**3. The precursory MHz EM activity as a second order phase transition phenomenon**

In natural rocks at large length scales there are long-range anti-correlations, in the sense that a high value of a rock property, e.g. threshold for breaking, is followed by a low value and

important observation in this approach is the fact that the distribution *P*(*l*) of the laminar

Are There Pre-Seismic Electromagnetic Precursors? A Multidisciplinary Approach 223

lengths *l* of the intermittent map (1) in the limit *�<sup>n</sup>* → 0 is given by the power law [53]

where the exponent *pl* is connected with the exponent *z* via *pl* = *<sup>z</sup>*

*pl* is related to the isothermal exponent *δ* by

with *δ* > 0.

critical behavior [52,53].

fitted by the relation:

*P*(*l*) ∼ *l*

*pl* = 1 +

Inversely, the existence of a power law such as relation (2), accompanied by a plateau form of the corresponding density *ρ*(*φ*), is a signature of underlying correlated dynamics similar to

*We emphasize that it is possible in the framework of universality, which is characteristic of critical*

The MCF is directly applied to time series or to segments of time series which appear to have a cumulative stationary behaviour. The main aim of the MCF is to estimate the exponent *pl*. The distribution of the laminar lengths, *l*, of fluctuations included in a stationary window is

<sup>−</sup>*p*<sup>2</sup> *e*

If *p*<sup>3</sup> is zero, then *p*<sup>2</sup> is equal to *pl*. Practically, as *p*<sup>3</sup> approaches zero, then *p*<sup>2</sup> approaches *pl* and the laminar lengths tend to follow a power-law type distribution. So, we expect a good fit to Eq. (4) with *p*<sup>2</sup> > 1 and *p*<sup>3</sup> ≈ 0 if the system is in a critical state [50]. In terms of physics this behaviour means that the system is characterized by a "strong criticality", e.g., the laminar lengths tend to follow a power-law type distribution: during this critical time window the

We stress that when the exponent *p*<sup>2</sup> is smaller than one, then, independently of the *p*3-value, the system is not in a critical state. Generally, the exponents *p*2, *p*<sup>3</sup> have a competitive character, namely, when the exponent *p*<sup>2</sup> decreases the associated exponent *p*<sup>3</sup> increases (they are mirror images of each other). To be more precise, as the exponent *p*<sup>2</sup> (*p*<sup>2</sup> < 1) is close to 1 and simultaneously the exponent *p*<sup>3</sup> is close to zero, then the system is in a sub-critical state. As the system moves away from the critical state, then the exponent *p*<sup>2</sup> further decreases while simultaneously *p*<sup>3</sup> increases, reinforcing in this way the exponential character of the laminar length distribution: the EM fluctuations show short range correlations. In this way, we can

On 13 May 1995 (8:47:13 UT) the Kozani-Grevena EQ (40.17◦N, 21.68◦E) occurred with magnitude *M* = 6.6. Fig. 1 shows the associated 41 MHz EM time series [25,28,29]. The

*phenomena, to give meaning to the exponent pl beyond the thermal phase transitions [53]*.

*P*(*l*) ∼ *l*

opening cracks (EM-emitters) are well correlated even at large distances [50].

identify the deviation from the critical state [50,52,53].

**3.2 Application of the MCF method**

data are sampled at 1 Hz.

1

<sup>−</sup>*pl* (2)

*<sup>δ</sup>* (3)

<sup>−</sup>*p*3*<sup>l</sup>* (4)

*<sup>z</sup>*−<sup>1</sup> . Therefore the exponent

vice versa. Failure nucleation begins to occur at a region where the resistance to rupture growth has the minimum value. An EM event is emitted during this fracture. The fracture process continues in the same weak region until a much stronger region is encountered in its neighborhood. When this happens, fracture stops, and thus the emitted EM emission ceases. The stresses are redistributed, while the applied stress in the focal area increases. A new population of cracks nucleates in the weaker of the unbroken regions, and thus a new EM event appears, and so on. Therefore, the associated precursory MHz EM activity should be characterized by antipersistent behaviour and the interplay between the heterogeneities and the stress field should be responsible for this behaviour. This crucial feature is included in the recorded MHz EM precursors.

Physically, the presence of anti-persistency implies a set of EM fluctuations tending to induce stability to the system, essentially the existence of a non-linear negative feedback mechanism that "kicks" the opening rate of cracks away from extremes. The existence of such a mechanism leads to the next step: it has been proposed that the fracture of heterogeneous materials can be described in analogy with a continuous second order phase transition in equilibrium [40,41]. Thus, a seismogenic MHz EM activity, which is rooted in the fracture of the highly heterogeneous system that surrounds the family of large high-strength asperities, should be described as critical phenomenon. This critical signature is also hidden in the recorded MHz EM precursors [28-29,34-36,39]. The relevant analysis is based on the recently introduced Method of Critical Fluctuations (MCF) [52,53].

#### **3.1 The method of critical fluctuations**

The MCF, which constitutes a statistical method of analysis for the critical fluctuations in systems that undergo a continuous phase transition at equilibrium, has been recently introduced [52,53]. The authors have shown that the fluctuations of the order parameter *φ*, obey a dynamical law of intermittency which can be described in terms of a 1-d nonlinear map. The invariant density *ρ*(*φ*) for such a map is characterized by a plateau which decays in a super-exponential way (see Fig. 1 in [52]). For small values of *φ*, this critical map can be approximated as

$$
\phi\_{n+1} = \phi\_n + \iota \phi\_n^z + \epsilon\_n \tag{1}
$$

The shift parameter *�n* introduces a non-universal stochastic noise: each physical system has its characteristic "noise", which is expressed through the shift parameter *�n*. For thermal systems the exponent *z* is introduced, which is related to the isothermal critical exponent *δ* by *z* = *δ* + 1.

The plateau region of the invariant density *ρ*(*φ*) corresponds to the laminar region of the critical map where fully correlated dynamics take place [29 and references therein]. The laminar region ends when the second term in Eq. (1) becomes relevant. However, due to the fact that the dynamical law (1) changes continuously with *φ*, the end of the laminar region cannot be easily defined based on a strictly quantitative criterion. Thus, the end of the laminar region should be generally treated as a variable parameter.

