Clustering macroseismic fields by statistical data depth functions

The macroseismic intensity is an ordinal variable that describes the seismic damage effects of an earthquake. The collection of intensity values recorded at sites of an area hit by an earthquake is called macroseismic field; it constitutes the only information on historical earthquakes for which no instrumental recordings are available. Around the area of the epicenter, lines bounding points of equal seismic intensity (isoseismal lines) are used to represent the spatial distribution of the macroseismic intensities, and their shapes can suggest the characteristics of the earthquake source. Our aim is to identify clusters of macroseismic fields according to the size and shape of their isoseismal lines, or in seismological terms, according to the trend of macroseismic attenuation. First, the isoseismal lines of some fields are approximated by convex hulls. Then, fixed an intensity value, the set of the corresponding convex hulls are analyzed on the basis of statistical data depth functions. This nonparametric method ranks the convex hulls according to their statistical depth values, which in the present study are defined by the modified local half-region depth function. A similarity measure based on the same depth function allows us to compare all pairs of hulls and build a dissimilarity matrix to which a clustering procedure is applied in order to detect clusters of fields homogenoeus from the attenuation viewpoint. This method is illustrated on both simulated and real macroseismic data.