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28
2004), Are seismic waiting time distributions universal
 Geophysical Research Letters
"... We show that seismic waiting time distributions in California and Iceland have many features in common as, for example, a powerlaw decay with exponent α ≈ 1.1 for intermediate and with exponent γ ≈ 0.6 for short waiting times. While the transition point between these two regimes scales proportional ..."
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Cited by 21 (2 self)
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We show that seismic waiting time distributions in California and Iceland have many features in common as, for example, a powerlaw decay with exponent α ≈ 1.1 for intermediate and with exponent γ ≈ 0.6 for short waiting times. While the transition point between these two regimes scales proportionally with the size of the considered area, the full distribution is not universal and depends in a nontrivial way on the geological area under consideration and its size. This is due to the spatial distribution of epicenters which does not form a simple monofractal. Yet, the dependence of the waiting time distributions on the threshold magnitude seems to be universal. 1. Introduction: Scaling
Earthquake recurrence as a record breaking process, Geophys
 Res. Lett
, 2006
"... We extend the notion of waiting times for a point process to recurrent events in spacetime. Earthquake B is a recurrence of a previous one, A, if no intervening earthquake happens after A and before B in the spatial disc centered on A with radius AB. The cascade of recurrent events, where each late ..."
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Cited by 10 (1 self)
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We extend the notion of waiting times for a point process to recurrent events in spacetime. Earthquake B is a recurrence of a previous one, A, if no intervening earthquake happens after A and before B in the spatial disc centered on A with radius AB. The cascade of recurrent events, where each later recurrence to an event is closer in space than all previous ones, forms a sequence of records. Representing each record by a directed link between nodes defines a network of earthquakes. For Southern California, this network exhibits robust scaling laws. The rupture length emerges as a fundamental scale for distance between recurrent events. Also, the distribution of relative separations for the next record in space (time) ∼ r −δr ( ∼ t −δt), with δr ≈ δt ≈ 0.6. While the indegree distribution agrees with a random network, the outdegree distribution shows large deviations from Poisson
Theory of Earthquake Recurrence Times
, 2006
"... Abstract: The statistics of recurrence times in broad areas have been reported to obey universal scaling laws, both for single homogeneous regions (Corral, 2003) and when averaged over multiple regions (Bak et al.,2002). These unified scaling laws are characterized by intermediate power law asymptot ..."
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Abstract: The statistics of recurrence times in broad areas have been reported to obey universal scaling laws, both for single homogeneous regions (Corral, 2003) and when averaged over multiple regions (Bak et al.,2002). These unified scaling laws are characterized by intermediate power law asymptotics. On the other hand, Molchan (2005) has presented a mathematical proof that, if such a universal law exists, it is necessarily an exponential, in obvious contradiction with the data. First, we generalize Molchan’s argument to show that an approximate unified law can be found which is compatible with the empirical observations when incorporating the impact of the Omori law of earthquake triggering. We then develop the full theory of the statistics of interevent times in the framework of the ETAS model of triggered seismicity and show that the empirical observations can be fully explained. Our theoretical expression fits well the empirical statistics over the whole range of recurrence times, accounting for different regimes by using only the physics of triggering quantified by Omori’s law. The description of the statistics of recurrence times over multiple regions requires an additional subtle statistical derivation that maps the fractal geometry of earthquake epicenters onto the distribution of the average seismic rates in multiple regions. This yields a prediction in excellent agreement with the empirical data for reasonable values of the fractal dimension d ≈ 1.8, the average clustering ratio n ≈ 0.9, and the productivity exponent α ≈ 0.9 times the bvalue of the GutenbergRichter law. Our predictions are remarkably robust with respect to the magnitude threshold used to select observable events. These results extend previous works which have shown that much of the empirical phenomenology of seismicity can be explained by carefully taking into account the physics of triggering between earthquakes. 1 1
Spike Avalanches Exhibit Universal Dynamics across the SleepWake Cycle
"... Background: Scaleinvariant neuronal avalanches have been observed in cell cultures and slices as well as anesthetized and awake brains, suggesting that the brain operates near criticality, i.e. within a narrow margin between avalanche propagation and extinction. In theory, criticality provides many ..."