Based on the foregoing description of the critical fluctuations, the MCF develops an algorithm permitting the extraction of the critical fluctuations, if any, in a recorded time series. The

important observation in this approach is the fact that the distribution *P*(*l*) of the laminar lengths *l* of the intermittent map (1) in the limit *�<sup>n</sup>* → 0 is given by the power law [53]

$$P(l) \sim l^{-p\_l} \tag{2}$$

where the exponent *pl* is connected with the exponent *z* via *pl* = *<sup>z</sup> <sup>z</sup>*−<sup>1</sup> . Therefore the exponent *pl* is related to the isothermal exponent *δ* by

$$p\_l = 1 + \frac{1}{\delta} \tag{3}$$

with *δ* > 0.

6 Will-be-set-by-IN-TECH

vice versa. Failure nucleation begins to occur at a region where the resistance to rupture growth has the minimum value. An EM event is emitted during this fracture. The fracture process continues in the same weak region until a much stronger region is encountered in its neighborhood. When this happens, fracture stops, and thus the emitted EM emission ceases. The stresses are redistributed, while the applied stress in the focal area increases. A new population of cracks nucleates in the weaker of the unbroken regions, and thus a new EM event appears, and so on. Therefore, the associated precursory MHz EM activity should be characterized by antipersistent behaviour and the interplay between the heterogeneities and the stress field should be responsible for this behaviour. This crucial feature is included in the

Physically, the presence of anti-persistency implies a set of EM fluctuations tending to induce stability to the system, essentially the existence of a non-linear negative feedback mechanism that "kicks" the opening rate of cracks away from extremes. The existence of such a mechanism leads to the next step: it has been proposed that the fracture of heterogeneous materials can be described in analogy with a continuous second order phase transition in equilibrium [40,41]. Thus, a seismogenic MHz EM activity, which is rooted in the fracture of the highly heterogeneous system that surrounds the family of large high-strength asperities, should be described as critical phenomenon. This critical signature is also hidden in the recorded MHz EM precursors [28-29,34-36,39]. The relevant analysis is based on the recently

The MCF, which constitutes a statistical method of analysis for the critical fluctuations in systems that undergo a continuous phase transition at equilibrium, has been recently introduced [52,53]. The authors have shown that the fluctuations of the order parameter *φ*, obey a dynamical law of intermittency which can be described in terms of a 1-d nonlinear map. The invariant density *ρ*(*φ*) for such a map is characterized by a plateau which decays in a super-exponential way (see Fig. 1 in [52]). For small values of *φ*, this critical map can be

*<sup>φ</sup>n*+<sup>1</sup> = *<sup>φ</sup><sup>n</sup>* + *<sup>u</sup>φ<sup>z</sup>*

The shift parameter *�n* introduces a non-universal stochastic noise: each physical system has its characteristic "noise", which is expressed through the shift parameter *�n*. For thermal systems the exponent *z* is introduced, which is related to the isothermal critical exponent *δ* by

The plateau region of the invariant density *ρ*(*φ*) corresponds to the laminar region of the critical map where fully correlated dynamics take place [29 and references therein]. The laminar region ends when the second term in Eq. (1) becomes relevant. However, due to the fact that the dynamical law (1) changes continuously with *φ*, the end of the laminar region cannot be easily defined based on a strictly quantitative criterion. Thus, the end of the laminar

Based on the foregoing description of the critical fluctuations, the MCF develops an algorithm permitting the extraction of the critical fluctuations, if any, in a recorded time series. The

*<sup>n</sup>* + *�<sup>n</sup>* (1)

recorded MHz EM precursors.

introduced Method of Critical Fluctuations (MCF) [52,53].

region should be generally treated as a variable parameter.

**3.1 The method of critical fluctuations**

approximated as

*z* = *δ* + 1.

Inversely, the existence of a power law such as relation (2), accompanied by a plateau form of the corresponding density *ρ*(*φ*), is a signature of underlying correlated dynamics similar to critical behavior [52,53].

*We emphasize that it is possible in the framework of universality, which is characteristic of critical phenomena, to give meaning to the exponent pl beyond the thermal phase transitions [53]*.

The MCF is directly applied to time series or to segments of time series which appear to have a cumulative stationary behaviour. The main aim of the MCF is to estimate the exponent *pl*. The distribution of the laminar lengths, *l*, of fluctuations included in a stationary window is fitted by the relation:

$$P(l) \sim l^{-p\_2} e^{-p\_3 l} \tag{4}$$

If *p*<sup>3</sup> is zero, then *p*<sup>2</sup> is equal to *pl*. Practically, as *p*<sup>3</sup> approaches zero, then *p*<sup>2</sup> approaches *pl* and the laminar lengths tend to follow a power-law type distribution. So, we expect a good fit to Eq. (4) with *p*<sup>2</sup> > 1 and *p*<sup>3</sup> ≈ 0 if the system is in a critical state [50]. In terms of physics this behaviour means that the system is characterized by a "strong criticality", e.g., the laminar lengths tend to follow a power-law type distribution: during this critical time window the opening cracks (EM-emitters) are well correlated even at large distances [50].

We stress that when the exponent *p*<sup>2</sup> is smaller than one, then, independently of the *p*3-value, the system is not in a critical state. Generally, the exponents *p*2, *p*<sup>3</sup> have a competitive character, namely, when the exponent *p*<sup>2</sup> decreases the associated exponent *p*<sup>3</sup> increases (they are mirror images of each other). To be more precise, as the exponent *p*<sup>2</sup> (*p*<sup>2</sup> < 1) is close to 1 and simultaneously the exponent *p*<sup>3</sup> is close to zero, then the system is in a sub-critical state. As the system moves away from the critical state, then the exponent *p*<sup>2</sup> further decreases while simultaneously *p*<sup>3</sup> increases, reinforcing in this way the exponential character of the laminar length distribution: the EM fluctuations show short range correlations. In this way, we can identify the deviation from the critical state [50,52,53].

#### **3.2 Application of the MCF method**

On 13 May 1995 (8:47:13 UT) the Kozani-Grevena EQ (40.17◦N, 21.68◦E) occurred with magnitude *M* = 6.6. Fig. 1 shows the associated 41 MHz EM time series [25,28,29]. The data are sampled at 1 Hz.

