see Eqs(11) and (12)) In this case, we use the reference state as the initial one (see the previous subsection) Under the above conditions, the equation of motion of a stationary block Eq(8) can be simplified to ẍ = vt x 1 36

39 31 THEORETICAL PRELIMINARIES CHAPTER 3 BLOCK The block slips if ẍ > 0 We start from the reference state, where t = 0, x = 0 As the driving plate moves, vt increases, while x remains zero The block becomes unstable when ẍ = 0, that is, when vt x 1 = vt 1 = 0 It means, that the system remains stable for t = 1/v time In general, it is true, that v changes only the system time elapsed between two given events Hence, changing the parameter v has no real impact on the system, so we can assume, that v = 1 From this point, we keep ourselves to this assumption As a result, the shift of the driving plate is vt = 1 at the time t = 1 Analytically thinking, we can say, that t needs to exceed 1 by only an infinitesimal amount When this happens, the block slips, making its velocity greater than 0 Because of this, the static friction (ϕ(0) = 1) changes to kinetic friction, which is 0 in our case At this point, an event is triggered, and because of the quasi-static nature of the events, the system time stops until the event is resolved When we examine the motion of the block during an event, we measure the time with τ, which is called the event time The event starts at τ = 0, when r(τ = 0) = r 0 = 1, where r is the displacement of the block from the equilibrium position of the spring of the driving plate At the moment of the slipping, the block does not move, resulting in v(τ = 0) = v 0 = 0 The block is now only affected by the elastic force of a single spring, which can be written as r = r This is the differential equation of the harmonic oscillator Its general solution is r(τ) = r 0 cos ωτ + v 0 ω sin ωτ With the initial conditions v 0 = 0, r 0 = 1, and with the general settings (ω = 1), we obtain r(τ) = r 0 cos τ = cos τ, as the equation of motion of the block So, the spring pulls the block by 2 r 0 = 2 to the right At this point, the spring would pull the block backwards, but moving to the left is prohibited, so the block sticks, and it is affected by the force of static friction again 37

40 31 THEORETICAL PRELIMINARIES CHAPTER 3 BLOCK Since the new position of the block is x = 2, for the driving plate it will take 2 units of system time to move it again The period of the event in event time can be expressed as T = 2π ω = 2π Damped harmonic oscillator We can realize the damped harmonic oscillator with the following assumptions: N = 1, there is only one block, constant kinetic friction affects the moving block 1, for ẋ = 0, ϕ(ẋ) = c, for ẋ > 0, where c is an arbitrary fixed constant in the interval (0, 1] The parameters of the model have to be the following: l = 0 k c = 0, σ = 1 c, α = 0 Since l = 0, the initial state will be the reference state The moving block is now affected by both the force of the spring of the driving plate and the kinetic friction (13) r = r c = (r + c) The equation of motion is similar to the equation of the harmonic oscillator, but it is shifted by the constant c In order to get the solution, we introduce z(τ) z(τ) := r(τ) + c ż = ṙ z = r, z 0 := z(τ = 0) = r 0 + c With this simplification, Eq(13) takes the form z = z, and the solution reads as z(τ) = z 0 cosτ, r(τ) = z(τ) c = (r 0 + c) cosτ c 38

41 32 TECHNICAL BACKGROUND CHAPTER 3 BLOCK It can be seen that the period remains the same, T = 2π/ω = 2π The only thing changes is the amplitude of the oscillation, which is smaller than the amplitude of the harmonic oscillator A = r 0 c < r 0 In the examples below, c = 03, so the spring moves the block by 2A = = 14 In the figures below we plotted the functions of the harmonic oscillator in red, while the functions of the damped harmonic oscillator are blue In Fig7 we show the analytical solution of the problem Due to the sticking, motion belongs only to the left side of the solution Figure 8 shows the numerical solution In this graph, we plotted the displacement x as a function of event time τ The event time is zero at the beginning of the first event Since event and system time are not on the same scale, we measure the time with two clocks Event time increases only during an event, while system time increases only when the driving plate moves For this, if we plot the events in event time, we cannot see static states, and when we plot the events against system time, we cannot see the exact motion of the blocks, only the jumps in their positions In the figure four events are presented Note that we cannot handle infinitesimal quantities numerically, so t has to exceed the critical time by a certain finite amount This amount is a parameter of the simulation, and practically it is equal to the minimal time interval that can be handled by the differential equation solver This parameter is the so-called time slice In our case, the time slice is set to 002 As we have seen in chapter 2, the lower the time slice is, the better the approximation of the solution is (if we disregard the effect of the roundoff error) In Fig9 the displacement of the block is plotted as a function of system time t When the damping is turned on, there are more events in a given time interval At the same time, the events get smaller, due to the dissipation of energy 32 Technical background BlocK is a software package dealing with the simulation of the one-dimensional Burridge Knopoff (or spring-block) model, and the processing and visualization of the data obtained by the computer simulation The latest version in 2007 is version 5 (v5) To take a glimpse into the development process, see the concise description of versions in 39

42 32 TECHNICAL BACKGROUND CHAPTER 3 BLOCK r(τ) without damping with damping τ Figure 7 Analytical solution of the oscillators 8 7 π 2π 3π 6 5 x(τ) without damping with damping τ Figure 8 The displacement of the block against event time The vertical lines indicate the end of an event Appendix B Version 5 consists of six major parts: common data and libraries (lib), BlocK simulator (sim), PaintState (bkps), StateStat (sstat), MagStat (magstat) and DistStat (diststat) We describe the parts of the package in this chapter The source code of BlocK is written in standard (ANSI) C/C++ The development philosophy behind the software package is classical modular development C++ is only used in the programs using CSV ( Comma Separated Value, semicolon (;) separated 40

