Brownian Passage Time Distribution Research Articles

Comments on Earthquake Probabilities Estimated by the Geometrical Mean Likelihood of the Brownian Passage Time Distribution Model for Recurrent Paleoearthquakes

Comments on Earthquake Probabilities Estimated by the Geometrical Mean Likelihood of the Brownian Passage Time Distribution Model for Recurrent Paleoearthquakes

Temporal distribution characteristics of earthquakes in Taiwan, China

The characteristics of seismic temporal distribution represent an important basis for earthquake prediction and seismic hazard analysis. In this paper, based on the seismic catalogs in Taiwan, and using Poisson (exponential distribution), Gamma, Lognormal, Weibull, and Brownian passage time distributions as target models, we adopt the maximum likelihood method for estimating model parameters. The optimal model for describing the temporal distribution of earthquakes in Taiwan is determined according to the Akaike Information Criterion (AIC), Bayesian Information Criterion (BIC), K-S test, Chi-square test, and coefficient of determination R2 results. The results show that for moderate-strong earthquakes events (MW < 7.0), the Gamma distribution model can well describe the temporal distribution characteristics of earthquakes, while large earthquakes (MW ≥ 7.0) can be described entirely by exponential distribution. In addition, the temporal correlation between earthquakes is also examined through diffusion entropy analysis. The results show that seismic activity features temporal correlation, and earthquakes with relatively small magnitude (MW < 7) are affected by larger events (MW ≥ 7.0), thus suggesting long-term memory in time. In this study, the probability of the occurrence of a major earthquake in Taiwan is also calculated. The results show that the probability of an MW ≥ 7.0 earthquake in Taiwan in the next 10 years reaches 91.3%. The results may be used to inform the selection of seismic time distribution models and the calculation of seismic activity parameters in earthquake prediction and seismic hazard calculation, and hold scientific significance for understanding the mechanism of earthquake genesis.

Forecasting tectonic tremor activity using a renewal process model

In many tectonically active regions of the world, a variety of slow deformation phenomena have been discovered and collectively termed slow earthquakes. Tectonic tremor is the high-frequency component of slow earthquakes and can be analyzed to monitor the overall slow deformation process, both spatially and temporally. Although tectonic tremor activity is complex, it does possess some characteristic patterns, such as spatial segmentation, a quasi-periodic recurrence, migration, and tidal modulation. These features are helpful for forecasting future activity if they are properly modeled in a quantitative manner. Here, we propose a stochastic renewal process to standardize and forecast tectonic tremor activity in the Nankai subduction zone, southwest Japan, using a 12.5-year tremor catalog that is divided into a 10-year estimation period and 2.5-year forecasting period. We group the tremor events into small rectangular 10-km regions and observe that the distribution of inter-event times is nearly bimodal, with the short and long inter-event times representing the characteristic times of nearby tremor interactions and long-term stress accumulation processes, respectively. Therefore, as the probabilistic distribution for the renewal process, we adopt a mixture distribution of log-normal and Brownian passage time distributions for the short and long inter-event times, respectively. The model parameters are successfully estimated for 72% of the entire tremor zone using a maximum likelihood method. This standard model can be used to extract anomalous tremor activity, such as that associated with long-term slow-slip events. We derive a scaling relationship between two characteristic times, the relative plate motion, episodicity of tremor activity, and tremor duration by characterizing the spatial differences in tremor activity. We confirm that the model can forecast the occurrence of the next tremor event at a given reference time for a certain prediction interval. This study can serve as a first step for implementing more complex models to improve the space–time forecasting of slow earthquakes.

A Statistical Model for Earthquake Recurrence Based on the Assimilation of Paleoseismicity, Historic Seismicity, and Instrumental Seismicity

AbstractPaleoearthquakes and historic earthquakes are the most important source of information for the estimation of long‐term earthquake recurrence intervals in fault zones, because corresponding sequences cover more than one seismic cycle. However, these events are often rare, dating uncertainties are enormous, and missing or misinterpreted events lead to additional problems. In the present study, I assume that the time to the next major earthquake depends on the rate of small and intermediate events between the large ones in terms of a “clock change” model. Mathematically, this leads to a Brownian passage time distribution for recurrence intervals. I take advantage of an earlier finding that under certain assumptions the aperiodicity of this distribution can be related to the Gutenberg‐Richter b value, which can be estimated easily from instrumental seismicity in the region under consideration. In this way, both parameters of the Brownian passage time distribution can be attributed with accessible seismological quantities. This allows to reduce the uncertainties in the estimation of the mean recurrence interval, especially for short paleoearthquake sequences and high dating errors. Using a Bayesian framework for parameter estimation results in a statistical model for earthquake recurrence intervals that assimilates in a simple way paleoearthquake sequences and instrumental data. I present illustrative case studies from Southern California and compare the method with the commonly used approach of exponentially distributed recurrence times based on a stationary Poisson process.

