# Geosystemics View of Earthquakes

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Geosystemics

^{15}Joule is released in some seconds), lightning strikes in the atmosphere (around 10

^{9}J in microseconds), etcetera. For instance, the information exchanged between contiguous parts of the Earth system producing increased entropy would allow us to better recognize and understand those irreversible processes occurring in the Earth’s interior. As said by [21], “geosystemics has the objective to observe, study, represent and interpret those aspects of geophysics that determine the structural characteristics and dynamics of our planet and the complex interactions of the elements that compose it” by means of some entropic measures.

## 3. Main Seismological Diagnostic Tools

- Present a physical model that can explain the proposed precursor anomaly.
- Exactly define the anomaly and describe how it can be observed.
- Explain how precursory information can be translated into a forecast and specify such a forecast in terms of probabilities for given space/time/magnitude windows.
- Perform a test over some time that allows us to evaluate the proposed precursor and its forecasting power.
- Report on successful prediction, missed earthquakes, and false predictions.

#### 3.1. M8

#### 3.2. The Reverse Tracing of Precursors (RTP)

#### 3.3. Pattern Informatics (PI)

## 4. Shannon Entropy and Shannon Information

## 5. Gutenberg-Richter Law and b-Value

_{min}is the minimum magnitude used in the b-value evaluation; Δ is the resolution involved in the magnitude estimation, normally Δ = 0.1. Usually, the M

_{min}is the magnitude of completeness of a seismic catalog, i.e., the magnitude threshold at which or above the corresponding seismic catalog includes all occurred EQs in the region.

## 6. Entropy and EQs

_{i}to have activated a certain i-th class of seismicity characterized by some range of magnitudes, the associated non-negative Shannon entropy h can be defined as [58]:

_{min}to infinity) then the discrete definition (4a) becomes an integral definition [30,59]:

## 7. Entropy and Critical Point Theory

_{λ}degree, when the system approaches the critical temperature as a power law. In addition, if the system changes its temperature linearly in time, the same plot is expected versus time [65].

_{1}is taken as an order parameter (e.g., [66]). As we defined (and applied) the Shannon entropy, we will show that it is a reliable parameter to characterize the critical point in both the two Italian case studies. It can be considered a parameter similar to other order parameters, with the difference the latter are usually approaching a minimum value while the Shannon entropy gets the largest one.

## 8. Entropy Studies of Two Italian Seismic Sequences

#### 8.1. The 2009 L’Aquila Seismic Sequence

_{t}= 2.5 σ (the mean value of entropy, <H>, is practically zero). To better visualize the mean behavior of entropy, the gray curve defines a reasonable smoothing of the entropy values: 15-point FFT before the main-shock and 50-point FFT smoothing after the main-shock. The different kind of smoothing is related to the different rate of seismicity before and after the main-shock. It is interesting to notice that the smoothed gray curve of the Shannon entropy reproduces the expected behavior of a critical system around its critical point, with the main-shock as a critical point.

#### 8.2. The 2012 Emilia Seismic Sequence

## 9. Accelerated Moment Release Revisited: The Case of L’Aquila and Emilia EQs

_{i:}

_{i}is the energy released by the EQ, i.e., 10

^{αM + β}(α = 1.5, β = 4.8 for energy expressed in Joule, although Benioff used slightly different values), and k

_{i}= (μPV

_{i}/2)

^{0.5}(μ = shear or rigidity modulus, V

_{i}= volume of the i-th fault rocks, P is the fraction of energy transmitted in terms of seismic waves; usually it is considered P ≈ 1). This theory is based on [72] arguments of the elastic rebound.

_{f}, i.e., the theoretic time of occurrence of the main shock: s(t) = A + B(t

_{f}− t)

^{m}, where A, B and m are appropriate empirical constants (m is expected between 0 and 1: typical value is 0.3; [37]). The fitting process gives as an outcome the time t

_{f}together with the expected magnitude, which is related to either A or B:

_{last}are the cumulative Benioff strain at the last precursory event considered (namely the N-th EQ). In this expression, one speculates that the main-shock will be the next EQ striking after the N-th, but the occurrence of many smaller EQs after the last analyzed shock and before the predicted time t

_{f}cannot be excluded.

^{0.43M}with M = EQ magnitude [16].

