^{1}, N. V. Sarlis

^{1}and P. A. Varotsos

^{1,a)}

### Abstract

Electric field variations that appear before rupture have been recently studied by employing the detrended fluctuation analysis (DFA) to quantify their long-range temporal correlations. These studies revealed that seismic electric signal (SES) activities exhibit a scale invariant feature with an exponent over all scales investigated (around five orders of magnitude). Here, we study what happens upon significant data loss, which is a question of primary practical importance, and show that the DFA applied to the natural time representation of the remaining data still reveals for SES activities an exponent close to 1.0, which markedly exceeds the exponent found in artificial (man-made) noises. This enables the identification of a SES activity with probability of 75% even after a significant (70%) data loss. The probability increases to 90% or larger for 50% data loss.

Complex systems usually exhibit scale-invariant features characterized by long-range power-law correlations, which are often difficult to quantify due to various types of nonstationarities observed in the signals emitted. This also happens when monitoring geoelectric field changes aiming at detecting seismic electric signal (SES) activities that appear before major earthquakes. To overcome this difficulty the novel method of detrended fluctuation analysis (DFA) has been employed, which when combined with a newly introduced time domain termed natural time, allows a reliable distinction of true SES activities from artificial (man-made) noises. This is so because the SES activities are characterized by “infinitely” ranged temporal correlations (resulting in DFA exponents close to unity) while the artificial noises are not (since all the noises studied to date have DFA exponents at most around 0.8). The analysis of SES observations often meets the difficulty of significant data loss caused either by failure of the data collection system or by removal of seriously noise-contaminated data segments. Here we focus on the effect of significant data loss on the long-range correlated SES activities quantified by DFA. We find that the remaining data, even after a considerable percentage of data loss (which may reach ), may still be revealing the scaling properties of SES activities. This is achieved by applying DFA

*not*to the original time-series of the remaining data but to those resulted when employing natural time.

I. INTRODUCTION

II. DETRENDED FLUCTUATION ANALYSIS AND NATURAL TIME

III. THE EXPERIMENTAL DATA

IV. DATA ANALYSIS AND RESULTS

V. CONCLUSIONS

### Key Topics

- Time series analysis
- 12.0
- Data analysis
- 11.0
- Scaling
- 9.0
- Earthquakes
- 5.0
- Dynamic mechanical analysis
- 3.0

## Figures

Examples of the electric field recordings in normalized units, i.e., by subtracting the mean value and dividing by the standard deviation . The following SES activities are depicted: (a) the one recorded on 18 April 1995 at Ioannina station; (b) the long duration SES activity recorded from 27 December 2010 to 30 December 2009 at Lamia station. (c) is an excerpt of (b) showing that, after long periods of quiescence, the electric field exhibits measurable excursions (transient pulses).

Examples of the electric field recordings in normalized units, i.e., by subtracting the mean value and dividing by the standard deviation . The following SES activities are depicted: (a) the one recorded on 18 April 1995 at Ioannina station; (b) the long duration SES activity recorded from 27 December 2010 to 30 December 2009 at Lamia station. (c) is an excerpt of (b) showing that, after long periods of quiescence, the electric field exhibits measurable excursions (transient pulses).

(a) Example of a surrogate time-series (in normalized units as in Fig. 1) obtained by removing segments of length from the signal of Fig. 1(a) with 50% data loss (i.e., ). (b) The natural time representation of (a). The values obtained from the analysis of (b) in natural time are , , , and .

(a) Example of a surrogate time-series (in normalized units as in Fig. 1) obtained by removing segments of length from the signal of Fig. 1(a) with 50% data loss (i.e., ). (b) The natural time representation of (a). The values obtained from the analysis of (b) in natural time are , , , and .

The dependence of the DFA measure vs the scale in natural time: we increase the percentage of data loss by removing segments of length samples from the signal of Fig. 1(a). The black (plus) symbols correspond to no data loss , the red (crosses) to 30% data loss , the green (asterisks) to 50% data loss , and the blue (squares) to 70% data loss . Except for the case , the data have been shifted vertically for the sake of clarity. The slopes of the corresponding straight lines that fit the data lead to , 0.94, 0.88, and 0.84 from top to bottom, respectively. They correspond to the average values of obtained from 5000 surrogate time-series that were generated with the method of surrogate by Ma *et al.* (Ref. 72) (see the text).

The dependence of the DFA measure vs the scale in natural time: we increase the percentage of data loss by removing segments of length samples from the signal of Fig. 1(a). The black (plus) symbols correspond to no data loss , the red (crosses) to 30% data loss , the green (asterisks) to 50% data loss , and the blue (squares) to 70% data loss . Except for the case , the data have been shifted vertically for the sake of clarity. The slopes of the corresponding straight lines that fit the data lead to , 0.94, 0.88, and 0.84 from top to bottom, respectively. They correspond to the average values of obtained from 5000 surrogate time-series that were generated with the method of surrogate by Ma *et al.* (Ref. 72) (see the text).

The probabilities (a) , (b) , and (c) to recognize the signal of Fig. 1(a) as true SES activity when considering various percentages of data loss , 0.3, 0.5, 0.7, and 0.8 as a function of the length of the contiguous samples removed. The removal of large segments leads to better results when using DFA in natural time (a), whereas the opposite holds when using the conditions of Eqs. (6) and (7) for , , and (b). The optimum selection (c) for the identification of a signal as SES activity consists of a proper combination of the aforementioned procedures in (a) and (b), see the text. The values presented have been obtained from 5000 surrogate time-series (for a given value of and ), and hence they have a plausible error of 1.4% .

The probabilities (a) , (b) , and (c) to recognize the signal of Fig. 1(a) as true SES activity when considering various percentages of data loss , 0.3, 0.5, 0.7, and 0.8 as a function of the length of the contiguous samples removed. The removal of large segments leads to better results when using DFA in natural time (a), whereas the opposite holds when using the conditions of Eqs. (6) and (7) for , , and (b). The optimum selection (c) for the identification of a signal as SES activity consists of a proper combination of the aforementioned procedures in (a) and (b), see the text. The values presented have been obtained from 5000 surrogate time-series (for a given value of and ), and hence they have a plausible error of 1.4% .

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