43 32 TECHNICAL BACKGROUND CHAPTER 3 BLOCK 8 7 without damping with damping 6 5 x(t) t Figure 9 The displacement of the block against system time in this particular case) input files through the CSVRead library (see the next subsection) If the input is not CSV, it is simple text with one value standing at each beginning of line, possibly followed by a comment until the end of line Empty and comment lines are allowed, the comment is denoted by using the semicolon character The output files are tabulator delimited text files, where exclamation marks (!) denotes comments The output files are designed to be processed with GLE (Graphics Layout Engine, [37]) The running of the executables can be parametrized by arguments If no argument is given to an executable, it displays a brief help on the parameters, files and general usage of the program BlocK has been developed for one-processor home computers, and has been written, compiled and tested under Linux platforms We used the GNU Compiler Collection (GCC and G++) to compile our programs [38] Only free softwares have been used in the development process All parts of the package, including the libraries described below, are written by the author Common data and libraries The common data of the package is stored in the bkdatah header file, which is included in all source files It contains data types (ex 41

44 32 TECHNICAL BACKGROUND CHAPTER 3 BLOCK event), commonly used macros (ex absolute value, square) and commonly used data (ex maximum length of file names, file identifier header strings) bkerror is the common error handling library of BlocK It realizes a usual C errorhandling strategy It contains the external integer variable errcode, the available error codes and a procedure writing the error messages to the standard error (stderr) The executables always return the error code to the caller when they terminate 0 error code stands for No error occured prandom is the library of the pseudo-random number generator It is the implementation of the congruent modulo random generator presented in the previous section prandom works both on 32- and 64-bit machines It uses compile-time code to determine the type of the processor, so it needs to be recompiled if one does not have the object code to the given type of computer CSVRead (csvread) is the C++ library written to read CSV files It had been designed to read charts in CSV format CSVRead allows one to change the delimiter and the line comment characters used, even if the given file is under process It had been originally used in the MACALC project [31] The source code of the common files consist of about 300 lines of C/C++ code BlocK simulator (sim) sim is the heart of the package, as it simulates the BKmodel by using the Runge Kutta method of the fourth order described in section 21 It consists of about 500 C++ lines (20 kbytes) The size of the compiled binary is 30 kbytes At runtime, it uses approximately N bytes memory for data, where N is the number of blocks For 800 blocks, the used memory (without the program code) is about 140 kbytes, for 6400 blocks it reaches 11 Mbytes sim needs only one argument, the name of the parameter file (with extension) The parameter file is a CSV file, where one can set the parameters of the model (namely l, α, σ, ν), the simulation (the N number of blocks, the number of events to be generated, the number of events to be discarded, and the discrete displacement of the driving plate, which is the time slice of the numerical method) The user can also set the type of the outputs to be generated and the name of the output files 42

45 32 TECHNICAL BACKGROUND CHAPTER 3 BLOCK The simulator generates all the output needed to investigate the model The output files are processed later by the other parts of the BlocK package There are four files sim can write as its output: system state data, event time data, event global data and event local data files These cover almost all the direct data one can measure in the model, so all other derived attributes of the state and dynamics of the system can be calculated from them We discuss sim s output files at the processing programs, where they appear as input files StateStat (sstat) StateStat is one of the data processing programs dealing with system state data files (SSDF) SSDF stores a stable state of the system, thus it can be saved between events The file begins with a header identifying the file type:!system STATE DATA FILE After it, there are the parameters of the simulation, followed by the serial number and the system time of the last event before the given system state It is followed by the data itself, consisting of two columns and N rows, where N is the number of blocks The first column shows the serial number of the blocks, and the second column shows the position of the block The SSDF files are in GLE format, so a state can be examined directly by plotting the position of the blocks against their serial number (see Fig4) sstat is a C++ program, and consists of about 190 lines (7 kbytes) The size of the compiled executable is 17 kbytes It uses a negligible amount of memory (a few bytes) StateStat opens SSDF files which names begins with the same string, and ends with a fixed seven digit serial number (which is the number of the state s preceding event if the file had been generated by sim) and a dat extension This common file name string can be given as the first argument of the program, while the second and third arguments give the range of the serial numbers with which the files are processed In the current version, all the input files with subsequent serial numbers will be processed, and there can be no missing files in the given range StateStat calculates the mean elongation of the plate and the connecting springs, and the potential energy stored in the system for each state The output file will be saved in GLE format, in which each line of data corresponds to one input state We save the 43

46 32 TECHNICAL BACKGROUND CHAPTER 3 BLOCK system time of the state, the energy stored in the two types of springs and the mean spring elongation for both spring types DistStat (diststat) DistStat is the other program that processes SSDF It had been designed to investigate the correlation between the position of blocks in different distances from each other Distance in this sense means the number of blocks between the two given blocks, as one could see in section 14 DistStat is written in C, and its source code has about 270 lines (9 kbytes) The compiled binary occupies 17 kbytes The runtime memory used by diststat is usually very low, but it depends on the output histogram s bin size (see below) The program needs a few hundred bytes by default, plus 8 bytes per bin DistStat waits for two arguments, the serial number of the two blocks to be examined It creates a histogram of the absolute difference of the positions of the given blocks, which we call a positional distance ( x) The parameters of the processing can be given in the parameter file diststatin In this the user can give the name and numbers of the input system state data files, just as in StateStat One has to provide the steps in the numbers of the file name serials, so the input files do not have to have subsequent serials The looks of the output histogram is heavily affected by the bin size, which can also be given in the input parameter file The program writes the results of the processing into the output file diststatout This file begins with the usual data of the simulation and the serials of the two examined blocks It is followed by general statistical data: the minimum, maximum, mean and the 0th, 1st and 2nd statistical moments of the positional distance The 2nd/1st moment ratio is also provided for future phase transition investigations The 0th moment serves test purposes The tail of the file contains the histogram described above It consists of three columns, where the first column is the center of the bin, the second column is the probability and the third column is the 10th logarithm of the probability The histogram is normalized to one Only the bins containing nonzero values are saved Because of the output format, the histogram can be easily plotted by GLE or other plotting software 44

