Algorithms based on Empirical Mode Decomposition (EMD) and Iterative Filtering (IF) are largely implemented for representing a signal as superposition of simpler well-behaved components called Intrinsic Mode Functions (IMFs). Although they are more suitable than traditional methods for the analysis of nonlinear and nonstationary signals, they could be easily misused if their known limitations, together with the assumptions they rely on, are not carefully considered. In this work, we examine the main pitfalls and provide caveats for the proper use of the EMD- and IF-based algorithms. Specifically, we address the problems related to boundary errors, to the presence of spikes or jumps in the signal and to the decomposition of highly-stochastic signals. The consequences of an improper usage of these techniques are discussed and clarified also by analysing real data and performing numerical simulations. Finally, we provide the reader with the best practices to maximize the quality and meaningfulness of the decomposition produced by these techniques. In particular, a technique for the extension of signal to reduce the boundary effects is proposed; a careful handling of spikes and jumps in the signal is suggested; the concept of multi-scale statistical analysis is presented to treat highly stochastic signals.
SUMMARY An unbiased estimation of the b-value and of its variability is essential to verify empirically its physical contribution to the earthquake generation process, and the capability to improve earthquake forecasting and seismic hazard. Notwithstanding the vast literature on the b-value estimation, we note that some potential sources of bias that may lead to non-physical b-value variations are too often ignored in seismological common practice. The aim of this paper is to discuss some of them in detail, when the b-value is estimated through the popular Aki’s formula. Specifically, we describe how a finite data set can lead to biased evaluations of the b-value and its uncertainty, which are caused by the correlation between the b-value and the maximum magnitude of the data set; we quantify analytically the bias on the b-value caused by the magnitude binning; we show how departures from the exponential distribution of the magnitude, caused by a truncated Gutenberg–Richter law and by catalogue incompleteness, can affect the b-value estimation and the search for statistically significant variations; we derive explicitly the statistical distribution of the magnitude affected by random symmetrical error, showing that the magnitude error does not induce any further significant bias, at least for reasonable amplitude of the measurement error. Finally, we provide some recipes to minimize the impact of these potential sources of bias.
The most common earthquake forecasting models assume that the magnitude of the next earthquake is independent from the past. This feature severely limits the capability to forecast large earthquakes with high probabilities. Here we investigate empirically on the magnitude-independence assumption, exploring if: (i) background and triggered earthquakes have the same frequency-magnitude distribution, (ii) variations of seismicity in the spacetime-magnitude domain encode some information on the future earthquakes size. For this purpose, and to verify the stability of the findings, we consider seismic catalogues covering different space-time-magnitude windows, such as the Alto Tiberina Near Fault Observatory (TABOO), the California and Japanese seismic catalogues. Our approach is inspired by the nearest-neighbour method proposed by Baiesi & Paczuski and elaborated by Zaliapin et al. to distinguish between triggered and background earthquakes. Here we implement the same metric-based correlation to identify the precursory seismicity of any triggered earthquake; this allows us to analyse, for each triggered earthquake, the space-time-magnitude distribution of the seismicity that likely contributed to its occurrence. Our results show that the magnitude-independence assumption holds reasonably well in all catalogues, with a remarkable exception that is consistent with a previous independent study; this departure from the magnitude-independence assumption shows that larger events tend to nucleate at a higher distance from the ongoing sequence. We also notice that the reliability of this assumption may depend on the spatial scale considered; it holds for seismic catalogues of large areas, but we identify possible departures in small areas, reflecting different ways to release locally seismic energy. Finally, we come across an important issue that may lead to misleading results in similar studies, that is, if a seismic catalogue appears overall complete above a fixed magnitude threshold, it may still yield spurious signals into the analysis. Specifically, we show that some significant departures from the magnitude-independence assumption do not survive when considering spatiotemporal variations of the magnitude of completeness.
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