Using polynomial regression modeland aftershocks classification patternto predict the magnitude and occurrence timeof the main aftershock occurring after large earthquakes

In recent years, studies in seismology have mainly focused on temporal and spatial analysis of earthquakes. This is important for crisis management due to a variety of reasons, including the necessity to estimate the magnitude and the occurrence time of the main aftershock in a given periodafter the mainshock.The present study seeks to identify the relationship between the magnitude and the occurrence time of the main aftershock in the first few hours after the mainshockusing the aftershock classification patterns. Various workshave been performed to model aftershocks of whichthe last centurystudies have yielded good results. However, providing a comprehensive description of how energy is released from seismic sources during an earthquakein different regionsis not easy. As a result, modeling of aftershocks is very complex and a precise model has not yet been provided for estimatingfeatures of the main aftershocks.  

During the preliminary investigation, magnitude of the mainshock and the number of aftershocks in the initial 12 hours were identified as two important parameters affecting the magnitude and the occurrence time of the main aftershock. However, this simple model lacks sufficient accuracy (accuracy of 0.5 in magnitude estimation and 5.8 hours in the estimation of the main aftershock occurrence time). Therefore, a polynomial function with higher number ofparameterswas used in the present study to reach a more accurate modeling. A linear polynomial model with 15 different parameters was introduced. These parameters includemagnitude of the mainshock, number of aftershocks during the initial time period, and in half and quarter of the period, and the number of aftershocks and mean temporal interval between aftershocks occurringin classes of 2.5 to 3.5, 3.5 to 4.5, 4.5 to 5.5 and greater than 5.5 Richter. The initial time period refers to the minimum number of hours needed after the mainshockto collect information about the aftershocks. Coefficients of occurrence time and magnitude of the main aftershock were calculated in the two proposed modelsusing 32 earthquake events and the least square method. These earthquakeshad occurred with a magnitude of greater than 5.6 from 2006 to 2020.In order to select the best model using the least mean square error (MSE), several models have been considered with a change in their initial time period (using for classification of the aftershocks) and secondary time period(the time duration at which the features of the main aftershock are estimated).  

Based on the mean square error, three models were introduced to estimate features of the main aftershock in short, mid and long-term. These models can be used to estimate features of the main aftershocks occurring 2, 8 and 20 days after the main shock, respectively. The short-term prediction model use aftershocks occurring during the first hour after the main shock to predict the magnitude of the main aftershock with a precision of 0.21 (MN) and its occurrence time with a precision of 3.1 hours. Mid-term prediction model also useaftershocks occurring during the first 3hoursafter the main shock to predict the magnitude of the main aftershock with a precision of 0.23 (MN) and the occurrence time with a precision of 19.3 hours. Finally, the long-term prediction model use aftershocks occurring during the first9hoursafter the main shock to predict the magnitude of the main aftershock with a precision of 0.22 (MN) and the occurrence time with a precision of 38.5 hours.  

  To evaluate errorsof the proposed models, information collected from 9 recent earthquakes in Iran and Turkey was used. Magnitude and occurrence time of the main aftershock ineach selected earthquake were calculated using short, mid and long term prediction models. Results demonstrate that these models can predict the magnitude of the main aftershock with an average error of 0.18 (MN). They also can predict the occurrence time of the main aftershock with an average error of 18.1 hours. It is worth noting that the proposed models havepredicted themagnitudeof these recentnine earthquakes with a mean error less than their accuracy estimated using the 32 earthquake events.