<sup>100</sup> <sup>200</sup> <sup>300</sup> <sup>400</sup> <sup>500</sup> <sup>0</sup>

**b**

from normal behaviour, revealing the presence of an EM anomaly.

**2**

100 200 300 400 500 Amplitude (mV)

EQ. The lower part elucidates the evolution of symmetry breaking with time. **4. How can we recognize a kHz EM anomaly as a pre-seismic one?**

**c**

Fig. 1. The upper part shows the 41 MHz EM time series associated with the Kozani-Grevena

An anomaly in a recorded time series is defined as a deviation from normal (background) behaviour. In order to develop a quantitative identification of EM precursors, tools of information theory and concepts of entropy are used in order to identify statistical patterns. Entropy and information are seen to be complementary quantities, in a sense: entropy "drops" have as a counterpart information "peaks" in a more ordered state. The seismicity is a critical phenomenon [41,54] , thus, it is expected that a significant change in the statistical pattern, namely the appearance of entropy "drops" or information "peaks", represents a deviation

*It is important to note that one cannot find an optimum organization or complexity measure. Thus, a combination of some such quantities which refer to different aspects, such as structural or dynamical*

Several well-known techniques have been applied to extract EM precursors hidden in kHz

(i) *T*-entropy: It is based on the intellectual economy one makes when rewriting a string

(ii) Approximate entropy: It provides a measure of the degree of irregularity or randomness within a series of data. More precisely, this examines the presence of similar epochs in time series; more similar and more frequent epochs lead to lower values of approximate

(iii) Fisher Information: It represents the amount of information that can be extracted from a

(iv) Correlation Dimension: It measures the probability that two points chosen at random will be within a certain distance of each other, and examines how this probability

**3**

04:00 08:00 12:00 16:00 20:00 00:00 04:00 08:00

Time (UT)

**1 2 3 4**

41MHz electromagnetic timeíseries (raw data)

Are There Pre-Seismic Electromagnetic Precursors? A Multidisciplinary Approach 225

100 200 300 400 500 Amplitude (mV)

**d**

**4**

100 200 300 400 500 Amplitude (mV)

**e**

**EQ**

**a**

Amplitude (mV)

0.05 0.1 0.15

*properties, is the most promising way*.

according to some rule [55].

set of measurements [56].

entropy [35 and references therein].

changes as the distance is increased [57].

EM time series:

200 400

(mV)

**1**

Probability

A critical window (CW) has been identified including 23000 points (Fig. 1a) starting almost 11 hours before the time of the EQ occurrence. The corresponding distribution of the amplitude *P*(*ψ*) of the emerged EM pulses in this CW is shown in Fig. (1c). It is characteristic the appearance of the plateau region in the top of distribution, as it is provided for the invariant density of critical map [52]. The laminar lengths *l* follow a power-law distribution *P*(*l*). This feature indicated that the underlying fracture mechanism is characterized by fluctuations which are extended at many different time scales as well as the presence of long-range correlations. We note that the amplitude *ψ<sup>i</sup>* of the preseismic MHz EM time series behaves as a kind of the order parameter [29]. Therefore, in the CW the fluctuations of the amplitude *ψ<sup>i</sup>* of the recorded EM time series have an intermittent behaviour similar to the dynamics of the order parameter's fluctuations of a thermal critical system at the critical point. It is for this reason that this window is characterized as *critical window*.

A thermal phase transition is associated with a *symmetry breaking*. To gain inside into the temporal evolution of fracture, as the EQ is approaching, we elucidate the evolution of the *symmetry breaking* with time by making an analogy to a thermal continuous phase transition [29]. In the latter, the distribution of the fluctuations of the order parameter with temperature reveals the progress of the *symmetry breaking*. This distribution is almost a *δ* function at high temperature and evolves to a Gaussian with mean value zero as the system approaches the critical point. At the critical point, a characteristic plateau in the distribution appears, and the *symmetry breaking* emerges as the temperature further decreases. Below the critical temperature the distribution becomes again Gaussian, but its mean shifts to higher values associated with the *symmetry breaking*. As temperature approaches to 0◦*K*, where the *symmetry breaking* is completed, it becomes a *δ* function again. We look for these characteristic features in the preseismic time series, with stress taking on the role of temperature [29].

Let us look specifically at the precursor under study. Figs. (1b-1e) exhibit the distribution of the recorded EM fluctuations in successive time windows. As is was mentioned, the distribution of the amplitude (order parameter) in Fig. (1c) indicates the appearance of the CW. Fig. (1b) shows the distribution before the emergence of CW: the laminar lengths, *l*, do not follow a power-law-type distribution *P*(*l*). The system is characterized by a spare almost symmetrical distributed in space random cracking with short-range correlations.

During the CW the sort-range correlation evolve to long-range; the corresponding distribution (Fig. 1c) might be considered as a precursor of the impending *symmetry breaking*. The *symmetry breaking* is readily observable in the subsequent time interval (Fig. 1d). The cracking is restricted in the narrow zone that includes the backbone of strong asperities distributed along the activated fault sustaining the system [29]. The distribution of the order parameter in Fig. (1e) is very similar to that of Fig. (1b). However, here there is an upward shift of the values to the range of the second lobe of the distribution in Fig. (1d). The laminar lengths does not follow a power-law distribution *P*(*l*). The appearance of this window indicates that the *symmetry breaking* in the underlying fracto-EM process has been almost completed [29]. *The siege of strong asperities begins* [29]. However, the prepared EQ will occur if and when the local stress exceeds fracture stresses of asperities. The lounge of the kHz EM activity shows the fracture of asperities sustaining the fault [28,29,32-36]. Indeed, a very strong kHz EM burst appeared a few hours later and after that face the EQ occurred [29].

