Based on established experimental seismological criteria, the ETAS model provides a reliable estimate of future events in the seismic zone. In this model, using the catalogue of earthquakes and valid seismological hypotheses, standard functions are calculated to identify background earthquakes and the cluster earthquakes. They are then structured to obtain comprehensible diagrams. Different hypotheses have been investigated and accepted in seismology and are also compatible with the condition of the seismic zone. These empirical laws are used within the framework of disaster management and the significant challenges of modern seismology. In seismology, each of the experimental laws has a specific application. With the logical combination of these rules and the specific calculations at the end, the ETAS model also provides an estimate of future events that is also consistent with the observed facts (Zhuang et al. 2002, 2004; Zhuang and Ogata 2004; and Ogata 1988, 1989, 1992).

The ETAS model can be considered an embodiment of the total seismicity estimation in a region formed by the simultaneous estimates of three empirical laws; these laws include Omori, Gutenberg-Richter, and a law that describes how ruptures grow on faults. In fact, the future estimate of seismicity is based more on the analysis of past events (Helmstetter and Sornette 2002; Zhuang et al. 2002; Guo et al. 2015; Ogata 2013; Nanjo et al. 2011, 2012).

The researchers try to use the ETAS model to recognize different space-time-dependent seismicity patterns formed by aftershocks in an area. These sequences generally occur following the occurrence of background earthquakes, and the common denominator of all these models is the application of Gutenberg-Richter experimental law to determine the magnitude factor. In ETAS model, seismic events are classified into two classes: background earthquakes and cluster earthquakes. The identification of aftershock sequences as a Poisson point process or Hawkes process model guarantees the results obtained from the model. In this process, the seismic zone is impacted by aftershock sequences resulting from earthquakes of constant magnitude. In other words The only way to understand the prediction of earthquakes in a seismic zone by ETAS model is through drawings, diagrams, calculations, plans, and tolerances (Kolev and Ross 2019; Zhuang et al. 2002; Zechar et al. 2010).

Current knowledge is far from having a full understanding of the phenomenology of earthquakes. At the same time, with the help of various technical methods and achievements of modern technology, advanced knowledge has been obtained in the instrumental recording of events as well as the exploration of the background earthquakes and cluster earthquakes in a seismic zone. Although this knowledge is behind the revelation of the earthquake event; but it is very important in terms of its impact in providing a statistical set analysis set of events in a seismic zone (Kagan 1991; Musmeci and Vere-Jones 1992; Rathbun 1993; Ogata 1998).

The statistical analyses carried out using the catalogue of earthquakes without having to calculate the details of this phenomenon, increases the level of knowledge of the seismic behavior in an area. The use of these methods for quantifying how earthquakes occur is an invaluable advantage. Then Omri and Gutenberg-Richter laws are among the oldest known experimental laws that have been used in the statistical analysis of earthquakes; the two laws that form the basic structure of the ETAS model and also provide the scientific support for this model (Kagan 1991; Musmeci and Vere-Jones 1992; Rathbun 1993; Ogata 1998).

In the ETAS model, the standard statistical and probability functions are calculated as a conditional intensity function. This is done to simulate and obtain an overview of the existing dependency between future and past events. In these calculations, the conditional intensity function is considered as a time-space dependent point process as Eq. (1) (Daley and Vere-Jones 2003):

$${\lambda }\left(t,x,y|{\mathcal{H}}_{t}\right)\underset{\varDelta t,\varDelta \text{x},\varDelta \text{y}\to 0}{\text{lim}}\frac{{p}_{\varDelta t,\varDelta x,\varDelta y}(t,x,y|{\mathcal{H}}_{t})}{\varDelta t\varDelta x\varDelta y} \left(1\right)$$

In further studies by Ogata, while developing the model within the Hawks model point process is shown; normally seismic behavior in an area can be obtained from the sum of the constant intensity of the background earthquakes and the rate of cluster earthquakes (Eq. 2):

$${\lambda }\left(\text{t},\text{x},\text{y}|{\mathcal{H}}_{\text{t}}\right)={\mu }\left(\text{x},\text{y}\right)+\sum _{\left\{\text{t}:{\text{t}}_{\text{i}}<\text{t}\right\}}\text{g}(\text{t}-{\text{t}}_{\text{i}},\text{x}-{\text{x}}_{\text{i}},\text{y}-{\text{y}}_{\text{i}};{\text{M}}_{\text{i}}\left) \right(2)$$

The first part of Eq. (2) includes probabilistic calculations of the occurrence of background earthquakes. In other words, this term is responsible for estimating the probability of the occurrence of future strong earthquakes; so that with the occurrence of background earthquakes in this area, the creation of aftershock sequences in the seismic zone will not be unexpected. Calculations related to estimating the number and intensity of these sequences also followed in the second semester of Eq. (2). The calculations examine the probabilities of the rate of events due to the stress-induced instabilities caused by the background earthquakes and, at the same time represent a fundamental principle of the ETAS model. This principle states that each background earthquake, depending on its size (or magnitude), can be the source of aftershock sequences within the seismic zone and the aftershock sequences are also responsible for Power Reduction at a given time according to Omri's law.

The conditional intensity function in Eq. (2) is the expression of the behavior of an N-point process modeled as λ. This conditional the rate of is conditional on the background earthquakes of the point process and can be used to estimate the recurrence of future events.

Perform statistical mathematical calculations, details of which are available in (Ogata 1985, 1988, 1989, 1992; Zhuang et al. 2002, 2004; Zhuang and Ogata 2004); Eq. (2) is delivered to Eq. (3) and then Eq. (4):

$${\lambda }\left(\text{t},\text{x},\text{y},\text{M}\right){\lambda }\left(\text{t},\text{x},\text{y}\right)\text{j}\left(\text{M}\right) \left(3\right)$$

$$\lambda \left(\text{t},\text{x},\text{y}\right)={\mu }\left(\text{x},\text{y}\right)+\sum _{\text{i}:{\text{t}}_{\text{i}}<\text{t}}{\kappa }\left({\text{M}}_{\text{i}}\right)\text{g}\left(\text{t}-{\text{t}}_{\text{i}}\right)\text{f}\left(\text{x}-{\text{x}}_{\text{i}},\text{y}-{\text{y}}_{\text{i}}|{\text{M}}_{\text{i}}\right) \left(4\right)$$

Five significant and influential factors in the seismicity of an area structured in the ETAS model: include background seismicity (µ), the expected number of events triggered by a background earthquake of magnitude M (κ (M)), the probability density function (pdf) form of the modified Omori law (g (t)), location distribution of the triggered events, (f ( x, y, M), and the probability density of magnitudes of all the events, independent from other components and taking the form of the Gutenberg- Richter law (J (M)) empirical law, each of which is initially functioned by complex statistical mathematical calculations. Probabilities become standard, quantitative, and countable. These five components are then logically formulated together in a composite structure to present the final result in the form of intelligible graphs of time and space so that the probabilistic zone of the background earthquakes, as well as the aftershock sequences, are determined (Zhuang et al. 2002, 2005, 2011, 2012; Zhuang and Ogata 2004; Ogata 1988, 1992).