Alvaro Corral

Long-Term Clustering, Scaling, and Universality in the Temporal Occurrence of Earthquakes

Analyzing diverse seismic catalogs, we have determined that the probability densities of the earthquake recurrence times for different spatial areas and magnitude ranges can be described by a unique universal distribution if the time is rescaled with the rate of seismic occurrence, which therefore fully governs seismicity. The shape of the distribution shows the existence of clustering beyond the duration of aftershock bursts, and scaling reveals the self-similarity of the clustering structure in the space-time-magnitude domain. This holds from worldwide to local scales, for quite different tectonic environments and for all the magnitude ranges considered.

Local distributions and rate fluctuations in a unified scaling law for earthquakes.

A recently proposed unified scaling law for interoccurrence times of earthquakes is analyzed, both theoretically and with data from Southern California. We decompose the corresponding probability density into local-instantaneous distributions, which scale with the rate of earthquake occurrence. The fluctuations of the rate, characterizing the nonstationarity of the process, show a double power-law distribution and are fundamental to determine the overall behavior, described by a double power law as well.

Statistical similarity between the compression of a porous material and earthquakes

It has long been stated that there are profound analogies between fracture experiments and earthquakes; however, few works attempt a complete characterization of the parallels between these so separate phenomena. We study the acoustic emission events produced during the compression of Vycor (SiO(2)). The Gutenberg-Richter law, the modified Omoris law, and the law of aftershock productivity hold for a minimum of 5 decades, are independent of the compression rate, and keep stationary for all the duration of the experiments. The waiting-time distribution fulfills a unified scaling law with a power-law exponent close to 2.45 for long times, which is explained in terms of the temporal variations of the activity rate.

Universal Local Versus Unified Global Scaling Laws in the Statistics of Seismicity

The unified scaling law for earthquakes, proposed by Bak, Christensen, Danon and Scanlon, is shown to hold worldwide, as well as for areas as diverse as Japan, New Zealand, Spain or New Madrid. The scaling functions that account for the rescaled recurrence-time probability densities show a power-law behavior for long times, with a universal exponent about (minus) 2.2. Another decreasing power law governs short times, but with an exponent that may change from one area to another. This is in contrast with a local, time-homogenized version of Bak et al.s procedure, which seems to present a universal scaling behavior.

Universality of rain event size distributions

We compare rain event size distributions derived from measurements in climatically different regions, which we find to be well approximated by power laws of similar exponents over broad ranges. Differences can be seen in the large-scale cutoffs of the distributions. Event duration distributions suggest that the scale-free aspects are related to the absence of characteristic scales in the meteorological mesoscale.

Universal Earthquake-Occurrence Jumps, Correlations with Time, and Anomalous Diffusion

Spatiotemporal properties of seismicity are investigated for a worldwide (WW) catalog and for southern California in the stationary case (SC), showing a nearly universal scaling behavior. Distributions of distances between consecutive earthquakes (jumps) are magnitude independent and show two power-law regimes, separated by jump values about 200 (WW) and 15 km (SC). Distributions of waiting times conditioned to the value of jumps show that both variables are correlated, in general, but turn out to be independent when only short or long jumps are considered. Finally, diffusion profiles are found to be independent on the magnitude, contrary to what the waiting-time distributions suggest.

Fitting and Goodness-of-Fit Test of Non-Truncated and Truncated Power-Law Distributions

Recently, Clauset, Shalizi, and Newman have proposed a systematic method to find over which range (if any) a certain distribution behaves as a power law. However, their method has been found to fail, in the sense that true (simulated) power-law tails are not recognized as such in some instances, and then the power-law hypothesis is rejected. Moreover, the method does not work well when extended to power-law distributions with an upper truncation. We explain in detail a similar but alternative procedure, valid for truncated as well as for non-truncated power-law distributions, based in maximum likelihood estimation, the Kolmogorov-Smirnov goodness-of-fit test, and Monte Carlo simulations. An overview of the main concepts as well as a recipe for their practical implementation is provided. The performance of our method is put to test on several empirical data which were previously analyzed with less systematic approaches. We find the functioning of the method very satisfactory.

Renormalization-group transformations and correlations of seismicity

The effect of transformations analogous to those of the real-space renormalization group are analyzed for the temporal occurrence of earthquakes. The distribution of recurrence times turns out to be invariant under such transformations, for which the role of the correlations between the magnitudes and the recurrence times are fundamental. A general form for the distribution is derived imposing only the self-similarity of the process, which also yields a scaling relation between the Gutenberg-Richter b-value, the exponent characterizing the correlations, and the recurrence-time exponent. This approach puts the study of the structure of seismicity in the context of critical phenomena.

Time-decreasing hazard and increasing time until the next earthquake.

The existence of a slowly always decreasing probability density for the recurrence times of earthquakes in the stationary case implies that the occurrence of an event at a given instant becomes more unlikely as time since the previous event increases. Consequently, the expected waiting time to the next earthquake increases with the elapsed time, that is, the event moves away fast to the future. We have found direct empirical evidence of this counterintuitive behavior in two worldwide catalogs as well as in diverse regional catalogs. Universal scaling functions describe the phenomenon well.

Statistical Features of Earthquake Temporal Occurrence

A review of the statistical properties of earthquakes is provided, centered mainly in the work of the author (apologies for that). We explain the scaling law for the recurrence-time distributions, its universal character for stationary seismicity and for Omori sequences, the counterintuitive phenomenon of decreasing hazard with time and the increasing of the expected residual recurrence time, the relation of the scaling law with a renormalization-group transformation, and the correlations between recurrence times and magnitudes. Finally, the connections with Bak et al.s unified scaling law are analyzed.