47 32 TECHNICAL BACKGROUND CHAPTER 3 BLOCK MagStat (magstat) MagStat is the main data processing program of the BlocK package It processes event global data files (EGDF) An EGDF contains global information about the simulated events which had not been discarded The file starts with a header string:!event GLOBAL DATA FILE Then the general parameters of the simulation follows The data of the events are written in the tail of the file (which can be quite long, depending on the number of non-discarded events), where each line stands for one event sim saves the serial number and the system time of an event, followed by the serial number of the leftmost and rightmost blocks involved in the event, and the number of the epicenter block, which is the block where the event started The last data of the event is the total displacement of the blocks, called the moment (see section 14) MagStat reads one EGDF as input, and uses the data there to make five types of histograms The name of the input EGDF file (without extension, which is automatically dat) has to be given as magstat s only argument All the other parameters are stored in the file magstatparam In this file the first five parameters are the bin sizes of the output histograms Then one can give the upper and lower boundaries of the magnitudes of the events to be processed During the processing, MagStat completely ignores the events with magnitudes out of the given range With the last parameter one can set the degree of locality This parameter tells the program the maximum distance (in number of blocks) between two subsequent events If an event s magnitude is in range, but it is further from the previous event than it is allowed, the event will not be considered MagStat saves five output files The name of the output files will be the same as the name of the input file with a different extension All outputs include a file ID header, the name of the input file and the parameters of both the simulation and the processing, including the number of the accepted events, and their ratio to all the events contained in the input file It also shows the minimum, maximum, mean and the 0th, 1st and 2nd moments of the investigated quantity The program produces the blocks involved statistics in the output files ending with bi extension These files contain the number of blocks involved in the simulated events, which can be interpreted as the geographical extent of an event, according to the model The moment statistics are written into the files which s names end with 45

48 32 TECHNICAL BACKGROUND CHAPTER 3 BLOCK mom MagStat saves magnitude statistics into the files with names ending with mag Magnitude statistics play a central part in the science of earthquakes, that s why the utility itself received its name after the mag files it can produce The last two types of output is in connection with the recurrence of events In the recurrence time files ( rect ) the statistics of event recurrence times are saved These files contain auxiliary data which make possible to plot the histogram easily against recurrence time or against mean recurrence time The recurrence distance is stored in the recd files This shows how frequently two subsequent accepted events occur at a given distance from each other All the histograms produced by MagStat are normalized to 1 MagStat is composed of about 530 lines of C code (23 kbytes) It occupies 28 kbytes when compiled It uses about 1 kbyte memory for data, plus 8 bytes for each bin The processing of an input with about 43 Mbytes size takes approximately 6 seconds on a computer with an intel Pentium 4 CPU Both MagStat and DistStat realizes dynamical histogram drawing, which means they set the boundaries of the histograms according to the data It is an efficient way to calculate histograms, and it has the advantage that the user does not need to set the boundaries beforehand PaintState (bkps) PaintState is a visualizational accessory, which had been made to draw images and animations of events It processes event time data files (ETDF) This type of file is the only output of sim which uses event time instead of system time An ETDF contains information about the dynamics of each block during an event Besides the serial number and system time of the event, the file contains one line for each event time slice of the event Each line contains the position, velocity and acceleration of the blocks, starting from the leftmost one This type of file can only be used to test sim, and to make the events more spectacular by bkps PaintState is a C++ program, which has a source code of about 600 lines (23 kbytes) The compiled executable has a size of 30 kbytes It uses approximately 15 kbytes plus 40 bytes/block memory for data, which results in a 335 kbytes memory occupation for 46

49 33 SIMULATIONAL RESULTS CHAPTER 3 BLOCK the typical 800 number of blocks, and 025 Mbytes for a large system consisting of 6400 blocks On has to give the name of the input ETDF as the first argument of bkps The second and third arguments are the serial numbers of the leftmost and rightmost blocks of the array of blocks to be drawn If the last two parameters are omitted, the program shows the entire system The other parameters are given in the input parameter file bkpsbkp In this file, one can set the appearance of the output images/animation Since the BK model does not include the relaxed length of the springs, it is impossible to draw a state perfectly (see section 14) Some additional information is needed for reference We call a state the default state when the image of the state drawn by PaintState looks like a state of tranquility (i e the distances between the neighboring blocks are the same) We used the initial state for this purpose, stored in the file istate25600dat This way bkps draws the states relative to the initial state One can use another default state (for example a real state of tranquility) For this, we only need to give the name of the default state file in bkpsbkp bkps uses GLE to draw the images and MPEG Encode (written at the University of California, Berkeley) to make the animation [37, 39] It does not draw the plate springs, because there are usually orders of magnitude difference between the length of the plate and the connecting springs (see Fig7) This way if we draw them on the same image, it spoils the perspicuity of the image We used PaintState to draw the images of the states in Appendix C 33 Simulational results Burridge and Knopoff in [9] were intended to create a simple model of earthquakes, where the small events represent tremors in the crust of the Earth, while large events are the earthquakes To decide whether the BK model is applicable, we have to examine if its behavior is similar to that of real earthquakes In this section we test our software package by analyzing the results of the simulations Furthermore, we investigate the properties of the model and compare it to our expectations and/or previous research results We begin 47

