# On the Omori Law in the Physics of Earthquakes

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## Abstract

**:**

## 1. Introduction

## 2. Elementary Master Equation

## 3. Logistic Equation

## 4. Stochastic Equation

## 5. Nonlinear Diffusion Equation

## 6. Discussion

#### 6.1. Inverse Problem

_{0}, has been reliably established. Figure 3 shows the result of measurements of σ at different values of M

_{0}. To measure the deactivation factor, we used the USGS/NEIC earthquake catalog and the technique developed during the compilation of the Atlas of Aftershocks [20]. We see that, on average, σ decreases monotonically with the increase in M

_{0}. The dependence σ(M

_{0}) is approximated by the formula

^{2}= 0.82. Thus, the theoretical inequality dσ/dM

_{0}is reliably confirmed by direct measurements. We have a wonderful harmony between theory and experiment.

#### 6.2. Triggers

#### 6.3. Triads

_{0}is always greater than the maximum magnitudes of foreshocks and aftershocks. The classical triad satisfies the inequalities

**Mirror triads.**Extensive literature is devoted to the experimental study of the classical triads. We point here to work [21], since in the study of anomalous triads we used a database and general methods of analysis similar to those used here in the study of classical triads (see also [9]).

**GTS.**So, we found that there is a rare but rather interesting subclass of tectonic earthquakes, in which the number of aftershocks in the interval of 24 h after the main shock is significantly less than the number of foreshocks in the same interval before the main shock. In many cases, there are no aftershocks at all. We asked the question: Are there earthquakes with magnitudes M

_{0}≥ 6, neither before nor after which there are neither foreshocks nor aftershocks? The search result was amazing. We have discovered a wide variety of this kind of earthquake and named it Grande terremoto solitario (Italian), or GTS for short [47]. In Figure 7, we see that the number of GTS (2460) is approximately equal to the number of classical triads (2398).

_{0}≥ 6 and ${N}_{+}={N}_{-}$: ${N}_{-}=186$, ${N}_{0}=121$, ${N}_{+}=186$. It is interesting to note that, formally, GTS can be related to a variety of symmetric triads, since for them ${N}_{+}={N}_{-}=0$.

**Activation factor.**Figure 5 and Figure 6 shows that foreshocks in the mirror triad appear to have a temporal distribution similar to the Omori distribution for aftershocks in the classical triad. Let us dwell on this in more detail. We represent the classical Omori law [1] in the simplest differential form

_{0}< 6. Here ${N}_{-}=1050$, ${N}_{0}=742$, ${N}_{+}=1050$. The top panel shows an amazing mirror image. In the bottom panel, we have shown the variations of the ${\sigma}_{-}$ and ${\sigma}_{+}$ functions as the first step towards studying the activation and deactivation coefficients of an earthquake source in mirror triads. (For the procedure for calculating ${\sigma}_{\pm}$, see [8]).

**Origin of mirror triads.**In conclusion of this section, we would like, with all the necessary reservations, to express a careful judgment on the question of the origin of mirror triads. Let us assume that a system of faults in a certain volume of rocks is under the influence of a slowly growing total shear stress τ. Threshold tension ${\tau}_{*}$ at which destruction occurs, i.e., the sides of the fault shift and a rupture occurs, generally speaking, is the lower, the larger the linear dimensions of the fault l:

_{0}.

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Phase portrait on the phase plane of Equation (6). The red, green, and blue phase trajectories are plotted at ${X}_{\infty}=0,0.5\mathrm{and}1$, respectively (see text).

**Figure 2.**Logistic curve (on left) and aftershocks curve (on right) (second branch of logistic equation) at ${X}_{\infty}=0.2$. Dimensionless time $T=\gamma t$ is plotted along the horizontal axis.

**Figure 3.**Dependence of the deactivation factor of the earthquake source on the magnitude of the main shock. Other explanations see in the text below.

**Figure 4.**Generalized picture of a shortened classical triad. Zero moment of time corresponds to the moment of the main shock.

**Figure 5.**Generalized view of truncated mirror triads. Zero moment of time corresponds to the moment of the main shock. The red line is obtained by averaging over a sliding window of 20 min, with the step 1 min.

**Figure 6.**Time distribution of foreshocks and aftershocks of mirror triads in the range of magnitudes of the main shocks 5 ≤ M

_{0}< 6.

**Figure 8.**Time dependence of the properties of symmetric triads. From top to bottom: earthquake frequency, activation (blue), and deactivation (red) factors.

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**MDPI and ACS Style**

Zavyalov, A.; Zotov, O.; Guglielmi, A.; Klain, B.
On the Omori Law in the Physics of Earthquakes. *Appl. Sci.* **2022**, *12*, 9965.
https://doi.org/10.3390/app12199965

**AMA Style**

Zavyalov A, Zotov O, Guglielmi A, Klain B.
On the Omori Law in the Physics of Earthquakes. *Applied Sciences*. 2022; 12(19):9965.
https://doi.org/10.3390/app12199965

**Chicago/Turabian Style**

Zavyalov, Alexey, Oleg Zotov, Anatol Guglielmi, and Boris Klain.
2022. "On the Omori Law in the Physics of Earthquakes" *Applied Sciences* 12, no. 19: 9965.
https://doi.org/10.3390/app12199965