![]() For the proper rotation matrix, the determinant is 1.0. 2 Double-Couple Representationħ where I 3 is the third invariant of the matrix and is equivalent to the determinant. These rotation variables may be used in more extensive investigations of earthquake occurrence geometry and mechanics. Thereafter, we consider those transformations necessary to obtain a more complete and accurate description of the 3-D rotations. Initially, we describe algorithms that can be used with the original focal mechanisms presented in earthquake catalogues. Then we obtain expressions for all four angles and rotation pole positions. We first describe how to calculate the minimum 3-D rotation angle between two focal mechanism solutions, starting with the simplest methods. Moreover, since the knowledge of quaternions is not common among geophysicists, we largely avoid their use here. ) In contrast, the equations in this paper can be easily programmed in any software by a few lines of code. Bird recently reworked the programme in fortran90, it is available from his Web site. The relatively complex original quaternion programme ( Kagan 1991) was created in fortran, a computer language not widely used now. This angle can be obtained by a simple formula. Kagan (1991, section 4) mentioned other possible uses of the algorithms for calculating DC rotation.įor most of these studies, only the minimum rotation angle between two focal mechanism solutions was considered. Among these applications are finding a difference in focal mechanism solutions obtained through diverse methods analysis of seismicity patterns investigating connections between tectonic stress fields and earthquake source mechanisms, and so on. Frohlich & Davis 1999 Kagan & Jackson 2000 Kagan 2003 Bird & Kagan 2004 Okal 2005 Matsumoto et al. The quaternion method ( Kagan 1991) has been used to evaluate these rotations in many investigations of earthquake focal mechanisms (see, e.g. Two spherical coordinates on a reference sphere can be used to describe each pole. Thus, four different rotation angles and four rotation pole positions (the intersection of the rotation axis with a reference sphere) need to be found. This symmetry also means that the standard methods for determining 3-D rotation (see the citations above) need modifications to accommodate it. ![]() Kuipers (2002) and Hanson (2005) present a more accessible mathematical treatment of 3-D rotations with many explicit formulae.ĭue to the symmetry of DC sources, there are four possible rotations with the angle less or equal to 180° between two different mechanisms. That method was based on transforming a focal mechanism solution to a normalized quaternion, and the problem of the 3-D rotation was solved by applying quaternion algebra ( Altmann 1986). Kagan (1991) published an algorithm to determine the 3-D rotation parameters of a DC earthquake source. Presently different orientations of earthquake focal mechanisms are studied by separately displaying two or three DC axes on an equal-area plot (see, e.g. In this paper, we consider three-dimensional (3-D) rotations by which one double-couple (DC) earthquake source can be turned into another arbitrary DC. ![]() ![]() Methods for determining focal mechanism are discussed by Ekström (2005, and references therein) and by Snoke (2003). For accessible discussion of earthquake focal mechanisms see Jost & Herrmann (1989) and Pujol & Herrmann (1990). Additionally, a simple method is needed to compare focal mechanism solutions obtained by different methods. Thus, for individual events we can study varying focal mechanisms to see whether they yield any information regarding the spatial orientations of the earthquakes and microearthquakes that comprise a fault system ( Kagan 2006). Earthquake-source mechanism, fault-plane solutions, inverse problem, seismic moment, seismotectonics, statistical methods 1 IntroductionĪ large number of traditional fault plane and moment tensor solutions is presently available.
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