# Hazard Model: Epidemic-Type Aftershock Sequence (ETAS) Model for Hungary
## 3 Epidemic-Type Aftershock Sequence (ETAS) Modelling Methods
### Model Formulation
Furthermore, due to the heavy tail of the power law function, supercritical parameter fitting outcomes, which imply an unstable process, are sometimes obtained when the modified Omori law is used in an epidemic-type model [18]. The minimum delay parameter \(\Delta t_{0}\) in Equation (5) is motivated by experience with stochastic simulations based on the model. This truncation was introduced to avoid modelling an unrealistically high number of aftershocks immediately after the trigger event, which were never observed or identified as separate events in actual measurements since the point process model reaches its limits at very short time intervals. Regarding the space response function, observed aftershock sequences often show anisotropic patterns. Nonetheless, isotropic functions are widely used in ETAS models as a simplification. Both the power law function, e.g., in the form \(f(x,y|M)\propto\left[\left(x^{2}+y^{2}\right)/e^{\alpha(M-m_{0})}+d\right]^{-q}\) and the bivariate normal distribution have been discussed in the literature with a general preference for the power law based on goodness-of-fit diagnostics [6; 8]. The Gaussian form is chosen in this study mainly for its analytical simplicity and ease of implementation in the model. Since the exponential scaling parameter \(\alpha\) is the same in Equations (3) and (6), the space response formula assumes that the aftershock area grows in proportion to the number of events triggered, which is in line with the observed trend known as the Utsu-Seki law [6; 27]. This constraint from early ETAS model variants has been challenged in the literature, raising the suggestion for a separate spatial scaling parameter \(\gamma\)[8; 11]. However, due to the small number of major clusters available in the Hungarian catalogue, it is preferred to keep the number of fitted parameters as low as possible. Finally, for the moderately strong Hungarian events in the study period, a weak spatial scaling trend between trigger event magnitudes and aftershock radii is observed. It is assumed that the underlying effect is the epicenter error, which becomes a dominant factor at low magnitudes; hence, the motivation to include the \(\varepsilon\) parameter. We note that the formulation in Equation (6) is a simplified reflection of epicenter error when the space response function \(f\) is centred on the observed epicenters rather than the underlying ones. In a precise mathematical model, the conditional distributions of the error vectors would be neither isotropic nor independent. When an isotropic space-time model is fitted to non-isotropic aftershock data, a typical impact is a downward bias of the \(\alpha\) parameter; an indication of this effect is a significant gap between the \(\alpha\) values estimated from space-time versus time-only models [18; 28]. This effect is avoided by applying the constraint \(\alpha=\beta\), motivated by the assumption of self-similarity: when the magnitude threshold of the model is shifted from \(m_{0}\) to \(m_{1}\), the \(K\) parameter is rescaled to \(K_{1}=K_{0}\cdot e^{(\alpha-\beta)(m_{1}-m_{0})}\) while ignoring the upper magnitude bound for simplicity; therefore, the model parameters are almost unchanged across magnitude scales if \(\alpha=\beta\) except in the vicinity of the upper bound \(m_{x}\). Such a model also preserves [PERSON]'s law, i.e., the observation that the average magnitude difference between a trigger event and its biggest aftershock appears to be invariant [9; 14; 18] (S1.2.1) [29]. The branching ratio of the model characterizing the stability of the process is defined as the average number of the first-generation descendants triggered by an event and expressed as [8; 18]:  
\[\varrho=\int_{m=m_{0}}^{m_{x}}\int_{t=0}^{\infty}K\cdot e^{\alpha(m-m_{0})} \cdot g(t)\cdot f(m)\ dt\ dm. \tag{7}\]  
The process is subcritical (stable) if \(\varrho<1\), it is critical (unstable) if \(\varrho=1\), and it is supercritical (unstable, potentially explosive) if \(\varrho>1\).