8 Will-be-set-by-IN-TECH

A critical window (CW) has been identified including 23000 points (Fig. 1a) starting almost 11 hours before the time of the EQ occurrence. The corresponding distribution of the amplitude *P*(*ψ*) of the emerged EM pulses in this CW is shown in Fig. (1c). It is characteristic the appearance of the plateau region in the top of distribution, as it is provided for the invariant density of critical map [52]. The laminar lengths *l* follow a power-law distribution *P*(*l*). This feature indicated that the underlying fracture mechanism is characterized by fluctuations which are extended at many different time scales as well as the presence of long-range correlations. We note that the amplitude *ψ<sup>i</sup>* of the preseismic MHz EM time series behaves as a kind of the order parameter [29]. Therefore, in the CW the fluctuations of the amplitude *ψ<sup>i</sup>* of the recorded EM time series have an intermittent behaviour similar to the dynamics of the order parameter's fluctuations of a thermal critical system at the critical point. It is for this

A thermal phase transition is associated with a *symmetry breaking*. To gain inside into the temporal evolution of fracture, as the EQ is approaching, we elucidate the evolution of the *symmetry breaking* with time by making an analogy to a thermal continuous phase transition [29]. In the latter, the distribution of the fluctuations of the order parameter with temperature reveals the progress of the *symmetry breaking*. This distribution is almost a *δ* function at high temperature and evolves to a Gaussian with mean value zero as the system approaches the critical point. At the critical point, a characteristic plateau in the distribution appears, and the *symmetry breaking* emerges as the temperature further decreases. Below the critical temperature the distribution becomes again Gaussian, but its mean shifts to higher values associated with the *symmetry breaking*. As temperature approaches to 0◦*K*, where the *symmetry breaking* is completed, it becomes a *δ* function again. We look for these characteristic features

Let us look specifically at the precursor under study. Figs. (1b-1e) exhibit the distribution of the recorded EM fluctuations in successive time windows. As is was mentioned, the distribution of the amplitude (order parameter) in Fig. (1c) indicates the appearance of the CW. Fig. (1b) shows the distribution before the emergence of CW: the laminar lengths, *l*, do not follow a power-law-type distribution *P*(*l*). The system is characterized by a spare almost

During the CW the sort-range correlation evolve to long-range; the corresponding distribution (Fig. 1c) might be considered as a precursor of the impending *symmetry breaking*. The *symmetry breaking* is readily observable in the subsequent time interval (Fig. 1d). The cracking is restricted in the narrow zone that includes the backbone of strong asperities distributed along the activated fault sustaining the system [29]. The distribution of the order parameter in Fig. (1e) is very similar to that of Fig. (1b). However, here there is an upward shift of the values to the range of the second lobe of the distribution in Fig. (1d). The laminar lengths does not follow a power-law distribution *P*(*l*). The appearance of this window indicates that the *symmetry breaking* in the underlying fracto-EM process has been almost completed [29]. *The siege of strong asperities begins* [29]. However, the prepared EQ will occur if and when the local stress exceeds fracture stresses of asperities. The lounge of the kHz EM activity shows the fracture of asperities sustaining the fault [28,29,32-36]. Indeed, a very strong kHz EM burst

in the preseismic time series, with stress taking on the role of temperature [29].

symmetrical distributed in space random cracking with short-range correlations.

appeared a few hours later and after that face the EQ occurred [29].

reason that this window is characterized as *critical window*.

Fig. 1. The upper part shows the 41 MHz EM time series associated with the Kozani-Grevena EQ. The lower part elucidates the evolution of symmetry breaking with time.

#### **4. How can we recognize a kHz EM anomaly as a pre-seismic one?**

An anomaly in a recorded time series is defined as a deviation from normal (background) behaviour. In order to develop a quantitative identification of EM precursors, tools of information theory and concepts of entropy are used in order to identify statistical patterns. Entropy and information are seen to be complementary quantities, in a sense: entropy "drops" have as a counterpart information "peaks" in a more ordered state. The seismicity is a critical phenomenon [41,54] , thus, it is expected that a significant change in the statistical pattern, namely the appearance of entropy "drops" or information "peaks", represents a deviation from normal behaviour, revealing the presence of an EM anomaly.

*It is important to note that one cannot find an optimum organization or complexity measure. Thus, a combination of some such quantities which refer to different aspects, such as structural or dynamical properties, is the most promising way*.

Several well-known techniques have been applied to extract EM precursors hidden in kHz EM time series:


**5.1 The activation of a single fault as a self-affine image of the regional and laboratory**

**5.1.1 The activation of a single fault as a "reduced self-affine image" of the regional**

Gutenberg-Ricter (G-R) type law for the magnitude distribution of EQs:

<sup>2</sup> <sup>−</sup> *<sup>q</sup>* 1 − *q*

fault. Thus, we examine whether the kHz EM activity also follows the Eq. (5).

energy, the magnitude *m* of the candidate "EM-EQ" is given by the relation

*<sup>m</sup>* <sup>=</sup> log *<sup>ε</sup>* <sup>∼</sup> log

 log 1 −

where *N* is the total number of EQs, *N*(> *m*) the number of EQs with magnitude larger than *m*. Parameter *α* is the constant of proportionality between the EQ energy, *ε* and the size of fragment. The entropic index *q* describes the deviation of Tsallis entropy from the traditional Shannon one. The proposed non-extensive G-R type law (5) provides an excellent fit to seismicities generated in various large geographic areas, each of them covering many geological faults. We emphasize that the *q*-values are restricted in the narrow region from 1.6 to 1.8 [72-74]. Notice, the magnitude-frequency relationship for EQs do not say anything about a specific activated fault (EQ). A kHz EM precursors refers to the activation of a specific

*Definition of the "Electromagnetic earthquake":* We regard as amplitude *A* of a candidate "fracto-EM fluctuation" the difference *Af em*(*ti*) = *A*(*ti*) − *Anoise*, where *Anoise* is the background (noise) level of the EM time series. We consider that a sequence of *k* successively emerged "fracto-EM fluctuations" *Af em*(*ti*), *i* = 1, . . . , *k* represents the EM energy released, *ε*, during the damage of a fragment. We shall refer to this as an "electromagnetic earthquake" (EM-EQ). Since the squared amplitude of the fracto-EM emissions is proportional to their

> ∑

The Eq. (5) provides an excellent fit to the pre-seismic kHz EM experimental data incorporating the characteristics of nonextensivity statistics into the distribution of the

*Af em*(*ti*)

2 

<sup>1</sup> <sup>−</sup> *<sup>q</sup>* 2 − *q* 102*<sup>m</sup>*

*<sup>α</sup>*2/3 (5)

