Conventions

The definitions needed to read result.json and the report.

Angles and frame

Moment tensors are given in x = North, y = East, z = Down (NED; Aki and Richards 2002, Box 4.4). Strike is measured clockwise from north with the fault dipping to the right, dip from the horizontal (0 to 90 deg), and rake in the fault plane from the strike direction to the slip of the hanging wall (-180 to 180 deg).

Moment tensor and magnitude

mt_ned is \((M_{xx}, M_{xy}, M_{xz}, M_{yy}, M_{yz}, M_{zz})\) in NED, normalized to a scalar moment of 1, with

\[ M_0 = \sqrt{M_{ij} M_{ij} / 2}. \]

The tensor in N m is m0_nm times mt_ned; m0 is the same moment in units of \(10^{20}\) dyne cm. The moment magnitude is Kanamori’s (1977), with \(M_0\) in N m:

\[ M_w = \tfrac{2}{3} \left( \log_{10} M_0 - 9.1 \right). \]

In the Global CMT convention (r = Up, \(\theta\) = South, \(\varphi\) = East): \(M_{rr} = M_{zz}\), \(M_{\theta\theta} = M_{xx}\), \(M_{\varphi\varphi} = M_{yy}\), \(M_{r\theta} = M_{xz}\), \(M_{r\varphi} = -M_{yz}\), \(M_{\theta\varphi} = -M_{xy}\).

Isotropic and CLVD parts

With --mt deviatoric or --mt full, the non_dc block of the record gives the moment tensor in the parameterization of Zhu and Ben-Zion (2013):

\[ M_{ij} = \sqrt{2} \, M_0 \left[ \zeta \, \frac{\delta_{ij}}{\sqrt{3}} + \sqrt{1 - \zeta^2} \left( \sqrt{1 - \chi^2} \, D^{\mathrm{DC}}_{ij} + \chi \, D^{\mathrm{CLVD}}_{ij} \right) \right], \]

where \(D^{\mathrm{DC}}\) and \(D^{\mathrm{CLVD}}\) are unit double-couple and CLVD tensors sharing the principal axes. iso is \(\zeta \in [-1, 1]\) (\(\zeta = 1\) an explosion) and clvd is \(\chi \in [-\tfrac{1}{2}, \tfrac{1}{2}]\); --mt deviatoric keeps \(\zeta = 0\). The signed percentages are

\[ \mathrm{ISO\%} = 100 \, \operatorname{sgn}(\zeta) \, \zeta^2, \qquad \mathrm{CLVD\%} = 100 \, \operatorname{sgn}(\chi) (1 - \zeta^2) \chi^2, \qquad \mathrm{DC\%} = 100 - |\mathrm{ISO\%}| - |\mathrm{CLVD\%}|. \]

Lune. The report plots the solution on the lune of Tape and Tape (2012): longitude \(\gamma \in [-30^\circ, 30^\circ]\) and latitude \(\delta\) (gamma and delta in non_dc, in degrees), with

\[ \zeta = \sin\delta, \qquad \chi = \sin\gamma. \]

A double couple is at the center, an explosion at \(\delta = 90^\circ\).

Sign of the CLVD. \(\chi > 0\) means the intermediate eigenvalue is positive: \(\chi = +\tfrac{1}{2}\) has eigenvalues proportional to (1, 1, -2) and \(\chi = -\tfrac{1}{2}\) to (2, -1, -1). Catalogs that call (2, -1, -1) a positive CLVD have the opposite sign. So does the report’s lune, which names its edges as Tape and Tape (2012) do: its “+CLVD” is (2, -1, -1), at \(\gamma = -30^\circ\), \(\chi = -\tfrac{1}{2}\).

Data and fit

Components. Z is up, R points along the great circle away from the source, and T is R turned 90 deg clockwise seen from above. Data and synthetics are ground velocity in cm/s.

Scalar moment and misfit. For component \(c\), let \(d_c\) be the data and \(s_c\) the synthetic of the mechanism at unit moment, both with the distance scaling of the inversion; \(o_c\) the component’s weight (the station’s weights in the record, set by the QC); and \(b_c = 1\) if the component is in use (weight above 0), else 0. For every trial mechanism the moment matches the energy of the data,

\[ M_0 = \sqrt{\frac{\sum_c o_c b_c \|d_c\|^2}{\sum_c o_c b_c \|s_c\|^2}}, \]

and the misfit is

\[ E = \sum_c o_c \left( \|d_c\|^2 + M_0^2 \|s_c\|^2 - 2 M_0 b_c \max_\tau d_c \cdot s_c(\tau) \right), \]

with \(\tau\) the time shift of the component. The solution is the mechanism of least \(E\).

Variance reduction. vr is, in percent,

\[ \mathrm{VR} = 100 \left( 1 - E \Big/ \sum_c o_c \|d_c\|^2 \right). \]

Scaling every weight by the same factor changes neither \(M_0\) nor VR, and a component of weight 0 enters neither. settings.m0_vr is "weighted" for these definitions. A record without it has those of cap3D.c: \(b_c\) in place of \(o_c b_c\) in \(M_0\), and every loaded component in the VR denominator, \(\sum_c \|d_c\|^2\). With weights 0 or 1 that equal \(b_c\) the two give the same \(M_0\); when every weight is 1 they also give the same VR.

Kagan angle. The smallest rotation that takes one double couple to another (Kagan 1991).

Jackknife. Each station is left out in turn. uncertainty gives the median and 90th percentile of the Kagan angles to the full solution and the depth standard error \(\sqrt{\tfrac{n-1}{n} \sum_i (z_i - \bar{z})^2}\) over the \(n\) leave-one-out depths.

References

• Aki, K., and P. G. Richards (2002). Quantitative Seismology, 2nd ed., University Science Books, Sausalito, California.

• Kagan, Y. Y. (1991). 3-D rotation of double-couple earthquake sources. Geophysical Journal International, 106(3), 709-716. doi:10.1111/j.1365-246X.1991.tb06343.x

• Kanamori, H. (1977). The energy release in great earthquakes. Journal of Geophysical Research, 82(20), 2981-2987. doi:10.1029/JB082i020p02981

• Tape, W., and C. Tape (2012). A geometric setting for moment tensors. Geophysical Journal International, 190(1), 476-498. doi:10.1111/j.1365-246X.2012.05491.x

• Zhu, L., and Y. Ben-Zion (2013). Parametrization of general seismic potency and moment tensors for source inversion of seismic waveform data. Geophysical Journal International, 194(2), 839-843. doi:10.1093/gji/ggt137