A Synthetic Seismicity Model for the Northwestern Portion of the Xianshuihe Fault, Southwestern China: Simulation Using the Monte Carlo Method, Based on Historical Earthquake Data

Abstract In order to estimate distributions of recurrence intervals and segment interactions across the four fault segments along the northwestern portion of the Xianshuihe fault (NPXF) in southwestern China, I generate a large number of synthetic earthquake catalogs by sampling recurrence intervals and segment interactions for the four segments within the preferred ranges based on the seismogenic mechanism of fault rupture and interaction. The ranges are derived from previous studies. Those catalogs that are closest to the real historical earthquake catalog consisting of seven strong events of M ≥6.8 during 1792–1981 along the NPXF, are then selected to determine the possible distributions of the intervals and interactions. Finally, I create a synthetic earthquake catalog by using Monte Carlo sampling from the determined distributions of the intervals and interactions. The ∼40,000‐year simulated seismic activity shows good agreement with the observed data with respect to the following four statistical features. (1) Most events are single‐segment ruptures, with few rupturing two or three segments (∼8% of simulated earthquakes) and none breaking all four segments simultaneously. (2) The coefficient of variation C v of the recurrence interval tends to increase with the complexity of a segment or fault. (3) The Brownian passage time distribution represents the best fit for each of the four individual segments, whereas the Weibull distribution matches the entire NPXF best. (4) The next strong event is most likely to occur in the eastern part of the southernmost segment of the NPXF.

Reply to "Comment on 'Test of Seismic Hazard Map from 500 Years of Recorded Intensity Data in Japan' by Masatoshi Miyazawa and Jim Mori" by Celine Beauval, Pierre-Yves Bard, and John Douglas

Miyazawa and Mori (2009) focused on obtaining the recorded maximum seismic intensity in Japan over the last 500 years and made comparisons to the corresponding Japanese probabilistic seismic hazard map (PSHM) by the Headquarters for Earthquake Research Promotion (2005). The comments by Beauval et al. (2010) state that this comparison is not correct. In this reply, we try to further justify the validity of the comparison by clarifying that (1) many stations over Japan are used to test the PSHMs on average and (2) the PSHM assumes both a time-dependent model and a time-independent model for the probability of earthquake occurrences. The present PSHMs in Japan use both time-dependent and time-independent models. It is well-known that large earthquakes with magnitude larger than 7 repeatedly occur in subduction zones at intervals on the order of 100 years or less and with similar mechanisms. Also the recurrences of onshore earthquakes on major active faults have been investigated geologically, though their uncertainties are much larger compared to the subduction zone earthquakes. The recurrences of these earthquakes are modeled in the PSHM as a Brownian passage time distribution (Fujiwara et al. , 2006), and Miyazawa and Mori (2009, p. 3124) say that the “present PSHM is a time-dependent hazard map.” For some onshore earthquakes with unclear return periods and earthquakes with nonspecified source faults, the occurrence probability is modeled as a Poisson process, which is time independent. For details, see National Research Institute for Earth Science and Disaster Prevention (2009) and the Japan Seismic Hazard Information Station Web site (National Research Institute for Earth Science and Disaster Prevention [NIED], 2010). Beauval et al. (2010) point out …

Recurrent Large Earthquakes in a Fault Region: What Can Be Inferred from Small and Intermediate Events?

We present a renewal model for the recurrence of large earthquakes in a fault zone consisting of a major fault and surrounding smaller faults with Gutenberg–Richter-type seismicity represented by seismic moment release drawn from a truncated power-law distribution. The recurrence times of characteristic earthquakes for the major fault are explored. It is continuously loaded (plate motion) and undergoes positive and negative fluctuations due to adjacent smaller faults, with a large number N eq of such changes between two major earthquakes. Because the distribution has a finite variance, in the limit N eq→∞ the central limit theorem implies that the recurrence times follow a Brownian passage-time (BPT) distribution. This allows us to calculate individual recurrence-time distributions for specific fault zones without tuning free parameters: the mean recurrence time can be estimated from geological or paleoseismic data, and the standard deviation is determined from the frequency-size distribution, namely, the Richter b -value, of an earthquake catalog. The approach is demonstrated for the Parkfield segment of the San Andreas fault in California as well as for a long simulation of a numerical fault model. Assuming power-law distributed earthquake magnitudes up to the size of the recurrent Parkfield event ( M 6), we find a coefficient of variation that is higher than the value obtained by a direct fit of the BPT distribution to seven large earthquakes. We argue that the BPT distribution is a reasonable choice for seismic hazard assessment because it governs not only Brownian motion with drift but also models with power-law statistics for the recurrence of large earthquakes in an asymptotic limit. Finally, we find that the condition b =1 separates two regimes: in the first ( b 1) it is greater.

