A geometric perspective on seismic moment tensors
Below are some example figures related to the papers listed here.
I have posted online some matlab scripts, GMT scripts, and mathematica scripts relevant for interpreting and plotting moment tensors. These files are available within the compearth repository on github.
They can be downloaded from github as a zipped file or using git from the command line.
Please email me with corrections or suggestions.
Beachball representing slip on a fault plane (Tape and Tape, 2012, Figure 13).
Geometric representation of the probability distribution for ξ0, the minimum rotation angle between two sets of principal axes. The green surface is the portion of the sphere that is on or inside the cube. As t increases, the sphere grows and the surrounding cube shrinks. The probability density (above) is proportional to the area of the green surface. The letters in the plot above correspond to the letters (or t values) at the left. The fifth percentile, tau05 = 35.7o, is labeled. The angle varies from 0o to 120o (t3 = 2π/3). [Tape and Tape, 2012c]
Figures motivating the use of ωdc instead of ξ0 for quantifying the difference between two double couple moment tensors (Figures 14 and 15, Tape and Tape, 2012, "Angle between principal axis triples").
Left (Fig. 14). The moment tensor M0 is at the origin. The measure ξ0 considers the difference in orientation between M0 and M1 to be the same as that between M0 and M2, namely ξ0 = π/2 in both cases. Yet, intuitively, the beachball M1 resembles M0 more closely than does M2. In fact, since their colors are reversed, M2 and M0 are as different as moment tensors of the same size can be, whereas M0 and M1 are not so different from each other, especially in the x direction, where both are red. Since all three beachballs have the same eigenvalue triple, any differences in appearance ought to be attributable to differences in orientation. Since the parameter ξ0 does not capture these differences, ξ0 may not always be the best measure of separation for moment tensor orientations.
Right (Fig. 15). Comparison of ξ0 and ω for 90o rotations. The angle ξ0 is constant, ξ0 = π/2, whereas ωdc varies from π/3 at U1 to π at U2. The orientations U1 and U2 are the 90o rotations about the x- and y-axes, the same as in Fig. 14. The ωdc values for U1 and U2 seem to capture our intuition from Fig. 14 better than do the ξ0 values.
Probability density of random moment tensors, displayed on three different source-type plots. The distribution is uniform on the v-w rectangle. It is elevated on the center of the eigenvalue lune (the center is a double couple). It is peculiar on the T-k plot.