50 33 SIMULATIONAL RESULTS CHAPTER 3 BLOCK our investigations with the events, which are local and happen instantaneously, and we proceed by examining the attributes of more general phenomena The course of events Before we can analyse the statistical properties of many events, we have to make sure that the computation of each and every event is perfect For this, we need to follow the dynamics of the events, ie we have to examine the movement of the blocks during an event This can be done by plotting the data of the blocks against event time, and comparing the results to the analytical solutions of the most simple cases This can be seen in section 31 in Fig8 We also investigated the velocity and acceleration graphs of the blocks, and we found that their values vary according to our expectations Since the system is quasi-static, the results of an event (the movement of the blocks) seem to occur instantly in the terms of system time (see Fig9) After we checked the simulational results of the most simple cases, we have to take a look at the event time behavior of more complex systems To see whether these results are reasonable, we called the help of PaintState In Fig10 one can see the position event time graph of an event It is not trivial to decide whether it is right or not without seeing the blocks pushing and pulling each other via the springs PaintState solves this visualization Figure 11 shows the initial and final states of the event The figures drawn by PaintState show the blocks and the interconnecting springs between them The color of the springs indicate the direction of the force they exert on the blocks, where red stands for a pulling (elongated) spring and blue stands for a pushing (compressed) one The saturation of the color of the springs shows the strength of the force Grey indicates that the spring is relaxed, and a more saturated color belongs to a more elongated spring The saturation is normalized to the strongest force in the given event, no matter whether it belongs to a pulling or a pushing spring, and it is linear with elongation (so the perceived color is distorted by color sensing and sometimes by image compression effects) We have written the event time on the bottom of the images, both from the beginning of the event (relative event time, rel ) and from the beginning of the first event (absolute event time, abs ) In Fig36 in Appendix C, one can follow the event in Fig10 in more details We 48

51 33 SIMULATIONAL RESULTS CHAPTER 3 BLOCK provided the MPEG movie of this event on the home page of the project [40] There we show another interesting case study of a very large event (277th, magnitude 3546) of the same system, in which the compression wave of the event reflects four times from the ends of the system before its energy is dissipated x(τ) τ Figure 10 Graph of an event local data file (ELDF) The figure shows the displacement of the blocks of a 7-block system in event time This is the 321st event, with a magnitude of The critical block is block 1 l = 3, α = Event time: (abs) (rel) Event time: (abs) (rel) Figure 11 The state of the system when the event in Fig 10 occured (upper figure), and the new state which formed due to the event (lower figure) See Fig 36 in Appendix C for more details 49

52 33 SIMULATIONAL RESULTS CHAPTER 3 BLOCK Independence of the blocks It is obvious, that if there is connection between two blocks, their position cannot be independent It is also reasonable to assume, that the correlation between the positions of two blocks decreases if the two blocks are further from each other, ie if there are more blocks between them This implies that the correlation will be the greatest when the two blocks are adjacent, so they are directly connected to each other This can be used as a test to see if the simulated system works as it is expected Two distance concepts are used in the model The (block) distance of the i-th and j-th blocks is d = i j The positional distance of the i-th and j-th blocks is x = x i x j In Fig12 the distribution of the positional distance x of the blocks is presented In the graphs green indicates the normalized histograms obtained from the states of a simulation run The parameters of the system are N = 800, l = 3, α = 1, and we use the static states occuring after the events in the range of (10 4, 10 6 ) events The fitted curves show, that the histogram can be well approximated by Gaussian distribution In the upper right corner of each graph one can see the two blocks we examine We can summarize, that the further the blocks in question are, the more independent they are This can be seen from the increasing standard deviation of the fitted curve of standard normal distribution Statistics of the stable states In the previous two subsections we dealt with the local attributes of the system and the events From now on, we investigate the system and the events as a whole We start with the global properties of the system, which are derived from the local properties (the properties of the blocks and springs) and form statements about the system itself First we examine the mean elongation of the springs The elongations are treated as signed real numbers The elongation of a plate spring is positive if its block is on the right side of the fixed ending of the spring (which is on the driving plate) The elongation of a block spring is positive if the spring is elongated (pulling spring), and it is negative if it is compressed (pushing spring) Figure 7 in section 14 shows the changing of the mean elongation of the block and plate springs with system time The mean elongation of the 50

53 33 SIMULATIONAL RESULTS CHAPTER 3 BLOCK simulated m = 0, σ = simulated m = 0, σ = R( x) 15 R( x) x x 14 simulated m = 0, σ = simulated m = 0, σ = R( x) 08 R( x) x x Figure 12 Distributions of the positional distance of blocks at different distances, approximated by standard Gaussians The standard deviation σ increases, indicating increasing independence of the blocks with distance The parameters are N = 800, l = 3, α = 1 block springs starts from 0, which shows that the quality of our random number generator is adequate, since the blocks are distributed uniformly in the initial configuration On the other hand, the mean elongation of the plate springs starts from 0 of the method we use to connect the plate springs (see section 31) In the usual range of parameters we use throughout this work, the mean elongation of the plate springs is much larger than that of the block springs (that s why PaintState omits the drawing of the plate springs) The block springs instantiates the short range interactions between the parts of the system (the blocks), while the plate springs tell us something about the current 51

54 33 SIMULATIONAL RESULTS CHAPTER 3 BLOCK driving and positional state of the system Because of this, we write only about the plate springs here In the beginning of the time evolution of the system, the absolute value of the mean elongation increases as the driving plate begins to pull the springs This is the transient state of the system After we let the system rearrange itself, the mean elongation fluctuates around a constant value When this occurs, the system reaches its critical state The fluctuations represent events, where the size of a fluctuation is proportional to the moment of the given event We show in the next subsection, that the distribution of the moments (the magnitude distribution) follows a power law From this it can be seen, that the noise-like fluctuation we see in Fig7 is an example of the famous 1/f noise we mentioned in section 13 Figure 8 shows the total elastic potential energy stored in the springs of the system It is quadratic in elongation, so the curve belonging to the transient is a part of a parabola Magnitude distribution The cornerstone of testing earthquake models is to see whether they can reproduce the Gutenberg Richter law Computer simulation revealed that the BK model with the velocity weakening friction force does so, for a well-defined interval of magnitudes We get the longest interval if we set α = 1 With this setting, the shape of the power law is only distorted by finite-size effects [23, 27] We can see the magnitude distribution of this configuration with N = 800 blocks in Fig 13 The graph of the approximating power function had also been plotted with an exponent of B = 05 This is the same exponent as Mori and Kawamura found in [27] Following [27] we also investigated the effect of changing the model parameters on the magnitude distribution The results can be seen in Figs14, 15 and 16 Mori and Kawamura found that for α 2, so when the velocity-weakening property of the friction force is increased, the system exhibits a pronounced peak structure for large events, and the GR law remains for small events We can confirm this statement, but by Fig14 we have to remark, that the transition from the GR law to this pronounced peak state is continuous, and it does not have a sharp critical point at α = 2 It can also be seen that the α > 2 cases does not differ qualititavely from the α = 2 case, so in our further 52