(6)

log(*N*>*m*) = log *N* +

The self-affine nature of faulting and fracture predicts that the activation of a single fault is a reduced / magnified image of the regional/ laboratory seismicity, correspondingly (see Section 2.2.2). A fracto-EM precursor rooted in the activation of a single fault should be

Are There Pre-Seismic Electromagnetic Precursors? A Multidisciplinary Approach 227

A model for EQ dynamics coming from a non-extensive Tsallis formulation [66,67] has been recently introduced by Sotolongo-Costa and Posadas, [72]. Silva et al. [73] have revised this model. The authors assume that the mechanism of relative displacement of fault plates is the main cause of EQs. The space between fault planes is filled with the residues of the breakage of the tectonic plates, from where the faults have originated. The motion of the fault planes can be hindered not only by the overlapping of two irregularities of the profiles, but also by the eventual relative position of several fragments. Thus, the mechanism of triggering EQs is established through the combination of the irregularities of the fault planes on one hand and the fragments between them on the other hand. This nonextensive approach leads to a

**seismicity**

**seismicity**

consistent with the above mentioned requirement.


The application of all the above mentioned multidisciplinary statistical procedure [30,33,35,36,68-71] sensitively recognizes and discriminates the candidate EM precursors from the EM background: they are characterized by significantly higher organization in respect to that of the EM noise in the region of the station. However, we should keep in mind that though a sledge hammer may be wonderful for breaking rock, it is a poor choice for driving a tack into a picture frame!

#### **5. Focus on the possible seismogenic origin of the detected kHz EM anomaly by means of universally holding scaling laws of fracture**

As it is mentioned in the previous Section, all the applied techniques reveal that the kHz EM anomaly is characterized by a significant lower complexity (or higher organization). Importantly this anomaly is also characterized by strong persistency [28,29]. The simultaneous appearance of both these two crucial characteristics implies that the underlying fracture process is governed by a positive feedback mechanism which is consistent with an anomaly being a precursor of an ensuing catastrophic event.

However, we suggest that any multidisciplinary statistical analysis by itself is not sufficient to characterize an emerged kHz EM anomaly as a pre-earthquake one. Much remains to be done to tackle systematically real pre-seismic EM precursors.

As it is mentioned in Section 2.2, the Earth's crust is extremely complex. However, despite its complexity, there are several universally holding scaling relations. Such universal structural patterns of fracture and faulting process should be included into an EM precursor which is rooted in the activation of a single fault. Therefore an important pursuit is to investigate whether universal features of fractures and faulting are included in the recorded kHz EM precursors.

10 Will-be-set-by-IN-TECH

(v) R/S analysis: It provides a direct estimation of the Hurst Exponent which is a precious

(vi) Detrended Fluctuation Analysis: It has been proven useful in revealing the extent of

(vii) Shannon *n*-block entropies (conditional entropy, entropy of the source, Kolmogorov-Sinai entropy): They measure the uncertainty of predicting a state in the future, provided a history of the present state and the previous states [61-65]. (viii) Tsallis entopy: One of the crucial properties of the Boltzmann-Gibbs entropy in the context of classical thermodynamics is extensivity, namely proportionality with the number of elements of the system. The Boltzmann-Gibbs (B-G) entropy satisfies this prescription if the subsystems are statistically (quasi-) independent, or typically if the correlations within the system are essentially local. In such cases the energy of the system is typically extensive and the entropy is additive. In general, however, the situation is not of this type and correlations may be far from negligible at all scales. Inspired by multifractals concepts, Tsallis [66, 67] has proposed a generalization of the B-G statistical mechanics. He introduced an entropic index *q* which leads to a nonextensive statistics. The value of *q* is a measure of the nonextensivity of the system: *q* = 1 corresponds to the standard, extensive, B-G statistics. The order of organization of the nonextensive

The application of all the above mentioned multidisciplinary statistical procedure [30,33,35,36,68-71] sensitively recognizes and discriminates the candidate EM precursors from the EM background: they are characterized by significantly higher organization in respect to that of the EM noise in the region of the station. However, we should keep in mind that though a sledge hammer may be wonderful for breaking rock, it is a poor choice for driving a

**5. Focus on the possible seismogenic origin of the detected kHz EM anomaly by**

As it is mentioned in the previous Section, all the applied techniques reveal that the kHz EM anomaly is characterized by a significant lower complexity (or higher organization). Importantly this anomaly is also characterized by strong persistency [28,29]. The simultaneous appearance of both these two crucial characteristics implies that the underlying fracture process is governed by a positive feedback mechanism which is consistent with an

However, we suggest that any multidisciplinary statistical analysis by itself is not sufficient to characterize an emerged kHz EM anomaly as a pre-earthquake one. Much remains to be done

As it is mentioned in Section 2.2, the Earth's crust is extremely complex. However, despite its complexity, there are several universally holding scaling relations. Such universal structural patterns of fracture and faulting process should be included into an EM precursor which is rooted in the activation of a single fault. Therefore an important pursuit is to investigate whether universal features of fractures and faulting are included in the recorded kHz EM

indicator of the state of randomness of a time-series [58].

long-range correlations in time series [59, 60].

systems is measured by the Tsallis entropy.

**means of universally holding scaling laws of fracture**

anomaly being a precursor of an ensuing catastrophic event.

to tackle systematically real pre-seismic EM precursors.

tack into a picture frame!

precursors.

#### **5.1 The activation of a single fault as a self-affine image of the regional and laboratory seismicity**

The self-affine nature of faulting and fracture predicts that the activation of a single fault is a reduced / magnified image of the regional/ laboratory seismicity, correspondingly (see Section 2.2.2). A fracto-EM precursor rooted in the activation of a single fault should be consistent with the above mentioned requirement.