From regional seismic hazard to “scenario earthquakes” for seismic microzoning: A new methodological tool for the Celano Project

We present the results of a probabilistic seismic hazard assessment and disaggregation analysis aimed to understand the dominant magnitudes and source-to-site distances of earthquakes that control the hazard at the Celano site in the Abruzzo region of central Italy. Firstly, we calculated a peak ground acceleration map for the central Apennines area, by using a model of seismogenic sources defined on geological-structural basis. The source model definition and the probabilistic seismic hazard evaluation at the regional scale (central Apennines) were obtained using three different seismicity models (Gutenberg–Richter model; characteristic earthquake model; hybrid model), consistent with the available seismological information. Moreover, a simplified time-dependent hypothesis has been introduced, computing the conditional probability of earthquakes occurrence by Brownian passage time distributions. Subsequently, we carried out the disaggregation analysis, with a modified version of the SEISRISK III code, in order to separate the contribution of each source to the total hazard. The results show the percentage contribution to the Celano hazard of the various seismogenic sources, for different expected peak ground acceleration classes. The analysis was differentiated for close (distance from Celano <20 km) and distant (distance from Celano >20 km) seismogenic sources. We propose three different “scenario earthquakes”, useful for the site condition studies and for the seismic microzoning study: (1) large ( M=6.6) local (Celano-epicentre distance ∼16 km) earthquake, with mean recurrence time of ∼590 years; (2) moderate ( M=5.5) local (Celano-epicentre distance ∼7.5 km) earthquake, with mean recurrence time of ∼500 years; and (3) large ( M=6.6) distant (Celano-epicentre distance ∼24 km) earthquake, with mean recurrence time of ∼980 years. The probabilistic and time-dependent approach to the definition of the “scenario earthquakes” changes clearly the results in comparison to traditional deterministic analysis, with effects in terms of engineering design and seismic risk reduction.

A synthetic seismicity model for the Xianshuihe fault, southwestern China: simulation using a rate- and state-dependent friction law

SUMMARY We performed a numerical simulation of seismic activity along the Xianshuihe fault, a highly active strike-slip fault approximately 350 km long, located along the eastern margin area of the Tibetan plateau, southwestern China. Historical earthquake data over the last 300 yr indicate repeated periods of seismic activity, and migration of large earthquakes along the fault during active seismic periods. To understand the characteristics of historical seismicity and to obtain insight into seismic potential, we performed a numerical simulation of slip behaviour along the fault using a 2-D elastic plate model. The friction on the model fault obeys a laboratory-derived rate- and state-dependent friction law, while the long-term slip rates along the fault are assumed to be consistent with geologically and geodetically estimated slip rates. To simulate segmented rupture behaviour we introduced non-uniformity into friction parameters along the fault. The simulation results indicate that seismic rupture is arrested in regions with frictional properties that are highly velocity-strengthening or large values of characteristic slip distance over which frictional stress evolves. In the regions where seismic rupture is arrested, post-seismic sliding occurs, causing time-dependent stress transfer along the fault. Post-seismic slip histories in highly velocity-strengthening regions are well approximated by logarithmic time functions, while those in regions with large characteristic slip distances increase more rapidly in the initial stages of post-seismic sliding. This difference in post-seismic sliding produces the difference in time-dependent stress transfer and, in consequence, the statistical characteristics of earthquake recurrence. Fault interactions near a fault branch point produce complexity in the simulated earthquake sequence. Simulated rupture commonly triggers additional rupture in a neighbouring segment with a time delay of several years. The simulated seismic activity shows good agreement with observational data with respect to the following statistical features: (1) successive seismically active periods are separated by periods of relative quiescence of 100‐ 200 yr duration (2) and the duration of active period is a few tens of years. Strictly speaking, neither the time- nor slip-predictable models are able to describe the simulated local slip history for each fault segment. The ratio of the standard deviation to the average value of the recurrence intervals of simulated earthquakes ranges from 0.05 to 0.22, depending on the fault segment. The cumulative distribution function of the recurrence intervals of simulated earthquakes at each segment approximately obeys a Brownian passage time distribution or a lognormal distribution, which is commonly used in the statistical evaluation of earthquake occurrence.

