Yet another moment-tensor parameterization

The source mechanism of a microseismic event, or any small earthquake, can be described by its moment tensor. The source type is described by a 3-vector formed from the ordered, principal (eigen) values of the moment tensor and the source orientation from the normalized eigenvectors. The direction and magnitude of the principal-value vector describe the source type and scalar moment, respectively. The source type can be described by the position of the principal-value vector on the permitted lune of a unit sphere. As with any projection or cartography mapping, many parameterizations have been suggested to describe this position. Two dominate in the literature – the Riedesel–Jordan and Hudson–Pearce–Rogers parameters. All have advantages and disadvantages. In this paper, we review the most popular parameterizations, illustrating their similarities and the distortion they cause in the mapping between a uniform distribution of source types in the permitted lune and the parameter space. A new parameterization is suggested based on the solid angles formed by the principal-value vector. This has the advantage of being simple to define geometrically in the principal-value space (although the formulae are complicated), being naturally normalized and having a more uniform mapping than other parameterizations. However, we do not claim that this is the best or ideal parameterization, just a reasonable choice.