[Paper Review] Seismic Electric Signals and 1/f "noise" in natural time

P. Varotsos, N. V. Sarlis+1 more|ArXiv.org|Nov 23, 2007

Complex Systems and Time Series Analysis6 references10 citations

TL;DR

This paper proposes a stochastic model in natural time that reproduces the $1/f^a$ noise with $a \approx 1$, mimicking the power-law behavior observed in Seismic Electric Signals (SES). Despite similarities in spectral properties, the model predicts increased entropy under time reversal ($S_{-} > S$), contradicting SES data where $S$ can be either larger or smaller than $S_{-}$, suggesting SES exhibit critical dynamics beyond the model's scope.

ABSTRACT

By making use of the concept of natural time, a simple model is proposed which exhibits the $1/f^a$ behavior with $a$ close to unity. The properties of the model are compared to those of the Seismic Electric Signals (SES) activities that have been found to obey the ubiquitous $1/f^a$ behavior with $a \approx 1$. This comparison, which is made by using the most recent SES data, reveals certain similarities, but the following important difference is found: The model suggests that the entropy $S_-$ under time reversal becomes larger compared to the entropy $S$ in forward time, thus disagreeing with the experimental SES results which show that $S$ may be either smaller or larger than $S_-$. This might be due to the fact that SES activities exhibit {\em critical} dynamics, while the model cannot capture all the characteristics of such dynamics.

Motivation & Objective

• To model $1/f^a$ noise with $a \approx 1$ using natural time formalism.
• To compare the model's statistical properties with real Seismic Electric Signal (SES) data from three magnitude 6.0+ earthquakes.
• To investigate whether the model captures the critical dynamics observed in SES activities.
• To identify discrepancies in entropy behavior under time reversal between the model and experimental SES data.

Proposed method

• The model uses a stochastic process where events are indexed in natural time, with inter-event times drawn from an exponential distribution.
• The variable $\epsilon_n$ represents the number of renewals up to the $n$-th event, and its distribution is derived via combinatorial analysis of random permutations.
• Fourier power spectra and Detrended Fluctuation Analysis (DFA) are used to analyze the $1/f^a$ scaling of the time series.
• Entropy in forward ($S$) and reversed ($S_{-}$) time is computed to assess time irreversibility.
• The model is compared to a Voss-type $1/f$ generator using $k$-sided dice, highlighting differences in higher-order moments like skewness and kurtosis.
• Statistical properties such as mean, variance, skewness, and kurtosis of $\epsilon_n$ are analytically derived and compared to those of the Voss model.

Experimental results

Research questions

• RQ1 Can a simple stochastic model in natural time reproduce the $1/f^a$ noise with $a \approx 1$ observed in Seismic Electric Signals?
• RQ2 How do the entropy dynamics $S$ and $S_{-}$ in the model compare to those in real SES data?
• RQ3 What statistical differences exist between the proposed model and the Voss model in generating $1/f$ noise?
• RQ4 Why does the model fail to reproduce the time-reversal asymmetry observed in SES, despite matching spectral behavior?
• RQ5 To what extent do SES activities reflect critical dynamics not captured by the model?

Key findings

• The model successfully generates $1/f^a$ noise with $a \approx 1$, as confirmed by Fourier power spectra and DFA with $\alpha_{DFA} \approx 1$.
• The distribution of $\epsilon_n$ converges rapidly to the Cornish-Fisher approximation, enabling analysis for large $n$.
• The model predicts $S_{-} > S$, indicating increased entropy under time reversal, which contradicts experimental SES data where $S$ can be either larger or smaller than $S_{-}$.
• The Voss model produces symmetric, skewnessless distributions, while the $\epsilon_n$ distribution is skewed, indicating a fundamental statistical difference.
• The kurtosis of $\epsilon_n$ has the opposite sign compared to the Voss model, further distinguishing the two processes.
• The discrepancy in time-reversal entropy suggests that SES exhibit critical dynamics not captured by the current model, possibly due to complex stress accumulation and release near failure.