#### R-AMR for the 2009 L’Aquila and 2012 Emilia Seismic Sequences

_{i}) = d/R

_{i}

^{γ}, with d (normally 1km), R

_{i}in km and with γ ≈ 1, at the cost of considering a larger minimum magnitude threshold of around M4. Figure 5 and Figure 6 show the results for the cases of L’Aquila and Emilia sequences, respectively, where we apply to all shallow (depth h ≤ 40 and h ≤ 80 km, respectively) M4+ EQs both AMR and R-AMR analyses (top and bottom of each figure, respectively). Then, we consider a 300 km size for the regions where we applied R-AMR analysis. This size is comparable with the corresponding Dobrovolsky’s radius. Both the analyses stop well before the main-shocks that are not considered in the calculations. We notice that the time of preparation is rather long for both sequences, i.e., practically starting at the beginning of the whole period of investigation (May 2005). This fact could be simply interpreted as the larger foreshocks anticipate the beginning of the seismic acceleration with respect to the smaller ones, which were the most in the previous analyses in [38].

^{2}> 0.95 in both cases). In addition, the predicted magnitudes are comparable with (although lower than) the real ones. In both cases, the beginning of clear acceleration starts around 1.5 years before the main-shock.

## 10. Lithosphere-Atmosphere-Ionosphere Coupling (LAIC)

#### 10.1. Pre-EQ Ionospheric Evidences from Ground-Based Observations

^{−2}is required to produce the observed variation of ~3 TECU.

#### 10.2. Pre-EQ Ionospheric Evidence from In-Situ Measurements

**E×B**drifts responsible of the EIA.

#### 10.3. Pre-EQ Atmospheric Evidence

_{2}concentrations by more than one order of magnitude across Taiwan several hours prior to two (M6.8 and M7.2) significant EQs in the island.

#### 10.4. Physical Models

**Figure 8.**Freund model (adapted from [136]).

## 11. Examples of Thermal Coupling before L’Aquila and Emilia EQs

## 12. Mutual Information and Transfer Information: A Possible Future Direction

_{1}(x) and p

_{2}(y) are the corresponding probabilities and p(x,y) is the joint probability.

## 13. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Idealized Shannon entropy (above diagram) and a cumulative number of events (bottom diagram) for a dissipative system around its critical point, indicated by the vertical red line.

**Figure 2.**Shannon entropy for L’Aquila seismic sequence from around 1.5 years before the main-shock to around 1 year after, calculated for a circular area of 80 km around the main-shock epicenter. Each point is the entropy analysis based on non-overlapping windows, each composed by 30 foreshocks. The gray curve defines a reasonable smoothing of the entropy values: 15-point FFT before the main-shock and 50-point FFT smoothing after the main-shock. The different kind of smoothing is related to the different rate of seismicity before and after the main-shock. It is interesting how the smoothed curve reproduces the expected behavior of a critical system around its critical point. (Adapted from [67]).

**Figure 3.**Details of the Shannon entropy for L’Aquila seismic sequence from around 1.5 years before the main-shock to the main-shock occurrence. Each point is the entropy analysis based on non-overlapping windows, each composed by 30 foreshocks. The mean value of the entropy, <H>, which is almost zero, and one and two standard deviations are also shown. The gray curve defines a reasonable smoothing of the entropy values with 15-point FFT.

**Figure 4.**Shannon entropy for the Emilia seismic sequence from 2000 to 2014. The significant increase from around 2010, with the maximum at around the main-shock occurrence, is expected to be real. The gray area defines the statistically estimated (one standard deviation) error in computing the entropy.

**Figure 5.**Analyses of L’Aquila seismic sequence M ≥ 4 EQs (main-shock not shown and not used in the analysis): (top) ordinary AMR method; (bottom) R-AMR method. The dashed line represents the best linear fit, while solid gray curve is the best power law fit. Results of the fit are shown in the frame inside the graph at the bottom; r

^{2}is the coefficient of determination, providing a measure of the quality of the fit (the closer to 1, the better the fit).

**Figure 6.**Analyses of Emilia seismic sequence M ≥ 4 EQs (main-shock not shown and not used in the analysis): (top) ordinary AMR method; (bottom) R-AMR method. Here the ordinary AMR also showed a little acceleration (C-factor = 0.80) but the R-AMR version is much better (C-factor = 0.46). The dashed line represents the best linear fit, while the solid gray curve is the best power law fit. Results of the fit are shown in the frames inside the graphs; r

^{2}is the coefficient of determination, providing a measure of the quality of the fit (the closer to 1, the better the fit).

