# Hazard Model: Epidemic-Type Aftershock Sequence (ETAS) Model for Hungary
## 7 Discussion and Conclusions  
ETAS methodology has been applied to Hungarian earthquake data of the period 1996-2021. Despite the data challenges caused by moderate seismicity, we succeeded in fitting space-time ETAS parameters to recent instrumental data of the region, which include the 2020-2021 Croatian earthquake sequences. Our conclusions from the modelling exercise are the following:  
1. Poisson goodness-of-fit tests of the modelled versus the observed process show a satisfactory fit for events \(M\geq 3.0\), which is appropriate for application in insurance. On the other hand, the same tests show significant deviations from the model at the magnitude threshold \(M\geq 2.5\). These deviations at low magnitudes likely result from the inhomogeneities of the observed process between earthquake clusters, from data deficiencies, and possibly also from boundary effects at the edge of the modelled area or at the magnitude threshold. 2. A stretched exponential time response function is used in the model. Based on the results of the event accumulation and time lag tests, we conclude that this time response form is appropriately fitted to the observed data. Deviations from the theoretical curve in the time lag test occur at very short time periods due to early aftershock incompleteness and at long time periods due to the time bounds of the observed catalogue. The modelled branching ratio \(\varrho=0.47\) implies a stable process. A systematic comparison of the stretched exponential function against the more commonly used modified Omori law as an alternative, however, was not part of this study. 3. The model uses a Gaussian space response function with an allowance for epicenter error. The results of the standardized distance test show a good fit between the model and the observed data. The \(\varepsilon=2.34\) km value of the error parameter is consistent with the horizontal error estimates of the underlying catalogue. On the other hand, the fitted \(\varepsilon\) parameter possibly also absorbs a correction for a deviation of the data from the Gaussian space response model or from the \(\gamma=\alpha\) constraint. An indication for this is that the 42-m value of the \(D\) parameter is unlikely to reflect the aftershock radius of \(M=2.5\) events by itself. 4. After the overall ETAS parameter estimation, parameter adjustments for regional zones (downscaling) were also calculated. In this step, historical catalogue data going back to 1700 were incorporated. Despite the known limitations of the historical catalogue, we find it important that, in an insurance model for a moderately active region like Hungary, the most destructive historical events are taken into account for parameterization and for model testing. Even after the inclusion of historical events, the dataset used for downscaling is still dominated by recent instrumental data which have a lower magnitude threshold of completeness. The modelled event numbers were tested against long-term historical observations in the magnitude range relevant for insurance losses. For the total modelled area, the test outcomes show a good fit both with and without downscaling, with the modelled median numbers falling slightly closer to the observations with the non-downscaled parameters. However, downscaling provides clear improvements to the results by zone. While the parameters of the overall model are subject to significant uncertainties, the tests indicate that the downscaled parameters are only moderately sensitive to the variations of the overall parameters. 5. For a Hungarian model, the central Zone 01-02-03 is especially important. In this zone, the downscaled model overestimates long-term event numbers in the 4.0-4.5 magnitude range. On the other hand, it gives better-fitting and more conservative results than the original parameterization for magnitudes \(M>4.5\), which is desirable in the case of a model for insurance applications.