# The Relevance of Foreshocks in Earthquake Triggering: A Statistical Study

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## Abstract

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## 1. Introduction

- A1: The number of aftershocks ${n}_{a}$ depends on the mainshock magnitude ${m}_{M}$ according to the productivity law ${n}_{a}={K}_{a}{10}^{{\alpha}_{a}{m}_{M}}$;
- A2: The aftershock number decays as function of the time $\Delta t$ from the mainshock, consistently with the Omori law ${n}_{a}(\Delta t)\sim \Delta {t}^{-p}$ with $p\simeq 1$;
- A3: The distribution of epicentral distances between mainshock and aftershocks $G(\Delta r,{m}_{M})$ clearly depends on the mainshock magnitude ${m}_{M}$.

- F1: The average foreshock number in instrumental catalogs is significantly larger than the one expected according to the ETAS model;
- F2: The organization in space of instrumental foreshocks exhibit a dependence on the mainshock magnitude not predicted by the ETAS model.

## 2. Epidemic Models for Aftershocks and Foreshock Occurrence

#### 2.1. The ETAS Model

#### 2.2. The ETAS Incomplete Catalog

#### 2.3. The ETAFS Model

## 3. Results in the Instrumental Catalogs

#### 3.1. Data Sets and the Definitions of Mainshocks, Aftershocks and Foreshocks

#### 3.2. The Aftershock and Foreshock Number

#### 3.3. Aftershock and Foreshock Spatial Distribution

## 4. Results in Numerical Catalogs

#### 4.1. Results in the ETAS Catalog

#### The Aftershock and Foreshock Number in the ETAS Catalog

#### 4.2. Aftershock and Foreshock Spatial Distribution in the ETAS Catalog

#### 4.3. Results in the ETASI2 Catalog

#### 4.4. Results in ETAFS Catalogs

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**(Color online) The ratio ${n}_{aft}\left({m}_{M}\right)/{n}_{main}\left({m}_{M}\right)$ and ${n}_{fore}\left({m}_{M}\right)/{n}_{main}\left({m}_{M}\right)$ in instrumental and synthetic catalogs. Different panels correspond to different instrumental catalogs. We use open symbols for ${n}_{aft}\left({m}_{M}\right)/{n}_{main}\left({m}_{M}\right)$ and filled symbols for ${n}_{fore}\left({m}_{M}\right)/{n}_{main}\left({m}_{M}\right)$. Results from the instrumental data sets are indicated with black circles. Green triangles are results for the ETASI2 model and red squares for the ETAFS model. The error bars (of the same size of symbols) in numerical catalogs represent the standard deviation from 100 realization of synthetic catalogs. The best parameter of the ETAFS model are listed in Table 1 whereas for the ETASI2 model the best agreement is obtained with $A=0.084$, $\alpha =0.9$ and $\mu =5.85\times {10}^{-4}{\mathrm{s}}^{-1}$ for RSCEC, $A=0.082$, $\alpha =0.88$ and $\mu =4.98\times {10}^{-4}{\mathrm{s}}^{-1}$ for RNCEC, $A=0.082$, $\alpha =0.88$ and $\mu =5.21\times {10}^{-4}{\mathrm{s}}^{-1}$ for ItEC and $A=0.26$, $\alpha =0.60$ and $\mu =5.92\times {10}^{-3}{\mathrm{s}}^{-1}$ for JapEC.

**Figure 2.**(Color online) The average distance ${\zeta}_{a}(\Delta r,{m}_{M})$ of aftershocks (open symbols) and of foreshocks ${\zeta}_{f}(\Delta r,{m}_{M})$ (filled symbols) is plotted as function of $\Delta r$ for the different catalogs. Different mainshock magnitude classes are plotted with different colors and symbols.

**Figure 3.**(Color online) The ratio ${n}_{fore}\left({m}_{M}\right)/{n}_{aft}\left({m}_{M}\right)$ in the RSCEC catalog (black stars) is compared with the value obtained in synthetic ETAS (upper panel), ETASI2 (central panel) and ETAFS (lower panel) catalog. (Upper panel) The red open symbols are results for the ETAS model for different choices of the parameters $A\in [0.05,0.12]$, $p\in [1.1,1.25]$ and $c\in [0.001,0.1]$. (Central Panel) The blue filled symbols are results for the ETASI2 model implementing Equation (3) with $\varphi =0.75$ and $\Delta m=0.8$ and for different choices of the parameters $A\in [0.05,0.12]$, $p\in [1.1,1.25]$ and $c\in [0.001,0.1]$. Green symbols correspond to $A=0.084$, $c=0.01$ and $p=1.2$ used in Figure 1 for the RSCEC catalog and considering different values of $\varphi $, $\Delta m$ and $\sigma $. (Lower panel) The filled magenta up triangles are results of the ETAFS model with the best set of parameters listed in Table 1.

**Figure 4.**(Color online) (Left panel) The average distance of aftershocks ${\zeta}_{a}(\Delta r,{m}_{M})$ in the RSCEC (open symbols) and in the synthetic ETASI2 catalogs (continuous lines) is plotted as function of $\Delta r$ for different mainshock magnitude classes. (Right Panel) The average distance of foreshocks ${\zeta}_{f}(\Delta r,{m}_{M})$ in the RSCEC (filled symbols) and in the synthetic ETASI2 catalog (continuous lines) is plotted as function of $\Delta r$ for different mainshock magnitude classes. Results for the EATFS model, for the best set of model parameters listed in Table 1, are plotted with crosses.

Catalog | A | $\mathit{\alpha}$ | B | ${\mathit{\alpha}}_{\mathit{f}}$ | $\mathit{\mu}$ s^{−1} |
---|---|---|---|---|---|

RSCEC | 0.084 | 0.9 | 0.050 | 0.54 | 5.84 × 10^{−4} |

RNCEC | 0.082 | 0.88 | 0.033 | 0.59 | 4.98 × 10^{−4} |

ItEC | 0.086 | 0.88 | 0.052 | 0.60 | 5.21 × 10^{−4} |

JapEC | 0.234 | 0.6 | 0.160 | 0.36 | 5.92 × 10^{−3} |

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**MDPI and ACS Style**

Lippiello, E.; Godano, C.; de Arcangelis, L.
The Relevance of Foreshocks in Earthquake Triggering: A Statistical Study. *Entropy* **2019**, *21*, 173.
https://doi.org/10.3390/e21020173

**AMA Style**

Lippiello E, Godano C, de Arcangelis L.
The Relevance of Foreshocks in Earthquake Triggering: A Statistical Study. *Entropy*. 2019; 21(2):173.
https://doi.org/10.3390/e21020173

**Chicago/Turabian Style**

Lippiello, Eugenio, Cataldo Godano, and Lucilla de Arcangelis.
2019. "The Relevance of Foreshocks in Earthquake Triggering: A Statistical Study" *Entropy* 21, no. 2: 173.
https://doi.org/10.3390/e21020173