55 33 SIMULATIONAL RESULTS CHAPTER 3 BLOCK 2 α = log[r(µ)] -1 B = µ Figure 13 Magnitude distribution of a system with parameters N = 800, l = 3, α = 1 The line of the power law fits the histogram well through 4 orders of magnitude, with an exponent of B = 05 The peak on the left and the cutoff are finite-size effects investigations we usually use only the α = 2 setting to represent the system exhibiting the peak structure for large events As previous authors showed in [22, 27, 19] the model has a transition from an SOC to a creeping-like behavior when α < 1 In the latter state the large events are suppressed, and the system has a characteristic small event Clancy and Corcoran in [22] shows that this transition is continuous, and Mori and Kawamura in [27] puts the critical area near α = 025 Our simulations show that this area is in the interval α [03, 04] (see Fig14) 53

56 33 SIMULATIONAL RESULTS CHAPTER 3 BLOCK 2 1 α = 1 α = 075 α = 05 α = α = 1 α = 2 α = 3 α = 5 α = log[r(µ)] -1 log[r(µ)] µ µ α = 020 α = 025 α = 040 α = 050 α = 060 α = 080 α = α = 10 α = 12 α = 14 α = 16 α = 18 α = 20 log[r(µ)] -1 log[r(µ)] µ µ α = 005 α = 010 α = 015 α = 020 α = 025 α = 030 α = 035 α = 040 α = 100 log[r(µ)] µ Figure 14 Magnitude distribution with varying α value At the fixed values N = 800 and l = 3, the α = 1 case shows power law behavior on the widest range of magnitudes With increasing α values the frequency of large earthquakes increase, while decreasing α below 1 raises the frequency of middle-sized events 54

57 33 SIMULATIONAL RESULTS CHAPTER 3 BLOCK Figure 15 shows the dependence of the manitude distribution on parameter l in the cases of α = 1 and α = 2, when large events may occur One can conclude by these figures, that if one increases the stiffness parameter l, the GR law will be extended from less than four to almost six orders of magnitude in the case of α = 1, and the scale-free behavior becomes more spectacular in the case of α = 2 One can also notice that the magnitude of the largest possible event increases with l, since the close range interactions get stronger Figure 15 is the reproduction of Fig2 in [27] Comparing the two figures, we can see that the finite-size effects are greater if we use periodic boundary conditions as in [27] We can see from Fig16 that if we consider a system of N = 6400 blocks with l = 10, the steep tail disappears From this we know, that this steep tail is a finite-size effect, but it does not change the quality of the distribution Because of this, we can say that it is adequate to use 800 blocks One can also notice, that though the slope of the tail changes with N, the position of the local maximum remains the same, which means that the characteristic magnitude is really characteristic to the system, and it does not depend on the system size 2 1 l = 2 l = 3 l = 6 l = l = 2 l = 3 l = 6 l = log[r(µ)] -1 log[r(µ)] µ µ Figure 15 Systems of N = 800 blocks We vary the value of l, while α is fixed to α = 1 (left) and α = 2 (right) Critical behavior describes the system on wider ranges of magnitude if we increase the value of l Figure 17 shows the first and second statistical moments of the magnitude against the friction parameter α One can also see the ratio of these moments Statistical moments 55

58 33 SIMULATIONAL RESULTS CHAPTER 3 BLOCK 2 1 α = 1, N = 800 α = 1, N = 6400 α = 2, N = 800 α = 2, N = log[r(µ)] µ Figure 16 The behavior of the model does not significantly depend on the size of the system We fixed l = 10 and vary the values of α and N are usually used to detect critical points of the system where phase transitions occur Thus, it has importance in the future research Recurrence time distribution In this subsection we examine the distribution of the time elapsed between two subsequent events We can constrain our investigations with the magnitude and spatial distance of events, so the meaning of the term subsequent event changes with our constraint settings When we refer to all events, we mean that the observation is not constrained by event magnitude When we say the observation is global, we mean that there is no constraint on the distance of the subsequent event from the current event In Fig18 we show the global recurrence time distribution of all events One can see that the distribution follows an exponential function The exponent of the exponential function increases when α is decreased, decreasing the T mean recurrence time So the events become more frequent as the stiffness of the system increases Figure 19 shows that the distributions fall on top of each other as we plot them against T/ T, the time normalized by the mean time 56

59 33 SIMULATIONAL RESULTS CHAPTER 3 BLOCK 6 5 1st moment 2nd moment 2nd/1st moment 4 3 momenti(α) α Figure 17 The statistical moments of a system with N = 800 and l = 3 The peak on the left side is artificial, caused by the numerical division of two numbers close to 0 Statistical moments are used to detect critical points in the parameter space, where phase transitions occur α = 025 α = 075 α = 1 α = 2 α = 3 α = α = 025 α = 075 α = 1 α = 2 α = 3 α = R(T) 600 log[r(t)] T T Figure 18 Recurrence time distribution on linear scale (left) and on semilogarithmic scale (right) The exponent increases with decreasing α N = 800, l = 3 Figure 20 illustrates the shape of the recurrence time distribution depends on l When α = 1, we can see from the curve of the graph, that the distribution is not a simple exponential, but it becomes more off-exponential as l increases This is even more spectacular in the case of α = 2 57

