# Hazard Model: Epidemic-Type Aftershock Sequence (ETAS) Model for Hungary
## 3 Epidemic-Type Aftershock Sequence (ETAS) Modelling Methods  
### Model Formulation  
The ETAS model describes earthquake occurrences as a non-stationary Poisson point process where every event can trigger a wave of aftershocks. In a time-only model, the rate of occurrence at time \(t\), conditional on the process history \(H_{t}\), is expressed as [5]  
\[\lambda(t|H_{t})=\mu+\sum_{j:\ t_{j}<t}K\cdot e^{\alpha(M_{j}-m_{0})}.g\big{(}t- t_{j}\big{)}, \tag{2}\]  
which is extended to a space-time model as [6]  
\[\lambda(t,x,y|H_{t})=\mu\cdot u(x,y)+\sum_{j:\ t_{j}<t}K\cdot e^{\alpha(M_{j}- m_{0})}.g\big{(}t-t_{j}\big{)}\cdot f\big{(}x-x_{j},y-y_{j}|M_{j}\big{)}, \tag{3}\]  
where the first term represents background seismicity with a constant overall rate \(\mu\) and area density \(u(x,y)\), and the sum in the second term reflects the rate of activity triggered by events in the catalogue prior to time \(t\). An exponential relationship is assumed between the trigger event magnitude \(M_{j}\) and the triggered rate of activity, where \(K\) and \(\alpha\) are constant parameters and \(m_{0}\) is the magnitude threshold. The functions \(g\) and \(f\) describe the form of the time and space response, respectively, relative to the trigger event occurrence time and epicenter. The process history \(H_{t}=\big{\{}\big{(}t_{j},x_{j},y_{j},M_{j}\big{)}:t_{j}<t\big{\}}\) is the catalogue of events before \(t\), defined by their occurrence time, location, and magnitude. Depth is not included in this ETAS model as a third coordinate, and the question of the depth distribution is not discussed in this article. This is a reasonable simplification since all events in the study catalogue were shallow crustal earthquakes and all except nine of them had a focal depth of less than 20 km. It was assumed that event magnitudes follow a truncated exponential (Gutenberg-Richter) distribution independent of the process history [8; 13]. The probability density function (PDF) of magnitudes between the bounds \(m_{0}\) and \(m_{x}\) is given by the following:  
\[J(m)=\frac{\beta\cdot e^{-\beta(m-m_{0})}}{1-e^{-\beta(m_{x}-m_{0})}}, \tag{4}\]  
where \(\beta\) is the frequency decay parameter. According to the magnitude independence assumption, an aftershock can be larger than its trigger event. For this study, we used a stretched exponential function truncated for a minimum delay as the time response:  
\[g(t)= \begin{cases}t^{q-1}\cdot\exp(-\eta t^{q})&\text{if }t>\Delta t_{0},\\ 0&\text{if }t\leq\Delta t_{0}\end{cases}  \tag{5}\]  
where \(\Delta t_{0}\) is a minimum delay parameter, and the normal distribution with an allowance for epicenter error is the space response function:  
\[ \begin{array}{c}f(x,y|M)=\frac{1}{2\pi\sigma^{2}}\exp\big{[}-\big{(}x^{2}+y ^{2}\big{)}/2\sigma^{2}\big{]},\\ \text{with }\sigma^{2}=D^{2}\cdot e^{\alpha(M-m_{0})}+\varepsilon^{2}.\end{array}  \tag{6}\]  
where \(D\) is the scale parameter of the magnitude-dependent part of the variance, and \(\varepsilon\) is a parameter representing a variance add-on for epicenter error. Since both Equations (5) and (6) represent a departure from the mainstream techniques, we explain the rationale of these approaches below. The most often used time response function is the modified Omori law [27], which is a power law function in the form of \(g(t)=(t+c)^{-p}\). The stretched exponential alternative has been suggested by [PERSON] [12], who, besides quantitative tests showing that the stretched exponential form is better fitted to the observations from several regions than the power law, also expressed qualitative concerns about the modified Omori law. Firstly, the positive time shift constant \(c\) avoids the singularity at zero, but it is difficult to find aphysical explanation for it. Secondly, a parameter value of \(p\leq 1\) would mean an infinite number of triggered events unless the process is artificially truncated in time.