Simulation-Based Distributions of Earthquake Recurrence Times on the San Andreas Fault System

Earthquakes on a specified fault (or fault segment) with magnitudes greater than a specified value have a statistical distribution of recurrence times. The mean recurrence time can be related to the rate of strain accumulation and the strength of the fault. Very few faults have a recorded history of earthquakes that define a distribution well. For hazard assessment, in general, a statistical distribution of recurrence times is assumed along with parameter values. Assumed distributions include the Weibull (stretched exponential) distribution, the lognormal distribution, and the Brownian passage-time (inverse Gaussian) distribution. The distribution of earthquake waiting times is the conditional probability that an earthquake will occur at a time in the future if it has not occurred for a specified time in the past. The distribution of waiting times is very sensitive to the distribution of recurrence times. An exponential distribution of recurrence times is Poissonian, so there is no memory of the last event. The distribution of recurrence times must be thinner than the exponential if the mean waiting time is to decrease as the time since the last earthquake increases. Neither the lognormal or the Brownian passage time distribution satisfies this requirement. We use the “Virtual California” model for earthquake occurrence on the San Andreas fault system to produce a synthetic distribution of earthquake recurrence times on various faults in the San Andreas system. We find that the synthetic data are well represented by Weibull distributions. We also show that the Weibull distribution follows from both damage mechanics and statistical physics.

Layered Seismogenic Source Model and Probabilistic Seismic-Hazard Analyses in Central Italy

We defined a seismogenic model for central Italy based on three layers of sources and computed the relative seismic-hazard maps. One layer is constituted by individual structures liable to generate major earthquakes (M 5.5). We defined them as seismogenic boxes by using geological information in terms of plan projec- tion of active faults; the seismicity rates associated with an individual source are based on the geometry and kinematics of the fault; the recurrence model is controlled by the earthquake-source association, and, when possible, we defined the occurrence time of the last major event, using it in a time-dependent approach. Another layer is given by the instrumental seismicity analysis of the past two decades, which allows us to evaluate the background seismicity (M �� 5.5); using a sliding-window selec- tion of events, we defined a model of regular adjacent cells of variable a and b values of the Gutenberg-Richter relation. The last layer utilizes all the instrumental earth- quakes and the historical events not correlated to known structures (4.5 � M �� 6), by separating them into seismotectonic provinces shaped on a geological-structural basis. The seismic-hazard computations first use this layered model in a traditional probabilistic scheme. The results indicate a narrow belt of peak ground acceleration (PGA) higher than 0.30g (with standard deviation in attenuation functions) in the axial part of the Apennine chain, with a maximum spot of PGA � 0.40g southeast of the area damaged by the 1997-1998 Umbria-Marche sequence (PGA expected not to be exceeded in 50 years at 90% probability level). The background seismicity gives a nonnegligible contribution to the hazard, at least for first damage levels. Then, a simplified time-dependent hypothesis has been introduced for the individual sources alone, computing the conditional probability of occurrence of characteristic earth- quakes for each source by Brownian passage time distributions. Adopting equivalent fictitious seismicity rates, we obtained maps referring to the next 50 years by using traditional codes. These results show that the contribution of the recently active sources vanishes, and the most hazardous sites are now located south of L'Aquila and in the Sulmona area. We consider that the methodology and results obtained are useful for seismic risk reduction strategies.

A Brownian Model for Recurrent Earthquakes

We construct a probability model for rupture times on a recurrent earthquake source. Adding Brownian perturbations to steady tectonic loading produces a stochastic load-state process. Rupture is assumed to occur when this process reaches a critical-failure threshold. An earthquake relaxes the load state to a characteristic ground level and begins a new failure cycle. The load-state process is a Brownian relaxation oscillator. Intervals between events have a Brownian passage-time distribution that may serve as a temporal model for time-dependent, long-term seismic forecasting. This distribution has the following noteworthy properties: (1) the probability of immediate rerupture is zero; (2) the hazard rate increases steadily from zero at t = 0 to a finite maximum near the mean recurrence time and then decreases asymptotically to a quasi-stationary level, in which the conditional probability of an event becomes time independent; and (3) the quasi-stationary failure rate is greater than, equal to, or less than the mean failure rate because the coefficient of variation is less than, equal to, or greater than \batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \(1{/}\sqrt{2}{\approx}0.707\) \end{document}. In addition, the model provides expressions for the hazard rate and probability of rupture on faults for which only a bound can be placed on the time of the last rupture. The Brownian relaxation oscillator provides a connection between observable event times and a formal state variable that reflects the macromechanics of stress and strain accumulation. Analysis of this process reveals that the quasi-stationary distance to failure has a gamma distribution, and residual life has a related exponential distribution. It also enables calculation of “interaction” effects due to external perturbations to the state, such as stress-transfer effects from earthquakes outside the target source. The influence of interaction effects on recurrence times is transient and strongly dependent on when in the loading cycle step perturbations occur. Transient effects may be much stronger than would be predicted by the “clock change” method and characteristically decay inversely with elapsed time after the perturbation.