# Hazard Model: Epidemic-Type Aftershock Sequence (ETAS) Model for Hungary
## 5 Parameter Downscaling Methods
### Transformed Time Estimates  
To be able to perform the MLE with variable observation periods on a non-declustered catalogue, the \(t\longmapsto\Lambda(t)\) transformation was used, which converts the event counts into a stationary Poisson point process with unit rate [5], where \(\Lambda(t)\) is given by Equation (14). When the event catalogue is incomplete, the part of the transformed time triggered by missing events is unknown. Given the ETAS parameters \((\mu,K,\eta,q,\alpha)\) and the magnitude-frequency decay parameter \(\beta\), an estimate is provided for the transformed time intervallength \(\Lambda(T_{2})-\Lambda(T_{1})\) corresponding to the observation period \([T_{1},T_{2}]\) where the observations are complete at magnitude threshold \(m_{c}\). Here, the space-time region is defined by a geographical zone and an observation period, and, as a simplification, it is assumed that the triggering effects crossing the boundaries of this region are negligible in both directions, which is a reasonable assumption if there are no major events close to the zone and period boundaries. This simplification allows for disregarding of the space-time integration bounds of the response functions \(g\) and \(f\) and ignoring all events outside the selected region. The total transformed time is divided into the following parts: (1) \(\Lambda_{0}\) is the transformed time coming from the background process, (2) \(\Lambda_{1+}\) is the transformed time triggered by the above-threshold events, and (3) \(\Lambda_{1-}\) is the transformed time triggered by the below-threshold events. Given that the observed below-threshold events only set a lower bound, \(\Lambda_{1-}\) is unknown. \[\Lambda_{0}=\mu\cdot(T_{2}-T_{1}), \tag{20}\]  
\[\Lambda_{1+}=\frac{K}{\eta q}\cdot\exp\Bigl{(}-\eta\Lambda t_{0}^{q}\Bigr{)} \cdot\sum_{j:t_{j}\in[T_{1},T_{2}]_{\infty}\ M_{j}\geq m_{c}}e^{\alpha(M_{j}- m_{0})}, \tag{21}\]  
\[\Lambda_{1-}\geq\Lambda_{1-}^{\min}\ =\frac{K}{\eta q}\cdot\exp\Bigl{(}-\eta \Lambda t_{0}^{q}\Bigr{)}\cdot\sum_{j:t_{j}\in[T_{1},T_{2}]_{\infty}\ M_{j}<m_{ c}}e^{\beta(M_{j}-m_{0})}. \tag{22}\]  
In Equation (22), \(\Lambda_{1-}\) is estimated by emulating iterations of the triggering process by tracing back every below-threshold event either to some above-threshold event or to the background:  
\[ \begin{split}&\Lambda_{1-}^{\text{est}}=(\Lambda_{0}+\Lambda_{1+}) \cdot\Bigl{[}p(m_{c})\cdot\varrho(m_{c})+p(m_{c})^{2}\cdot\varrho(m_{c})^{2}+ \ldots\Bigr{]}=\\ &=(\Lambda_{0}+\Lambda_{1+})\cdot\frac{p(m_{c})\cdot\varrho(m_{c} )}{1-p(m_{c})\cdot\varrho(m_{c})},\end{split}  \tag{23}\]  
where \(p(m_{c})\) is the probability that the magnitude of an event is lower than \(m_{c}\) according to Equation (24), and \(\varrho(m_{c})\) is the average reproduction rate of events below magnitude \(m_{c}\) according to Equation (25). The truncation at \(m_{c}\) is carried out to avoid double counting with \(\Lambda_{1+}\). \[p(m_{c})=\frac{1-e^{-\beta(m_{c}-m_{0})}}{1-e^{-\beta(m_{x}-m_{0})}}, \tag{24}\]  
\[\varrho(m_{c})= \begin{cases}\frac{K}{\eta q}\cdot\exp\Bigl{(}-\eta\Lambda t_{0 }^{q}\Bigr{)}\cdot\beta\cdot\frac{m_{c}-m_{0}}{1-e^{-\beta(m_{x}-m_{0})}}& \text{if $\alpha=\beta$,}\\ \frac{K}{\eta q}\cdot\exp\Bigl{(}-\eta\Lambda t_{0}^{q}\Bigr{)}\cdot\frac{ \beta}{\beta-\alpha}\cdot\frac{1-e^{-\beta(-\alpha)(m_{c}-m_{0})}}{1-e^{-\beta (m_{x}-m_{0})}}&\text{if $\alpha\
eq\beta$.}\end{cases}  \tag{25}\]  
Considering the lower bound in (22), the total estimated transformed time is then  
\[\Lambda(T_{2})-\Lambda(T_{1})=\Lambda_{0}+\Lambda_{1+}=\max\Bigl{(}\Lambda_{1 -}^{\text{est}},\ \Lambda_{1-}^{\min}\ \Bigr{)}. \tag{26}\]  
The MLE framework with variable observation periods is used with the above time transformation for downscaling the ETAS parameters \((\mu,K,\eta,q,\alpha,D,\varepsilon)\) to zones.