Advancement of Aftershock Distribution Prediction Model Following a Main Shock &#8211; Examining Parameter Correction Methods for Predictive Formula Parameters Based on Main Shock Information &#8211;
Authors: Hashimoto Ryuku , Kuzuha Yasuhisa 
Published: March 2024

Challenges in Aftershock ForecastingThe probabilistic evaluation of aftershock activity relies on two empirical rules: the Gutenberg&#8211;Richter law (GR law) and the Modified Omori law (MO law). An important issue arises in aftershock observation, particularly when regarding the technical aspects of seismic monitoring, where smaller earthquakes are more challenging to detect than larger ones.&#160; In cases where records of smaller seismic events are absent, the b value of the GR law and the K value of the MO law tend to be underestimated. This study was conducted to develop a model that corrects underestimation of these parameters based on main shock information.&#160;Data and MethodologySeismic source data from the Japan Meteorological Agency were used, including the main shock &#8211; aftershock sequence magnitudes, latitude and longitude of the epicenter, and occurrence times. Using data for the periods immediately after the main shock to 3 hours, 1 day, 30 days, and 90 days, calculations were performed on data to ascertain the K value of the MO law and the b value of the GR law. The objective was to investigate the relation between these parameters and the main shock magnitude (hereinafter, M0). Based on these relations, methodologies for correcting parameters were explored.&#160;Results and DiscussionRelation between M0 and parameters
Significant negative correlation was found between the M0 and the b value, with larger M0 values associated with smaller b values. Furthermore, correlation was stronger for b values closer to the immediate aftermath of the main shock. This strong correlation suggests that larger M0 values are more likely to result in the omission of weaker seismic events from the data. The omission of earthquakes is particularly noticeable immediately following occurrence of the main shock. Similarly, a tendency was observed for the K value to be underestimated immediately after the main shock, when M0 is larger.Parameter corrections
We introduce new parameters, b' and K', defined as shown below.Larger values of b' and K' indicate underestimation of parameters at 3 hours after the main shock compared to 1 day after the main shock. Using these parameters, we perform linear regression analysis with M0 as the independent variable and b' and K' as dependent variables to estimate 1 day post-main-shock parameters from the 3 hours post-main-shock values.The precisions of the estimated values are compared as shown in Figure 1.Figure1:The precisions of the estimated valuesEstimation of b shows superior accuracy compared to that obtained using earlier methodologies and conventional approaches used by the Japan Meteorological Agency. Estimated values of K show that systematic errors have been improved with the methodology used for this study. Using these corrected parameters, Figure 2 presents a comparison of the predicted aftershock numbers from 3 hours to 1 day after the main shock with the actual values. As the figure shows, on average, the methodology used for this provides favorable accuracy of predictions.Figure2:The accuracy of Aftershock Predictions