**Figure 9.**Enomoto model (adapted from [137]).

**Figure 10.**Median behavior of 2009 from 1 March to 30 April, compared with all 1979–2008 medians, and particular comparison with 2003 and 2005 medians. All values have been estimated at the epicenter. The red oval indicates when the thermal anomaly in 2009 is larger than or equal to 2 standard deviations, σ (as computed from the previous 1979–2008 years) and persists for at least two days. The vertical line is the EQ occurrence.

**Figure 11.**Median behavior of 2012 from 1 April to 31 May, compared with all 1979–2011 medians, and particular comparison with 2004 and 2006 medians. All values have been estimated at the epicenter. The red ovals indicate when the thermal anomaly in 2012 is larger than or equal to 2 standard deviations, σ (as computed from the previous 1979–2012 years) and persists for at least two days. The vertical line is the EQ occurrence.

**Table 1.**Main data related to the two Italian seismic sequences under study: (from left to right) the label, the main-shock source parameters, the number of data points (foreshocks) used in the fitting stage; the maximum distance from the main-shock epicenter defining the selection area and the minimum threshold magnitude of the selected events there considered. We provide also a rough estimation of the predicted magnitude (within brackets) of the impending main-shock (see text). N and R in the Fault style column stand for Normal, and Reverse focal mechanism, respectively. R

_{max}and M

_{min}are the largest area and minimum magnitude, respectively, considered in the analyses of R-AMR, while for the entropy analyses we considered always the completeness magnitude (M1.4 and M2 for L’Aquila and Emilia Earthquakes).

Sequence ID | L’Aquila | Emilia | |
---|---|---|---|

Main-shock Parameters | Coordinate (lat lon, in degree) | 42.34N 13.38E | 44.89N 11.23E |

Depth (km) | 8.3 | 6.3 | |

Date | 6 Apr 2009 | 20 May 2012 | |

${t}_{f}$ (in days from 1 May 2005) (predicted) | 1436.06 (1437.4) | 2576.09 (2577.7) | |

Fault style | N | R | |

Magnitude (predicted) * | 5.9 (5.3 ± 0.5) | 5.9 (5.7 ± 0.5) | |

# data (foreshocks) | 17 | 38 | |

R_{max} (km) | 300 | 300 | |

M_{min} (*) | 4.0 | 4.0 |

**Table 2.**Confusion matrix for pre-earthquake anomaly detection obtained from ionospheric anomalies analysis in Greece from 2003 to 2015 (adapted from [36]).

Ionospheric Anomaly | Seismicity | |
---|---|---|

Yes | No | |

Yes | 5 | 9 |

No | 5 | 26 |

**Table 3.**Confusion matrix for pre-earthquake anomaly detection obtained from skt time series analysis from 1994 to 2016 in Central Italy (adapted from [35]).

Skin Temperature Anomaly | Seismicity | |
---|---|---|

Yes | No | |

Yes | 2 | 3 |

No | 3 | 15 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

De Santis, A.; Abbattista, C.; Alfonsi, L.; Amoruso, L.; Campuzano, S.A.; Carbone, M.; Cesaroni, C.; Cianchini, G.; De Franceschi, G.; De Santis, A.;
et al. Geosystemics View of Earthquakes. *Entropy* **2019**, *21*, 412.
https://doi.org/10.3390/e21040412

**AMA Style**

De Santis A, Abbattista C, Alfonsi L, Amoruso L, Campuzano SA, Carbone M, Cesaroni C, Cianchini G, De Franceschi G, De Santis A,
et al. Geosystemics View of Earthquakes. *Entropy*. 2019; 21(4):412.
https://doi.org/10.3390/e21040412

**Chicago/Turabian Style**

De Santis, Angelo, Cristoforo Abbattista, Lucilla Alfonsi, Leonardo Amoruso, Saioa A. Campuzano, Marianna Carbone, Claudio Cesaroni, Gianfranco Cianchini, Giorgiana De Franceschi, Anna De Santis,
and et al. 2019. "Geosystemics View of Earthquakes" *Entropy* 21, no. 4: 412.
https://doi.org/10.3390/e21040412