#### **5.1.1 The activation of a single fault as a "reduced self-affine image" of the regional seismicity**

A model for EQ dynamics coming from a non-extensive Tsallis formulation [66,67] has been recently introduced by Sotolongo-Costa and Posadas, [72]. Silva et al. [73] have revised this model. The authors assume that the mechanism of relative displacement of fault plates is the main cause of EQs. The space between fault planes is filled with the residues of the breakage of the tectonic plates, from where the faults have originated. The motion of the fault planes can be hindered not only by the overlapping of two irregularities of the profiles, but also by the eventual relative position of several fragments. Thus, the mechanism of triggering EQs is established through the combination of the irregularities of the fault planes on one hand and the fragments between them on the other hand. This nonextensive approach leads to a Gutenberg-Ricter (G-R) type law for the magnitude distribution of EQs:

$$\log(N\_{\ge m}) = \log N + \left(\frac{2-q}{1-q}\right) \log \left[1 - \left(\frac{1-q}{2-q}\right) \left(\frac{10^{2m}}{a^{2/3}}\right)\right] \tag{5}$$

where *N* is the total number of EQs, *N*(> *m*) the number of EQs with magnitude larger than *m*. Parameter *α* is the constant of proportionality between the EQ energy, *ε* and the size of fragment. The entropic index *q* describes the deviation of Tsallis entropy from the traditional Shannon one. The proposed non-extensive G-R type law (5) provides an excellent fit to seismicities generated in various large geographic areas, each of them covering many geological faults. We emphasize that the *q*-values are restricted in the narrow region from 1.6 to 1.8 [72-74]. Notice, the magnitude-frequency relationship for EQs do not say anything about a specific activated fault (EQ). A kHz EM precursors refers to the activation of a specific fault. Thus, we examine whether the kHz EM activity also follows the Eq. (5).

*Definition of the "Electromagnetic earthquake":* We regard as amplitude *A* of a candidate "fracto-EM fluctuation" the difference *Af em*(*ti*) = *A*(*ti*) − *Anoise*, where *Anoise* is the background (noise) level of the EM time series. We consider that a sequence of *k* successively emerged "fracto-EM fluctuations" *Af em*(*ti*), *i* = 1, . . . , *k* represents the EM energy released, *ε*, during the damage of a fragment. We shall refer to this as an "electromagnetic earthquake" (EM-EQ). Since the squared amplitude of the fracto-EM emissions is proportional to their energy, the magnitude *m* of the candidate "EM-EQ" is given by the relation

$$m = \log \varepsilon \sim \log \left( \sum \left[ A\_{fem}(t\_i) \right]^2 \right) \tag{6}$$

The Eq. (5) provides an excellent fit to the pre-seismic kHz EM experimental data incorporating the characteristics of nonextensivity statistics into the distribution of the

included into an kHz EM precursor. If a time series is a temporal fractal then a power-law of the form *S*(*f*) ∝ *f* <sup>−</sup>*<sup>β</sup>* is obeyed, with *S*(*f*) the power spectral density and *f* the frequency. The spectral scaling exponent *β* is a measure of the strength of time correlations. The goodness of the power-law fit to a time series is represented by a linear correlation coefficient, *r*. Based on a fractal spectral analysis, which has been performed by means of wavelets, it has been shown [27-30,35,36] that the emergent strong kHz EM precursors follow the law *S*(*f*) ∝ *f* <sup>−</sup>*β*; the coefficient *r* takes values very close to 1, i.e., the fit to the power-law is excellent. This

Are There Pre-Seismic Electromagnetic Precursors? A Multidisciplinary Approach 229

(i) The EM bursts have long-range temporal correlations, i.e. strong memory: the current value of the precursory signal is correlated not only with its most recent values but also

(ii) The spectrum manifests more power at lower frequencies than at high frequencies. The enhancement of lower frequency power physically reveals a predominance of larger fracture events. This footprint is also in harmony with the final step of EQ preparation. (iii) Two classes of signal have been widely used to model stochastic fractal time series, fractional Gaussian (fGn) and fractional Brownian motion (fBm) model [83]. The fGn-model the scaling exponent *β* lies between -1 and 1, while the fBm regime is indicated by *β* values from 1 to 3. The estimated *β* exponent successfully distinguishes the candidate precursory activities from the EM noise [27-31,35,36]. Indeed, the *β* values in the EM background are between 1 and 2 indicating that the time profile of the EM series during the quiet periods is qualitatively analogous to the fGn class. On the contrary, the *β* values in the candidate EM precursors are between 2 and 3, suggesting

In summary, the fBm nature of faulting and fracture is included in the kHz EM precursors.

with 0 < *H* < 1 (1 < *β* < 3) for the fractional Brownian motion (fBm) model. The exponent *H* characterizes the persistent / anti-persistent properties of the signal. The range 0.5 < *H* < 1 (2 < *β* < 3) indicates persistency, which means that if the amplitude of the fluctuations increases in a time interval it is likely to continue increasing in the next interval. We recall that we found *β* values in the candidate EM precursors to lie between 2 and 3. The *H* values are close to 0.7 in the strong segments of the kHz EM activity [27-31,35,36]. This means that the EM fluctuations are positively correlated: the underlying dynamics is governed by a positive feedback mechanism. External influences would then tend to lead the system out of equilibrium. The system acquires a self-regulating character and, to a great extent, the property of irreversibility, one of the important components of prediction reliability. Sammis and Sornette [85] have recently presented the most important positive feedback mechanisms.

*β* = 2*H* + 1 (8)

result shows the fractal character of the underlying processes and structures.

with its long-term history in a scale-invariant, fractal manner.

that they belong to the fBm class.

**5.2.1 Persistent behaviour of the detected kHz EM precursors**

The *β* exponent is related to the Hurst exponent *H* by the formula [83, 84]:

The *β* exponent takes high values, i.e., between 2 and 3. This finding reveals that:

detected precursory "EM-EQs" [32,33,36,75]. Herein, *N*(> *m*) is the number of "EM-EQs" with magnitude larger than *m*, *P*(> *m*) = *N*(> *m*)/*N* is the relative cumulative number of "EM-EQs" with magnitude larger than *m*, and *α* is the constant of proportionality between the EM energy released and the size of fragment. The best-fit *q* parameter for this analysis has been estimated to be approximately 1.8 [32,33,36,75].

It is very interesting to observe the similarity in the *q*-values associated with the non-extensive Eq. (5) for: (i) seismicities generated in various large geographic areas, and (ii) the precursory sequence of "EM-EQs". This finding indicates that the statistics of regional seismicity could be merely a macroscopic reflection of the physical processes in the EQ source, as it has been suggested by Huang and Turcotte [76].

