# Earthquakes in Barcelona: How big are the biggest earthquakes? It has been known since the 1950s that the number of earthquakes of large magnitude follows the law of Gutenberg and Richter: for a region (any) and a time interval (sufficient) the probability distribution of the magnitude of earthquakes will be exponential (decreasing) , with many small earthquakes and (fortunately) few large ones.

The simplicity of the exponential distribution  (which appears in elementary courses of probability)  hides a more difficult approach than it seems if one cares about the physical sense. The magnitude, m, is a logarithmic measure of radiated energy, given by E = A · 10 ^ (3m / 2), which implies that the size of an earthquake, in terms of energy, follows a power law, and it is this that presents exciting mathematical properties, since in some cases, these are distributions with divergent moments (they are infinite). In the case of earthquakes, adjustments of the Gutenberg-Richter law imply that all moments have to diverge, including the expected value, which does not violate any mathematical requirement but obviously it violates physical ones: the Earth contains a finite amount of energy. Therefore, Gutenberg-Ritcher’s law is physically unacceptable, or at least can not be happily extrapolated to infinity (note that this joy can not be given, in other contexts, we use the Gaussian distribution). So, what law governs the occurrence of the most extreme earthquakes? The problem is not easy to solve, due to their scarcity. Some researchers have even suggested that more than 200 years will be required  to gather the required data and have enough statistics. Isabel Serra and Álvaro Corral, from the Center for Mathematical Research in Barcelona, led by the Obra Social LaCaixa, have not had patience and have addressed the issue with “professional” statistical instruments (doi: 10.1038 / srep40045).

The authors compare several generations of the Gutenberg-Richter law, so that for moderate tremors the classical law remains valid, but for extreme events the density of the law  decreases more rapidly. The estimation by the maximum likelihood allows them to adjust the parameters of each statistical model reliably and the likelihood ratio test provides the comparison of the different models, using Monte Carlo simulations (the peculiarities of the power distributions make that the theoretical results of the asymptotic distribution of the statistical test can not be applied). Although several authors had indicated that after the great Sumatra-Andaman earthquake of 2004 the resulting energy distribution was indistinguishable from a power-law, Serra and Corral demonstrate that the gamma distribution (truncated and generalized to a parameter of negative form) improve the power-law. This result will allow better matching of seismic risk.

Isabel Serra & Alvaro Corral