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Cited by 6 (0 self)
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Background: Scaleinvariant neuronal avalanches have been observed in cell cultures and slices as well as anesthetized and awake brains, suggesting that the brain operates near criticality, i.e. within a narrow margin between avalanche propagation and extinction. In theory, criticality provides many desirable features for the behaving brain, optimizing computational capabilities, information transmission, sensitivity to sensory stimuli and size of memory repertoires. However, a thorough characterization of neuronal avalanches in freelybehaving (FB) animals is still missing, thus raising doubts about their relevance for brain function. Methodology/Principal Findings: To address this issue, we employed chronically implanted multielectrode arrays (MEA) to record avalanches of action potentials (spikes) from the cerebral cortex and hippocampus of 14 rats, as they spontaneously traversed the wakesleep cycle, explored novel objects or were subjected to anesthesia (AN). We then modeled spike avalanches to evaluate the impact of sparse MEA sampling on their statistics. We found that the size distribution of spike avalanches are well fit by lognormal distributions in FB animals, and by truncated power laws in the AN group. FB data
B.: Scaling properties of the Parkfield aftershock sequence
 Bull. Seis. Soc. Am
"... In Shcherbakov et al., 2006, material in the original manuscript was inadvertently altered during composition. We apologize for the errors introduced through no fault of the authors. The corrections are as follows. On page S378, in the section titled “Scaling Studies of the Sequence, ” the sentence ..."
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Cited by 4 (1 self)
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In Shcherbakov et al., 2006, material in the original manuscript was inadvertently altered during composition. We apologize for the errors introduced through no fault of the authors. The corrections are as follows. On page S378, in the section titled “Scaling Studies of the Sequence, ” the sentence at the end of paragraph 2 should read: Their average value was 0.92 close to our value¯b b 0.89 for the entire aftershock sequence. On page S379, there are two errors in the text following equation 6. The text should read: where.bp b 0 p 1 A limiting case is obtained when b bp ( 0). In this situation the initial rate s is a constant for all magnitude cutoffs mc, which implies that at the very beginning of the sequence, on average, no small aftershocks occur. The ratio
Recurrence and interoccurrence behavior of selforganized complex phenomena
 NONLINEAR PROCESSES IN GEOPHYSICS
, 2007
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Correlated earthquakes in a selforganized model
, 2009
"... Abstract. Motivated by the fact that empirical time series of earthquakes exhibit longrange correlations in space and time and the GutenbergRichter distribution of magnitudes, we propose a simple fault model that can account for these types of scaleinvariance. It is an avalanching process that di ..."
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Cited by 1 (0 self)
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Abstract. Motivated by the fact that empirical time series of earthquakes exhibit longrange correlations in space and time and the GutenbergRichter distribution of magnitudes, we propose a simple fault model that can account for these types of scaleinvariance. It is an avalanching process that displays powerlaws in the event sizes, in the epicenter distances as well as in the waitingtime distributions, and also aftershock rates obeying a generalized Omori law. We thus confirm that there is a relation between temporal and spatial clustering of the activity in this kind of models. The fluctuating boundaries of possible slipping areas show that the size of the largest possible earthquake is not always maximal, and the average correlation length is a fraction of the system size. This suggests that there is a concrete alternative to the extreme interpretation of selforganized criticality as a process in which every small event can cascade to an arbitrary large one: the new picture includes fluctuating domains of coherent stress field as part of the global selforganization. Moreover, this picture can be more easily compared with other scenarios discussing fluctuating correlations lengths in seismicity. 1
Edinburgh Research Explorer
"... Masking of earthquake triggering behavior by a high background rate and implications for epidemictype aftershock sequence inversions Citation for published version: Touati, S, Naylor, M, Main, IG & Christie, M 2011, 'Masking of earthquake triggering behavior by a high background rate and i ..."
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Masking of earthquake triggering behavior by a high background rate and implications for epidemictype aftershock sequence inversions Citation for published version: Touati, S, Naylor, M, Main, IG & Christie, M 2011, 'Masking of earthquake triggering behavior by a high background rate and implications for epidemictype aftershock sequence inversions ' Journal of Geophysical
Preservation of long range temporal correlations under extreme random dilution
"... Abstract Many natural time series exhibit long range temporal correlations that may be characterized by powerlaw scaling exponents. However, in many cases, the time series have uneven time intervals due to, for example, missing data points, noisy data, and outliers. Here we study the effect of ran ..."
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Abstract Many natural time series exhibit long range temporal correlations that may be characterized by powerlaw scaling exponents. However, in many cases, the time series have uneven time intervals due to, for example, missing data points, noisy data, and outliers. Here we study the effect of randomly missing data points on the powerlaw scaling exponents of time series that are long range temporally correlated. The Fourier transform and detrended fluctuation analysis techniques are used for scaling exponent estimation. We find that even under extreme dilution of more than 50%, the value of the scaling exponent remains almost unaffected. Random dilution is also applied on heart interbeat interval time series. It is found that dilution of 7080% of the data points leads to a reduction of only 8% in the scaling exponent; it is also found that it is possible to discriminate between healthy and heart failure subjects even under extreme dilution of more than 90%.