Bath's Law Research Articles

A tentative model for the explanation of Båth law using the order parameter of seismicity in natural time

Using the order parameter of seismicity defined in natural time, we suggest a simple model for the explanation of Bath law, according to which a mainshock differs in magnitude from its largest aftershock by approximately 1.2 regardless of the mainshock magnitude. In addition, the validity of Bath law is studied in the Global Centroid Moment Tensor catalogue by using two different aftershock definitions. It is found that the mean of this difference, when considering all the pairs mainshock-largest aftershock, does not markedly differ from 1.2 and the corresponding distributions do not depend on the mainshock’s magnitude threshold in a statistically significant manner. Finally, the analysis of the cumulative distribution functions provides evidence in favour of the proposed model.

Neural networks to predict earthquakes in Chile

A new earthquake prediction system is presented in this work. This method, based on the application of artificial neural networks, has been used to predict earthquakes in Chile, one of the countries with larger seismic activity. The input values are related to the b-value, the Bath's law, and the Omori–Utsu's law, parameters that are strongly correlated with seismicity, as shown in solid previous works. Two kind of prediction are provided in this study: The probability that an earthquake of magnitude larger than a threshold value happens, and the probability that an earthquake of a limited magnitude interval might occur, both during the next five days in the areas analyzed. For the four Chile's seismic regions examined, with epicenters placed on meshes with dimensions varying from 0.5°×0.5° to 1°×1°, a prototype of neuronal network is presented. The prototypes predict an earthquake every time the probability of an earthquake of magnitude larger than a threshold is sufficiently high. The threshold values have been adjusted with the aim of obtaining as few false positives as possible. The accuracy of the method has been assessed in retrospective experiments by means of statistical tests and compared with well-known machine learning classifiers. The high success rate achieved supports the suitability of applying soft computing in this field and poses new challenges to be addressed.

Damage spectra for the mainshock–aftershock sequence-type ground motions

Strong aftershocks have the potential to increase the damage state of the structures due to the damage accumulation. This paper investigates the damage spectra for the mainshock–aftershock sequence-type ground motions with Park–Ang damage index. A method of simulating the mainshock–aftershock sequence-type ground motions is proposed based on the modified form of Bath's law and NGA ground motion prediction equation. The damage spectra are computed using the recorded and simulated sequence-type ground motions, and the effects of period of vibration, strength reduction factor, site condition, seismic sequence, damping ratio and post-yield stiffness on damage spectra are studied statistically. The results indicate that the effect of aftershock on structural damage is significant and recorded sequence-type ground motions may underestimate the damage of long-period structures due to the incompleteness of dataset. A simplified equation is proposed to facilitate the application of damage spectra in the seismic practice for mainshock–aftershock sequence-type ground motions.

Market dynamics immediately before and after financial shocks: Quantifying the Omori, productivity, and Bath laws

We study the cascading dynamics immediately before and immediately after 219 market shocks. We define the time of a market shock T{c} to be the time for which the market volatility V(T{c}) has a peak that exceeds a predetermined threshold. The cascade of high volatility "aftershocks" triggered by the "main shock" is quantitatively similar to earthquakes and solar flares, which have been described by three empirical laws-the Omori law, the productivity law, and the Bath law. We analyze the most traded 531 stocks in U.S. markets during the 2 yr period of 2001-2002 at the 1 min time resolution. We find quantitative relations between the main shock magnitude M≡log{10} V(T{c}) and the parameters quantifying the decay of volatility aftershocks as well as the volatility preshocks. We also find that stocks with larger trading activity react more strongly and more quickly to market shocks than stocks with smaller trading activity. Our findings characterize the typical volatility response conditional on M , both at the market and the individual stock scale. We argue that there is potential utility in these three statistical quantitative relations with applications in option pricing and volatility trading.

A Modified Form of Bath's Law

Bath's law states that the differences in magnitudes between mainshocks and their largest aftershocks are approximately constant, independent of the magnitudes of mainshocks. In our modified form of Bath's law we introduce the notion of the inferred “largest” aftershock from an extrapolation of the Gutenberg-Richter frequency-magnitude statistics of the aftershock sequence of a given mainshock. To illustrate the application of this modified law we consider 10 large earthquakes that occurred in California between 1987 and 2003 with magnitudes equal to or greater than m ms ≥ 5.5. The mean difference in magnitudes between these mainshocks and their largest detected aftershocks is 1.16 with a standard deviation σ Δ m = 0.46 (Bath's law). Our estimated mean difference in magnitudes between the mainshocks and the inferred “largest” aftershocks is 1.11 with σ Δ m * = 0.29. The scaling associated with the modified Bath's law implies that the stress transfer responsible for the occurrence of aftershocks is a self-similar process. We also estimate the partitioning of energy during a mainshock-aftershock sequence and find that about 96% of the energy dissipated in a sequence is associated with the mainshock and the rest is due to aftershocks. We suggest that the observed partitioning of energy could play a crucial role in explaining the physical origin of Bath's law. Manuscript received 6 August 2003.

Spatial distribution of the difference between the magnitudes of the main shock and the largest aftershock in the circum-Pacific belt

AbstractThe distribution of the magnitude difference D1 between the main shock and the largest aftershock for large circum-Pacific earthquakes is observed to be peaked at 1.2 magnitude units, in accord with Bath's law, but also at 1.8 magnitude units. The peak at 1.8 has been not noted before and is shown here with the T-test to be statistically valid. The observed spatial distribution of the sequences is the basis to suggest that, along circum-Pacific convergent plate boundaries, values of D1, tend to be larger for earthquake sequences within back-arc areas as compared to sequences closer to the plate interface, although admittedly a good number of exceptions to this generalization are also observed to exist.