Fig. 3

Results from applying the replenishing algorithm to the earthquake data from the Kumamoto aftershock region. a Magnitudes versus occurrence times of the earthquake events. b Rescaled magnitudes versus empirical distribution of occurrence times of the recorded events transformed by using the bi-scale empirical transformation. c Rescaled magnitudes versus rescaled occurrence times of the combination of the observed events, with the rescaling based on the empirical distribution that is estimated based on the events outside of S. d Rescaled magnitudes versus rescaled occurrence times of the observed events and replenished events, i.e., newly generated events after removing events that are close to any of the observed in S, with the rescaling based on the empirical distributions of the events outside of S. e Magnitudes versus occurrence times of the observed synthetic events and the replenished events. f Cumulative numbers of events against occurrence times for the original dataset (gray curve) and for the replenished dataset (black curve). The blue polygons in a–d are the area S and its corresponding mappings in which the missing events fall. Green dots in d and e are the replenished events