On entropy and clustering in earthquake hypocentre distributions

The degree of clustering or disorder within earthquake distributions may be measured using the concept of entropy. A method for calculating the entropy of any 3-D point set (e.g. earthquake foci) is presented. This makes use of Voronoi cells (convex polyhedra representing nearest neighbour regions) to measure point density in three dimensions. An estimate of event density can be determined directly from the size of Voronoi cells. Normalizations are introduced to the definition of entropy that allow data sets containing different numbers of events and occupying different volumes to be compared quantitatively, for example, earthquake catalogues from different tectonic regimes. Our results show a clear correlation between earthquake entropy and tectonic regime. The most ordered are the mid-ocean ridges, followed by the subduction zones and finally intraplate seismicity. We show how entropy may be used to quantify the simplification of earthquake distributions, for example, due to relocation procedures. A recently published algorithm called the collapsing method is used as an example of a technique that reduces entropy while respecting data fit. Modifications to this method are made that reduce artefacts and use additional temporal information in the earthquake distribution. These methods are applied to a global catalogue of 85 000 events, and a local catalogue from the SIL network in Iceland containing 43 300 events. The entropy of both catalogues is reduced. Results from the Hengill region within the SIL network show lineations whose orientations agree with independent studies using relative location techniques and surface faulting.