60 33 SIMULATIONAL RESULTS CHAPTER 3 BLOCK 0-1 α = 025 α = 075 α = 1 α = 2 α = 3 α = 10 log[r(t/ T)] T/ T Figure 19 The graphs of the histograms in Fig 18 fall on each other when rescaled by the mean recurrence time 0-1 l = 2 l = 3 l = 6 l = l = 2 l = 3 l = 6 l = 10 log[r(t/ T)] -2-3 log[r(t/ T)] T/ T T/ T Figure 20 Recurrence time distribution of systems with N = 800 and α = 1 (left) and α = 2 (right) The histogram becomes non-exponential with increasing l Following Ref[27] we distinguish the small and large events by the peak structure of the magnitude distribution (Fig 14, right side) We call an event large if its magnitude is greater than 15, which is near the maximum of the peak We also investigate significant events with magnitude greater than 05, which is close to the minimum between the 58

61 33 SIMULATIONAL RESULTS CHAPTER 3 BLOCK scale-free and the characteristic regions of the magnitude distribution We simulated 10 6 events, which is one order of magnitude less than the events generated by Mori and Kawamura in [27] The source of the noise in Figs21 and 22 is the relatively small amount of large events Figure 21 shows the global recurrence time distribution of large events (µ > 15), and Fig 22 shows it for significant events (µ > 05) These results are in accordance with [27] It has been noticed in [15, 27], that for relatively large l values, like l = 10, one can observe a pronounced peak structure in the recurrence time distribution Mori and Kawamura recently showed, that this peak structure is a finite-size effect, and global recurrence time distribution does not show peak structure if the system size is large enough [27] We observed the same phenomena both for µ > 15 and µ > 05, but when we took all events into account, we did not observe the peak structure at all α = 075 α = 1 α = 2 α = 3 α = l = 2 l = 3 l = 6 l = log[r(t/ T)] log[r(t/ T)] T/ T T/ T Figure 21 Recurrence time distribution of large events with magnitude µ > µ c = 125 The distribution does not change significantly with α or l N = 800, l = 3 (left), N = 800, α = 1 (right) Statistical noise is more intense than in Fig 20 because of the relatively small number of large events The effect of changing the parameters l and α on the local recurrence distribution can be seen in Fig23 We were unable to take only large events into account, since we had too few events The figure shows the recurrence time distribution of the local (the subsequent event has to be closer than 31 blocks) significant (µ > 05) events One can see that the exponential tail of the distributions does not change in quality, but the extremum of the 59

62 33 SIMULATIONAL RESULTS CHAPTER 3 BLOCK 0-1 α = 025 α = 075 α = 1 α = 2 α = 3 α = l = 2 l = 3 l = 6 l = 10 log[r(t/ T)] -2-3 log[r(t/ T)] T/ T T/ T Figure 22 Recurrence time distribution of major events with magnitude µ > µ c = 05 N = 800, l = 3 (left), N = 800, α = 1 (right) peaks at short times shift in the different cases This suggests the presence of twin events at α = 1, and characteristic recurrence times at larger α values in our simulations, just as in [27] We mention though, that since we constrained ourselves to local investigations, the few number of valid events denies us to do more detailed evaluation α = 075 α = 1 α = 2 α = 3 α = l = 2 l = 3 l = 6 l = log[r(t/ T)] log[r(t/ T)] T/ T T/ T Figure 23 Local recurrence time distribution of major events with magnitude µ > µ c = 05 The maximum distance of the epicenter of the subsequent events from the current event is d max = 30 blocks N = 800, l = 3 (left), N = 800, α = 1 (right) Major events rarely occur in each other s neighborhood, hence we have only a few events, causing a high level of noise 60

63 33 SIMULATIONAL RESULTS CHAPTER 3 BLOCK Spatial extent of events The earthquake fault is represented by the N blocks in the BK model The distance between two blocks, measured in number of blocks corresponds to the distance in the real space of the fault Thus, the spatial extent of an event can be obtained by examining how many blocks are involved in the event Knowing this, we can ask about the distribution of the spatial extent of events in the model The answer to this question can be seen in Fig24 One can see, that the distribution of the spatial extent of the events follows a power law with an exponent of 165 when we set N = 800, l = 3, α = 1 The change of the distribution is illustrated in Figs25 and 27 One can see that the exponent is heavily affected by the value of α, while l modifies its value by only a small amount It seems though, that l has an effect on the position of the tail of the distribution For greater values of l (stiffer systems), the scale-free interval is increased compared to the systems with smaller l values This can be a finite-size effect, but we need to do more simulation runs before we can decide this The distribution suggests the best scale-free behavior for the values of l = 10, α = 1 In Figs26 and 28 one can see the role of α and l in the distribution of the spatial extent of large events We kept the scale of Figs25 and 27 in Figs26 and 28 to make the contribution of large events to the distribution of all events easier to see One can observe, that large events have characteristic spatial extent, which is modified by α when α < 1 (Fig26), and it is also affected by l (Fig28) The shifts in the exponent of the distribution of spatial extent of all events can be understood by the shifts in the characteristic spatial size of the events with a given magnitude Further investigations are needed in the future to get a detailed view of the role of α and l in the attributes of the distribution, and to sort out the possible finite-size effects Recurrence distance distribution The distance of two subsequent earthquakes above a given magnitude can by obtained if one takes the number of blocks between their epicenters We treat the critical block (the block which slides first, triggering the event) of an event as the epicenter of the event If the subsequent events are completely independent of each other, the problem can be solved analytically In this case, the 61

