Yet another moment‐tensor parameterization

Abstract

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.

Generalized orthonormal moment tensor decomposition and its source-type diagram

“…It attracts data points along the circumference of the circular boundary, whereas the standard method and the Euclidean method show a somewhat uniform distribution and follow their own permitted region, which is diamond shaped for the standard method and lune shaped for the Euclidean method. A lengthy discussion of the probability densities of the two previously developed methods can be found in previous studies (Chapman 2019;Tape and Tape 2012b).…”

“…However, there is no universal method for decomposition. In recent years, some interest in revisiting the decomposition methods has arisen due to progress in observing non-DC microseismic events (Miller et al 1998;Eaton and Forouhideh 2010;Chapman 2019).…”

“…There have been many recent developments and new variations. For example, there is an effort to pursue a uniformly distributed parametrization (Chapman 2019). In contrast, the Euclidean method family (Chapman and Leaney 2012;Zhu and Ben-Zion 2013) maintains the orthogonality of the three bases, and thus, the calculations can be performed through a series of projections along the three bases, thereby generating the exact expansion of a moment tensor.…”

“…In contrast, the Euclidean method family (Chapman and Leaney 2012;Zhu and Ben-Zion 2013) maintains the orthogonality of the three bases, and thus, the calculations can be performed through a series of projections along the three bases, thereby generating the exact expansion of a moment tensor. A series of papers discusses the 3D geometry of its unique lune-shaped parameter space (Tape and Tape 2012a;2015;2019). Finally, there are some efforts to reparametrize the decomposition results, which can also result in some inconsistencies.…”