#### **5.1.2 The activation of a single fault as a "magnified self-affine image" of the laboratory seismicity**

Rabinovitch et al. [77] have studied the fractal nature of EM radiation induced in rock fracture. The analysis of the prefracture EM time series reveals that the cumulative distribution of the amplitudes also follows a power law with exponent *b* = 0.62. A similar statistical analysis of kHz EM precursor associated with the Athens EQ reveals that this also follows the power law *<sup>N</sup>*(<sup>&</sup>gt; *<sup>A</sup>*) <sup>∼</sup> *<sup>A</sup>*−*<sup>b</sup>* , where *<sup>b</sup>* <sup>=</sup> 0.62 [78].

In seismology, a well known scaling relation between magnitude and the number of EQs is given by the Gutenberg-Richter (G-R) relationship:

$$
\log N(>M) = \alpha - bm \tag{7}
$$

where, *N*(> *M*) is the cumulative number of EQs with a magnitude greater than *M* occurring in a specified area and time and *b* and *α* are constants.

Importantly, the Gutemberg-Ricther law also holds for acoustic emission events in rock samples [79]. Laboratory experiments by means of acoustic emissions also show a significant decrease in the level of the observed b-values immediately before the global fracture [79]. Characteristically, Ponomarev et al. [80] have reported a significant fall of the observed b-values from ∼ 1 to ∼ 0.6 just before the global rupture. Recently, Lei and Satoh [81], based on acoustic emission events recorded during the catastrophic fracture of typical rock samples under differential compression, suggest that the pre-failure damage evolution is characterized by a dramatic decrease in b-value from ∼ 1.5 to ∼ 0.5 for hard rocks. There are increasing reports on premonitory decrease of *b*-value before EQs: foreshock sequences and main shocks are characterized by a much smaller exponent compared to aftershocks [82]. We emphasise the sequence of kHz EM-EQs associated with the Athens EQ also follows the Gutenberg-Richter law with *b* = 0.51 [32].

The above mentioned results verify that the activation of a single fault behaves as a magnified self-affine image of the laboratory seismicity and reduced image of the regional seismicity.

#### **5.2 Signatures of fractional-Brownian-motion nature of faulting and fracture in the candidate kHz EM precursor**

Fracture surfaces were found to be self-affine following the fractional Brownian motion model (see Section 2.2.2) [27-30,35,36 and references therein]. This universal feature should be 12 Will-be-set-by-IN-TECH

detected precursory "EM-EQs" [32,33,36,75]. Herein, *N*(> *m*) is the number of "EM-EQs" with magnitude larger than *m*, *P*(> *m*) = *N*(> *m*)/*N* is the relative cumulative number of "EM-EQs" with magnitude larger than *m*, and *α* is the constant of proportionality between the EM energy released and the size of fragment. The best-fit *q* parameter for this analysis has

It is very interesting to observe the similarity in the *q*-values associated with the non-extensive Eq. (5) for: (i) seismicities generated in various large geographic areas, and (ii) the precursory sequence of "EM-EQs". This finding indicates that the statistics of regional seismicity could be merely a macroscopic reflection of the physical processes in the EQ source, as it has been

**5.1.2 The activation of a single fault as a "magnified self-affine image" of the laboratory**

Rabinovitch et al. [77] have studied the fractal nature of EM radiation induced in rock fracture. The analysis of the prefracture EM time series reveals that the cumulative distribution of the amplitudes also follows a power law with exponent *b* = 0.62. A similar statistical analysis of kHz EM precursor associated with the Athens EQ reveals that this also follows the power law

In seismology, a well known scaling relation between magnitude and the number of EQs is

where, *N*(> *M*) is the cumulative number of EQs with a magnitude greater than *M* occurring

Importantly, the Gutemberg-Ricther law also holds for acoustic emission events in rock samples [79]. Laboratory experiments by means of acoustic emissions also show a significant decrease in the level of the observed b-values immediately before the global fracture [79]. Characteristically, Ponomarev et al. [80] have reported a significant fall of the observed b-values from ∼ 1 to ∼ 0.6 just before the global rupture. Recently, Lei and Satoh [81], based on acoustic emission events recorded during the catastrophic fracture of typical rock samples under differential compression, suggest that the pre-failure damage evolution is characterized by a dramatic decrease in b-value from ∼ 1.5 to ∼ 0.5 for hard rocks. There are increasing reports on premonitory decrease of *b*-value before EQs: foreshock sequences and main shocks are characterized by a much smaller exponent compared to aftershocks [82]. We emphasise the sequence of kHz EM-EQs associated with the Athens EQ also follows the Gutenberg-Richter

The above mentioned results verify that the activation of a single fault behaves as a magnified self-affine image of the laboratory seismicity and reduced image of the regional seismicity.

Fracture surfaces were found to be self-affine following the fractional Brownian motion model (see Section 2.2.2) [27-30,35,36 and references therein]. This universal feature should be

**5.2 Signatures of fractional-Brownian-motion nature of faulting and fracture in the**

log *N*(> *M*) = *α* − *bm* (7)

been estimated to be approximately 1.8 [32,33,36,75].

suggested by Huang and Turcotte [76].

*<sup>N</sup>*(<sup>&</sup>gt; *<sup>A</sup>*) <sup>∼</sup> *<sup>A</sup>*−*<sup>b</sup>* , where *<sup>b</sup>* <sup>=</sup> 0.62 [78].

given by the Gutenberg-Richter (G-R) relationship:

in a specified area and time and *b* and *α* are constants.

**seismicity**

law with *b* = 0.51 [32].

**candidate kHz EM precursor**

included into an kHz EM precursor. If a time series is a temporal fractal then a power-law of the form *S*(*f*) ∝ *f* <sup>−</sup>*<sup>β</sup>* is obeyed, with *S*(*f*) the power spectral density and *f* the frequency. The spectral scaling exponent *β* is a measure of the strength of time correlations. The goodness of the power-law fit to a time series is represented by a linear correlation coefficient, *r*. Based on a fractal spectral analysis, which has been performed by means of wavelets, it has been shown [27-30,35,36] that the emergent strong kHz EM precursors follow the law *S*(*f*) ∝ *f* <sup>−</sup>*β*; the coefficient *r* takes values very close to 1, i.e., the fit to the power-law is excellent. This result shows the fractal character of the underlying processes and structures.

