Clustering in multidimensional spaces with applications to statistical analysis of earthquake clustering

We present in this work a statistical methodology for cluster analysis of seismicity in the time-space-energy domain. Baiesi and Paczuski's (2004) metric for measuring the distance between earthquakes is considered. The metric consists of a product involving the time interval and spatial distance between two earthquakes, as well as the magnitude of the first one. In this work, the metric is formally proved to define an asymmetric pseudoquasi-distance. The metric can be expanded into two dimensions: a magnitude-normalized time component and a magnitude-normalized space component. A statistical description following a model of Poisson marked point process in spatial and temporal dimensions is given. In this model, nearest-neighbor distance between earthquakes based on Baiesi and Paczuski's metric is found to follow Weibull distribution. Clustering is defined as deviation from this independence model. An existing declustering program that separates dependent and independent earthquakes into fore- and aftershock clusters is examined. This declustering program is improved with the implementation of an Expectation-Maximization algorithm that is able to automatically detect the two populations in an earthquake catalog and reset the separating threshold for nearest-neighbor earthquake distance according to the parameters of the specific catalog. This technique is applied to observed seismicity regionally and world-wide. Bimodal distribution of nearest neighbor distances in these catalogs is shown. The declustering results are used to create maps of fore-shock and aftershock production and compare earthquake clustering to regional properties of the earth's crust.