Multifractality of Pseudo-Velocities and Seismic Quiescence Associated with the Tehuantepec M8.2 EQ

By using earthquake catalogs, previous studies have reported evidence that some changes in the spatial and temporal organization of earthquake activity are observed before and after of a main shock. These previous studies have used different approaches for detecting clustering behavior and distance-events density in order to point out the asymmetric behavior of foreshocks and aftershocks. Here, we present a statistical analysis of the seismic activity related to the M w = 8.2 earthquake that occurred on 7 September 2017 in Mexico. First, we calculated the inter-event time and distance between successive events for the period 1 January 1998 until 20 October 2017 in a circular region centered at the epicenter of the M w = 8.2 EQ. Next, we introduced the concept of pseudo-velocity as the ratio between the inter-event distance and inter-event time. A sliding window is considered to estimate some statistical features of the pseudo-velocity sequence before the main shock. Specifically, we applied the multifractal method to detect changes in the spectrum of singularities for the period before the main event on 7 September. Our results point out that the multifractality associated with the pseudo-velocities exhibits noticeable changes in the characteristics of the spectra (more narrower) for approximately three years, from 2013 until 2016, which is preceded and followed by periods with wider spectra. On the other hand, we present an analysis of patterns of seismic quiescence before the M w = 8.2 earthquake based on the Schreider algorithm over a period of 27 years. We report the existence of an important period of seismic quietude, for six to seven years, from 2008 to 2015 approximately, known as the alpha stage, and a beta stage of resumption of seismic activity, with a duration of approximately three years until the occurrence of the great earthquake of magnitude M w = 8.2 . Our results are in general concordance with previous results reported for statistics based on magnitude temporal sequences.

BOUNDS ON THE SUPPORT OF THE MULTIFRACTAL SPECTRUM OF STOCHASTIC PROCESSES

The multifractal analysis of stochastic processes deals with the fine scale properties of the sample paths and seeks for some global scaling property that would enable extracting the so-called spectrum of singularities. In this paper, we establish bounds on the support of the spectrum of singularities. To do this, we prove a theorem that complements the famous Kolmogorov’s continuity criterion. The nature of these bounds helps us to identify the quantities truly responsible for the support of the spectrum. We then make several conclusions from this. First, specifying global scaling in terms of moments is incomplete due to possible infinite moments, both of positive and negative orders. The divergence of negative order moments does not affect the spectrum in general. On the other hand, infinite positive order moments make the spectrum of self-similar processes nontrivial. In particular, we show that the self-similar stationary increments process with the nontrivial spectrum must be heavy-tailed. This shows that for determining the spectrum it is crucial to capture the divergence of moments. We show that the partition function is capable of doing this and also propose a robust variant of this method for negative order moments.

MULTIFRACTAL ANALYSIS OF SOME WEIGHTED QUASI-SELF-SIMILAR FUNCTIONS

In this paper, a multifractal analysis of some non-self-similar functions based on the superposition of finite number of weighted quasi-self-similar ones ∑iωiFi is developed. In general, such superpositions do not yield neither a self-similar nor a quasi-self-similar outcome. Furthermore, there are two main problems that appear. Firstly, a phenomenon of regularity compensation may exist. Secondly, the computation of the spectrum of singularities and therefore the validity of the multifractal formalism based on the possibility of constructing Gibbs measures fail. In this paper, we propose to study such problems by conducting a multifractal analysis of such combinations and to check the validity of the multifractal formalism in the case where there is no compensation of regularity. Furthermore, we compute the box dimension of the associated graphs and provide some examples. The paper in its full subject re-considers the results of Ref. 3 in the quasi-self-similar case.

Random multifractals: negative dimensions and the resulting limitations of the thermodynamic formalism

The story described in this paper has started with the ‘death or survival’ criterion, which the author published in 1972-1974 and had obtained in 1968 while investigating Kolmogorov’s hypothesis that the turbulent dissipation ϵ (d x ) in a box is log-normally distributed. Using this criterion, the present paper discusses the concrete significance of negative fractal dimensions. They arise in those random multifractal measures, for which the Cramèr function f ( α ) (the ‘spectrum of singularities’) satisfies f ( α ) < 0 for certain values of α . It is shown that in that case the strict ‘thermodynamical formalism’ solely involves the form of f ( α ) in the range where f ( α ) > 0, and concerns three aspects of such measures: (a) the fine-grained multifractal properties, which are non-random and the same for (almost) all realizations; (b) the properties obtained by using the ‘partition function’ formalism ; and (c) the ‘typical’ coarse-grained multifractal properties. However, the f ( α )s in the range where f ( α ) > 0 say nothing about the variability of coarse-grained properties between samples. A description of these fluctuations, hence a fuller multifractal description of the measure, is shown to be provided by the values of f( α ) in the range where f ( α ) < 0. We prefer to reserve the term ‘thermodynamic’ for the fine-grained and partition-functional properties, and to say that the coarse-grained properties go beyond the thermodynamics, i.e. are not macroscopic but ‘mesoscopic'.