Relative directivity inversion of small earthquake rupture

The rupture directivity of an earthquake provides important insights into its physical mechanisms and strong motion generation. However, it is difficult to determine the rupture directivity of small earthquakes (e.g. M_L ≤ 3.3) because of their small rupture dimension and complex path effects from source to stations. Here we develop a new method, named 'relative directivity inversion method' (or RDI method), to determine rupture direction of small earthquakes. Based on the assumption that the target earthquake ruptures unilaterally, RDI method uses an earthquake pair and uses the smaller event as reference to invert the directivity of the larger event by determining the locations of its initiation point and the centroid energy point relative to those of the reference event. The rupture direction of the target earthquake is determined as the direction from the initiation location to the centroid energy location, and the rupture length as twice the distance between the two locations. As an example of testing the method, we apply RDI method to determine the rupture directivity of a target earthquake in Japan (ML 3.2) and obtain results that are supported by the observed waveform features, synthetic test and the distribution of earthquakes in the region.