64 33 SIMULATIONAL RESULTS CHAPTER 3 BLOCK 1 α = R(bi) 10-3 B = bi Figure 24 The distribution of the spatial extent (the number of blocks involved) of the events follows a power law, with an exponent of B = 165 The cutoff is caused by the limited space, i e the finite number of blocks N = 800, l = 3, α = α = 1 α = 075 α = 050 α = 025 α = α = 1 α = 2 α = 3 α = 5 α = R(bi) 10-3 R(bi) bi bi Figure 25 Blocks involved in events in a system with parameters N = 800 and l = 3 question takes the following form: what is the distribution of the distance of two points randomly thrown on a unit length interval with uniform distribution? The problem can 62

65 33 SIMULATIONAL RESULTS CHAPTER 3 BLOCK α = 1 α = 075 α = α = 1 α = 2 α = 3 α = 5 α = R(bi) 10-3 R(bi) bi bi Figure 26 Blocks involved in large events (µ c = 125) N = 800, l = 3 Large events have a characteristic spatial size l = 2 l = 3 l = 6 l = l = 2 l = 3 l = 6 l = R(bi) 10-3 R(bi) bi bi Figure 27 Blocks involved in events in a system with parameters N = 800, α = 1 (left) and α = 2 (right) be solved with geometric probabilities, and the result is (14) P(ξ < q) = F(q) = q(2 q), f(q) = 2(1 q), where ξ is the random variable of the distance of the points (ξ, q [0, 1]), F(q) is the probability distribution function, and f(q) is the probability density function The analytical solution of the independent events can be seen as a grey line in Figs29-31 Figure 29 shows the distribution of the distance of subsequent events, regardless of their magnitude One can observe, that for large distances the numerical results fits well to the theoretical expectations, so the far away events can be treated as independent 63

66 33 SIMULATIONAL RESULTS CHAPTER 3 BLOCK l = 2 l = 3 l = 6 l = l = 2 l = 3 l = 6 l = R(bi) 10-3 R(bi) bi bi Figure 28 Blocks involved in large events (µ c = 125) N = 800, α = 1 (left) and α = 2 (right) ones Its not the case for close event pairs For α > 035, the events tend to occur in the vicinity of each other (0 < d < 50) On the other hand, for α < 40 it is unlikely that after an event the subsequent event occurs in its neighborhood These results can be explained with a better relaxation of the of the system in a larger area when the velocity-weakening property of the friction force is decreased The transition between these two kinds of system seems to be continuous in α Figure 30 shows the same phenomena for small events µ > 05, while Fig31 shows it for significant events µ > 05 One needs to simulate more events to investigate large ones From these figures it seems that for α > 05 big events have a characteristic distance on which no subsequent event is likely to occur The Mogi doughnut phenomena We introduced the Mogi doughnut phenomenon in section 12 Carlson claimed in [15] that they did not observe the Mogi doughnut in the BK model, while Mori and Kawamura in [27] reported that they found statistical evidence to its presence, but for only in a very close vicinity of the epicenter of the mainshock, and for a time very close to the time of the mainshock We did not make statistical evaluation of the question, but we observed the Mogi doughnut for randomly chosen large events We selected a large event from the time series of the moments of the events, and we plotted the spatial differential moment of the blocks The latter means, that we measured the total moment of each block in a 64

67 33 SIMULATIONAL RESULTS CHAPTER 3 BLOCK α = 025 α = 075 α = 1 α = 2 α = 3 α = 10 uniform α = 005 α = 025 α = 030 α = 035 α = 040 α = 050 uniform R(d) 0002 R(d) d d Figure 29 Distribution of the distance of the epicenters of two subsequent events The grey line indicates the theoretical expectation when the events are independent At about α = 03 the system behaves much like as in the independent case When α > 03, the events tend to be closer to each other and subsequent events have a characteristic distance N = 800, l = α = 025 α = 075 α = 1 α = 2 α = 3 α = 10 uniform α = 005 α = 025 α = 030 α = 035 α = 040 α = 050 uniform R(d) 0002 R(d) d d Figure 30 Distribution of the distance of two subsequent events having a magnitude µ > µ c = 05 Events tend to occur in pairs at α 05 N = 800, l = 3 given event, and plotted it as a function of the block serial number Then we went back in time, and gathered the spatial differential moment of the blocks in the many small events preceding the mainshock We added up the moments of each block in the events until we found that the integral area of the moments became approximately the same as 65

68 33 SIMULATIONAL RESULTS CHAPTER 3 BLOCK α = 025 α = 075 α = 1 α = 2 α = 3 α = 10 uniform α = 005 α = 025 α = 030 α = 035 α = 040 α = 050 uniform R(d) 0002 R(d) d d Figure 31 Distribution of the distance of two subsequent major events above the magnitude threshold µ c = 05 The resolution decreases due to the lower number of events above the threshold magnitude The results are similar to large events with µ 125, but the noise is even bigger N = 800, l = 3 the moment of the large event Using this method we could observe the Mogi doughnut for large distances (see the example in Fig32) It would be reasonable in the future to make a statistical analysis of the Mogi doughnut phenomena based on this method to see whether it is really present in the BK model for long distances, or only for short ones as Mori and Kawamura claimed 66

69 33 SIMULATIONAL RESULTS CHAPTER 3 BLOCK moment(t) t moment moment bi bi Figure 32 The upper figure shows the moment of events occured in the time interval (10016,10019) in a system with parameters N = 800, l = 3, α = 1 In the two lower figures we show the integral displacement of each block in a given time interval The left figure shows the moments of the blocks of the system in the time interval (10016, ), while the right one presents them in the interval ( , ) Thus, we can see the (spatial) differential moment of one large event on the right figure, while the left one shows the differential moment of many small events preceding the large one 67

