# Hazard Model: Epidemic-Type Aftershock Sequence (ETAS) Model for Hungary
## 5 Parameter Downscaling Methods
### Maximum Likelihood Estimation with Variable Observation Periods  
In this section, we refer to the maximum likelihood estimation (MLE) method developed by [PERSON] [33]. In a nutshell, the aim of the estimation is to determine recurrence parameters for mainshocks, defined as the largest earthquakes in each cluster. The accumulation of mainshocks is assumed to follow a stationary Poisson point process with a constant recurrence rate \(\
u\). It is assumed that event magnitudes are independent and follow a truncated exponential distribution with decay parameter \(\beta\) between the bounds \(m_{0}\) and \(m_{x}\), according to Equation (4). Magnitudes are rounded and grouped into the magnitude bins \([m_{k}-\delta,\,m_{k}+\delta)\). The length \(T_{k}\) of the time interval when the historical event catalogue is complete varies by the magnitude class \(m_{k}\). The number of events in the respective magnitude class and completeness period is denoted by \(n_{k}\). The parameters \(m_{0},\ m_{x},\ \delta\), and \(T_{k}\) are determined by prior expert judgement and are fixed in the estimation; the unknown parameters to be fitted are \(\
u\) and \(\beta\). Under these assumptions, the event count \(n_{k}\) is drawn from a Poisson distribution whose mean parameter is:  
\[\lambda_{k}(\
u,\beta)=\
u\cdot T_{k}\cdot\int_{m_{k}-\delta}^{m_{k}+\delta}J( m)\ dm=\
u\cdot T_{k}\cdot e^{-\beta(m_{k}-m_{0})}.\frac{2\sinh(\beta\delta)}{1-e^{- \beta(m_{x}-m_{0})}}. \tag{17}\]  
The calculus for finding the maximum of the corresponding likelihood function \(L(\
u,\beta)\) leads to the equations [33]:  
\[\frac{\sum_{k}T_{k}\cdot m_{k}\cdot\exp(-\beta m_{k})}{\sum_{k}T_{k}\cdot\exp (-\beta m_{k})}=\frac{\sum_{k}n_{k}\cdot m_{k}}{\sum_{k}n_{k}}=\overline{m}, \tag{18}\]  
\[\
u=\frac{1-e^{-\beta(m_{x}-m_{0})}}{2\sinh(\beta\delta)}\cdot\frac{\sum_{k}n_ {k}}{\sum_{k}T_{k}\cdot\exp(-\beta(m_{k}-m_{0}))}. \tag{19}\]  
The recurrence parameter estimates are obtained by solving Equation (18) for \(\beta\) and then calculating \(\
u\) from Equation (19). The strength of this MLE method is that it makes optimum use of available data by allowing magnitude-dependent observation periods. The method aims to find only temporal parameters. Its limitation is the reliance on a stationary Poisson event counting process, which normally restricts its application to declustered catalogues. Because the largest event in every cluster is selected, the estimated \(\beta\) is expected to be lower than the \(\beta\) parameter of the full population.  
In our ETAS model context, the above MLE technique with variable observation periods could be straightforwardly adapted to find the \((\mu,\beta)\) parameters of the stationary Poisson background process while focusing on the initial event in each cluster rather than the largest one. For that, the calculation of the background weights \(\xi_{i}\) and indicators \(\chi_{i}\) according to Equations (10) and (11) would need to be extended to the full historical catalogue. Such an extension is possible, with the following caveats: due to incomplete data, the process may identify some triggered earthquakes as background events. In addition, the high epicenter uncertainty of historical events, and especially the use of macrosiismic epicenters, may lead to implausible results, unless Equation (10) is adjusted to take this effect into account, e.g., by the convolution of the space distributions with error functions. However, we did not elaborate in this paper on this method and its results because this study took a different approach for the downscaling of ETAS parameters, as seen in Section 5.3.