The *β* exponent takes high values, i.e., between 2 and 3. This finding reveals that:


In summary, the fBm nature of faulting and fracture is included in the kHz EM precursors.

#### **5.2.1 Persistent behaviour of the detected kHz EM precursors**

The *β* exponent is related to the Hurst exponent *H* by the formula [83, 84]:

$$
\beta = 2H + 1\tag{8}
$$

with 0 < *H* < 1 (1 < *β* < 3) for the fractional Brownian motion (fBm) model. The exponent *H* characterizes the persistent / anti-persistent properties of the signal. The range 0.5 < *H* < 1 (2 < *β* < 3) indicates persistency, which means that if the amplitude of the fluctuations increases in a time interval it is likely to continue increasing in the next interval. We recall that we found *β* values in the candidate EM precursors to lie between 2 and 3. The *H* values are close to 0.7 in the strong segments of the kHz EM activity [27-31,35,36]. This means that the EM fluctuations are positively correlated: the underlying dynamics is governed by a positive feedback mechanism. External influences would then tend to lead the system out of equilibrium. The system acquires a self-regulating character and, to a great extent, the property of irreversibility, one of the important components of prediction reliability. Sammis and Sornette [85] have recently presented the most important positive feedback mechanisms.

power-law type increase [27,28,78]. This experimental fact supports the hypothesis that both the seismicity and the preseismic EM activity represent two cuts in the same underlying fracture mechanism. Moreover, the spectral scaling exponent *β* (see Section 5.2) is a measure of the strength of time correlations. The *β*-values are significantly shifted to higher values as the EQ is approaching [27,28,78], namely, the correlation length in the time series increases as the catastrophic event approaches. Consequently, the two basic signatures predicted by the

Are There Pre-Seismic Electromagnetic Precursors? A Multidisciplinary Approach 231

**7. Interpretation of MHz-kHz EM precursors in terms of fractal electrodynamics**

Recently, the research area known as "fractal electrodynamics" has been established. This term was first suggested by Jaggard [44,45] to identify the newly emerging branch of research, which combines fractal geometry with Maxwel's theory of electrodynamics. From the laboratory scale to the geophysical scale, fault displacements, fault and fracture trace length, and fracture apertures follow a power-law distribution. Thus a fault shows a fractal pattern: a network of line elements having a fractal distribution in space is formed as the event approaches. However, an active crack or rupture can be simulated as a radiating element. The idea is that a *Fractal EM Geo-Antenna* can be formed as an array of line elements having a fractal distribution on the ground surface as the significant EQ is approached. This idea has been tested in [27]: the precursors are governed by characteristics (e.g., scaling laws, temporal evolution of the spectrum content, broad-band spectrum region, and accelerating emission rate) predicted by fractal electrodynamics. Notice, the fractal tortuous structure can significantly increase the radiated power density, as compared to a single dipole antenna. The tortuous path increases the effective dipole moment, since the path length along the emission is now longer than the Euclidean distance, and thus the possibility to capture these preseismic

The fractal dimension of the *fractal EM geo-antenna* associated with the Athens EQ is *D* = 1.2 [27]. Seismological measurements as well as theoretical studies [101,102 and references therein] suggest that a surface trace of a single major fault might be characterized by *D* = 1.2. We clarify that the exponent *D* does not describe the geometrical setting of the rupture faults but it only gives the distribution of rupture fault lengths irrespective of their positions. More information is needed for a full geometrical interpretation of the faults, e.g. the position of the

**8. The science of EQ prediction should, from the start, be multi-disciplinary**

As it was mentioned in Introduction, EQ's preparatory process has various facets which may be observed before the final catastrophe, thus, a candidate preseismic EM activity should be consistent with other EM precursors or precursors that are imposed by data from other disciplines (Seismology, Infrared Remote Sensing, Synthetic Aperture Radars Interferometry

A well documented type of precursory signals is the so-called seismic electric signals (SES) [103]. They are transient low frequency (< 1*Hz*) electric signals and are consistent with the "pressure stimulated currents model", which suggests that, upon a gradual variation of the pressure (stress) on a solid, transient electric signals are emitted, from the reorientation of

IC-model are included in the candidate kHz EM precursors.

radiations by aerial antennas.

**8.1 Seismic electric signals**

rupture centers.

e.t.c.).

It is expected that a positive feedback mechanism results in a finite-time singularity. The kHz EM time series under study shows such a behaviour by means of the "cumulative Benioff type EM energy release". A clear finite-time singularity of this type has been reported in [27,28,78].

*Remark*: The estimated Hurst exponents through the R/S analysis are in harmony with those estimated from the fractal spectral analysis via the hypothesis that the time series follows the fBm-model [35,36]. This fact supports the hypothesis that the profile of kHz EM precursors follow the persistent fBm-model. The last hypothesis has been further verified by a DFA-analysis [35,36].

#### **5.3 Footprints of universal roughness value of fracture surfaces in the kHz EM activity**

The Hurst exponent, *H*, specifies the strength of the irregularity ("roughness") of the fBm surface topography: the fractal dimension is calculated from the relation *D* = (2 − *H*) [83].

The Hurst exponent *H* ∼ 0.7 has been interpreted as a universal indicator of surface fracture, weakly dependent on the nature of the material and on the failure mode [86-90]. Importantly, the surface roughness of a recently exhumed strike-slip fault plane has been measured by three independent 3D portable laser scanners [91]. Statistical scaling analyses show that the striated fault surface exhibits self-affine scaling invariance that can be described by a scaling roughness exponent, *H*<sup>1</sup> = 0.7 in the direction of slip. In Section 5.2.1 we showed that the "roughness" of the profile of the kHz EM precursors, as it is represented by the Hurst exponent, is distributed around the value 0.7. This result has been verified by means of both fractal spectral analysis and R/S analysis [35,36]. Thus, the universal spatial roughness of fracture surfaces nicely coincides with the roughness of the temporal profile of the recorded kHz EM precursors.