70 Summary Catastrophic phenomena are part of our life We all know the words earthquake, tornado, volcanic eruption, slump, mass extinction and such We know they exist, but we cannot influence them or their effects They arise suddenly and turn our world upside down, giving also rise to human loss That s why catastrophic phenomena act as a driving force of today s experimental and theoretical research It has been shown recently that statistical physics and the theory of complex systems provide an excellent framework to describe catastrophes It turned out that most of catastrophic events share certain common properties For example they cannot be predicted, their statistics can be described with scaling laws and they exhibit 1/f noise behavior Some of these properties can only be handled by modern theories of physics, like the theory of chaos or self-organized criticality My thesis covers the details on the development of the simulational software package BlocK, and the numerical investigations we performed with it BlocK implements the widely used spring-block model of earthquakes, the Burridge Knopoff model The model applies velocity-weakening friction force instead of Coulomb-friction in order to describe the system of blocks more realistically The model s stick-slip dynamics are stored in a system of differential equations with many degrees of freedom We tested our programs in various ways, and we found them to be reliable We are able to reproduce all results existing in the literature of the field Our simulations extend the recent work of Kawamura et al by adding results on a more smoothly resolved parameter space We also introduce novel ideas by investigating the independence of the blocks, the statistical moments of the system, the distribution of the spatial extent of the events and the recurrence distance distribution We have also found evidence that the 68

71 Mogi doughnut phenomena is present in the BK model even for long distances These new results worth further investigations in the future, for which we continue the development of BlocK, which had proven itself to be a useful tool in the research on earthquakes and SOC 69

72 Acknowledgment I express my gratitude to my supervisor, Ferenc Kun for his support and deep insights I thank Péter Rózsa and Árpád Csámer for introducing me to the basics of geology I appreciate the help of Zoltán Halász, who drawed Fig 4 I am grateful to Szabolcs Baják and Kornél Kovács for their friendship and our many discussions on the winding paths of science 70

73 Appendix A Example of an earthquake Here one can see an example of a major earthquake, the one occured at November 14th 2007 in the Antofagasta (Chile) region, near the shores of the Pacific Ocean The data and images are taken from the USGS (United States Geological Survey), and are freely accessible on the internet [36] An excerpt from the official release by the USGS National Earthquake Information Center: The earthquake near Antofagasta, Chile of November 14th 2007 results from the release of stresses generated by the subduction of the oceanic Nazca plate beneath the South American plate In this region, known as the Peru-Chile subduction zone, the Nazca Plate thrusts beneath South America at a rate of approximately 79mm/year in an east-north-east direction This earthquake indicates subduction-related thrusting, likely on the interface between these two plates [36] According to CNN ([42]), authorities reported some injuries but no deaths, and some houses and cars damaged from the earthquake It closed roads, toppled power lines and caused minor panic It was horribly strong It was very long and there was a lot of underground noise, said one witness Another witness reported I was very frightened It was very strong I ve never felt one that strong [42] According to the report of the MTV Híradó (News of the Hungarian National Television) one day later, the earthquake killed two people and destroyed four thousand houses, making 15 thousand people homeless [43] Data on the earthquake on November 19th 2007 [36]: Date: November 14th 2007 Time: 15:40:53 UTC 12:40:53 pm at the epicenter Region: Antofagasta, Chile 71

74 CHAPTER 3 APPENDIX A EXAMPLE OF AN EARTHQUAKE Location: S, W Location uncertainty: horizontal ±86km, depth fixed Depth: 40 km Magnitude: 77 Event ID: USGS us2007jsat Figure 33 The position of the main shock (star) and the strongest aftershocks (circles) of the earthquake of Antofagasta (Chile) at November 14th 2007 The main shock had a magnitude of 77 (USGS) 72

75 CHAPTER 3 APPENDIX A EXAMPLE OF AN EARTHQUAKE Figure 34 Seismographic plots of the earthquake of Antofagasta, November 14th 2007 Zero time corresponds to the origin time of the earthquake The code of the seismographic stations are on the right side, the upper the closer to the earthquake Each trace is normalized to its maximum amplitude (USGS) 73

76 Appendix B History of versions Version 1 The first working version of the program It can be used to examine the correctness of the equations and to estimate the efficiency of the program It uses the Euler method to solve the equations of motion CSV format is used both in the input and output files Only basic statistics and momentary system states can be written as output We process the output data by spreadsheet programs Version 2 It uses the Runge Kutta method of the 4th order The output contains a few more statistics Version 3 The output format accomplishes the GLE input requirements We now use GLE for visualization and to plot data We also compiled and tested this version under Microsoft Windows using the GNU MingW compiler Version 4 The program generates tremor data, all the statistics and GLE planar graphics for system state stills and animations We built 64-bit support into the source code (it needs recompilation) Version 5 The source code is now separated into four parts The output now contains all the statistics we investigated In order to efficiently use computer resources we separated the computer simulation of the system and the data evaluation We splitted the I/O files and changed their format 74

77 Appendix C Illustrations x(τ) τ Figure 35 Graph of an event local data file (ELDF) The figure shows the displacement of the blocks of a 7-block system in event time This is the 321th event, with a magnitude of The critical block is block 1 l = 3,α = Event time: (abs) (rel) Event time: (abs) (rel) Event time: (abs) (rel) 75

78 Event time: (abs) (rel) Event time: (abs) (rel) Event time: (abs) (rel) Event time: (abs) (rel) Event time: (abs) (rel) Figure 36 Stills from the event plotted in Fig 35 The red lines indicate the stretched springs, while the blue lines shows that the spring is pressed (see the animation at [40]) x(τ) τ Figure 37 The 277th event of the same system This is a large event, with a magnitude of 3546 One can see the elastic wave reflecting from the ends of the system four times, before the wave becomes totally damped by the friction The critical block is block 2 N = 7, l = 3, α = 1 (see the animation at [40]) 76

79 Figure 38 Topography of the continents and oceans of the Earth Compare this figure to Fig 39 (National Geophysical Data Center) 77

80 Figure 39 Map of the present location of the plates (NASA Goddard Space Flight Center) 78