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Double-couple earthquake focal mechanism: random rotation and display

Abstract

This paper addresses two problems: the random rotation of double-couple (DC) earthquake sources and the display of earthquake focal mechanisms. We consider several equivalent representations for DC sources and their properties and provide mathematical expressions for their mutual transformation. Obviously, a 3-D rotation of any object is more intricate than a 2-D rotation. Any rotation of a DC source is further complicated by its symmetry properties. Applying statistical tests to a DC distribution often requires one to compare it to a completely (or uniform) random DC pattern. We review several methods for obtaining random distribution of DC orientation; some of these seemingly natural techniques yield an incorrect result. The DC random rotation problem is closely connected to displays of focal mechanisms. In such displays, a strike or an azimuth of a focal mechanism can be neglected; hence, we are confronted with mapping a 2-D distribution onto a flat surface. We review different methods for such displays and discuss more specifically how to project a random focal mechanism distribution on a flat 2-D display with uniform probability density. Such displays can be used to analyse earthquake patterns statistically in various tectonic regions.

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